Patent Publication Number: US-7221778-B2

Title: Image processing apparatus and method, and image pickup apparatus

Description:
TECHNICAL FIELD 
   The present invention relates to image processing apparatuses and methods, and image-capturing apparatuses, and more particularly, to an image processing apparatus and method, and an image-capturing apparatus in which a difference between a signal detected by a sensor and the real world is taken into consideration. 
   BACKGROUND ART 
   A technique for detecting incidents occurring in the real world by a sensor and for processing sampled data output from the image sensor have been widely used. 
   For example, motion blur occurs in an image obtained by capturing an object moving in front of a predetermined stationary background with a video camera if the moving speed is relatively high. 
   However, it has not hitherto been allowed to detect a shutter time of an image that has already been captured. 
   DISCLOSURE OF INVENTION 
   The present invention has been made in view of the above situation, and an object thereof is to allow detection of an exposure time of an image that has already been captured. 
   An image processing apparatus according to the present invention includes amount-of-inter-frame-movement detection means for detecting an amount of movement between frames of a foreground object of a designated frame of image data based on the designated frame and a frame preceding or subsequent to the designated frame; mixture-ratio detection means for detecting a mixture ratio indicating a ratio of mixture of a foreground object component constituting the foreground object and a background object component constituting a background object in a mixed area in the designated frame, in which the foreground object component and the background object component are mixed; amount-of-movement-within-exposure-time detection means for detecting, in accordance with the mixture ratio, an amount of movement of the foreground object within an exposure time in which the pixel data constituting the image data is obtained; and exposure-time-ratio detection means for detecting a ratio of a time interval of the frames to the exposure time based on the amount of movement between the frames and the amount of movement within the exposure time of the foreground object. 
   The exposure-time-ratio detection means may detect the exposure time based on a ratio of the amount of movement between the frames to the amount of movement within the exposure time of the foreground object and based on the time interval of the frames. 
   The mixture-ratio detection means may include relational-expression generating means for extracting, with respect to a designated pixel of the designated frame of the image data, the pixel data of a peripheral frame around the designated frame as background pixel data corresponding to the background object, also extracting the designated pixel data of the designated pixel and proximity pixel data of a proximity pixel in a proximity of the designated pixel in the designated frame, and generating a plurality of relational expressions indicating relationships among the designated pixel data, the proximity pixel data, and the background pixel data corresponding to the designated pixel and the proximity pixel, the mixture-ratio detection means detecting a mixture ratio of the designated pixel and of the proximity pixel based on the relational expressions. 
   The relational-expression generating means may generate the relational expressions based on a first approximation in which values of components of the foreground object included in the designated pixel data and in the proximity pixel data are the same, and based on a second approximation in which the mixture ratio changes linearly in the mixed area in relation to the positions of the pixels. 
   The amount-of-movement-within-exposure-time detection means may detect a reciprocal of a gradient of the linear change of the mixture ratio in the mixed area in relation to the positions of the pixels as the amount of movement of the foreground object within the exposure time in which the image data is obtained, outputting the amount of movement detected. 
   An image processing method according to the present invention includes an amount-of-inter-frame-movement detection step of detecting an amount of movement between frames of a foreground object of a designated frame of image data based on the designated frame and a frame preceding or subsequent to the designated frame; a mixture-ratio detection step of detecting a mixture ratio indicating a ratio of mixture of a foreground object component constituting the foreground object and a background object component constituting a background object in a mixed area in the designated frame, in which the foreground object component and the background object component are mixed; an amount-of-movement-within-exposure-time detection step of detecting, in accordance with the mixture ratio, an amount of movement of the foreground object within an exposure time in which the pixel data constituting the image data is obtained; and an exposure-time-ratio detection step of detecting a ratio of a time interval of the frames to the exposure time based on the amount of movement between the frames and the amount of movement within the exposure time of the foreground object. 
   In the exposure-time-ratio detection step, the exposure time may be detected based on a ratio of the amount of movement between the frames to the amount of movement within the exposure time of the foreground object and based on the time interval of the frames. 
   The mixture-ratio detection step may include a relational-expression generating step of extracting, with respect to a designated pixel of the designated frame of the image data, the pixel data of a peripheral frame around the designated frame as background pixel data corresponding to the background object, also extracting the designated pixel data of the designated pixel and proximity pixel data of a proximity pixel in a proximity of the designated pixel in the designated frame, and generating a plurality of relational expressions indicating relationships among the designated pixel data, the proximity pixel data, and the background pixel data corresponding to the designated pixel and the proximity pixel, a mixture ratio of the designated pixel and of the proximity pixel being detected based on the relational expressions in the mixture-ratio detection step. 
   In the relational-expression generating step, the relational expressions may be generated based on a first approximation in which values of components of the foreground object included in the designated pixel data and in the proximity pixel data are the same, and based on a second approximation in which the mixture ratio changes linearly in the mixed area in relation to the positions of the pixels. 
   In the amount-of-movement-within-exposure-time detection step, a reciprocal of a gradient of the linear change of the mixture ratio in the mixed area in relation to the positions of the pixels may be detected as the amount of movement of the foreground object within the exposure time in which the image data is obtained, outputting the amount of movement detected. 
   A program in a recording medium according to the present invention includes an amount-of-inter-frame-movement detection step of detecting an amount of movement between frames of a foreground object of a designated frame of image data based on the designated frame and a frame preceding or subsequent to the designated frame; a mixture-ratio detection step of detecting a mixture ratio indicating a ratio of mixture of a foreground object component constituting the foreground object and a background object component constituting a background object in a mixed area in the designated frame, in which the foreground object component and the background object component are mixed; an amount-of-movement-within-exposure-time detection step of detecting, in accordance with the mixture ratio, an amount of movement of the foreground object within an exposure time in which the pixel data constituting the image data is obtained; and an exposure-time-ratio detection step of detecting a ratio of a time interval of the frames to the exposure time based on the amount of movement between the frames and the amount of movement within the exposure time of the foreground object. 
   In the exposure-time-ratio detection step, the exposure time may be detected based on a ratio of the amount of movement between the frames to the amount of movement within the exposure time of the foreground object and based on the time interval of the frames. 
   The mixture-ratio detection step may include a relational-expression generating step of extracting, with respect to a designated pixel of the designated frame of the image data, the pixel data of a peripheral frame around the designated frame as background pixel data corresponding to the background object, also extracting the designated pixel data of the designated pixel and proximity pixel data of a proximity pixel in a proximity of the designated pixel in the designated frame, and generating a plurality of relational expressions indicating relationships among the designated pixel data, the proximity pixel data, and the background pixel data corresponding to the designated pixel and the proximity pixel, a mixture ratio of the designated pixel and of the proximity pixel being detected based on the relational expressions in the mixture-ratio detection step. 
   In the relational-expression generating step, the relational expressions may be generated based on a first approximation in which values of components of the foreground object included in the designated pixel data and in the proximity pixel data are the same, and based on a second approximation in which the mixture ratio changes linearly in the mixed area in relation to the positions of the pixels. 
   In the amount-of-movement-within-exposure-time detection step, a reciprocal of a gradient of the linear change of the mixture ratio in the mixed area in relation to the positions of the pixels may be detected as the amount of movement of the foreground object within the exposure time in which the image data is obtained, outputting the amount of movement detected. 
   A program according to the present invention allows a computer to execute an amount-of-inter-frame-movement detection step of detecting an amount of movement between frames of a foreground object of a designated frame of the image data based on the designated frame and a frame preceding or subsequent to the designated frame; a mixture-ratio detection step of detecting a mixture ratio indicating a ratio of mixture of a foreground object component constituting the foreground object and a background object component constituting a background object in a mixed area in the designated frame, in which the foreground object component and the background object component are mixed; an amount-of-movement-within-exposure-time detection step of detecting, in accordance with the mixture ratio, an amount of movement of the foreground object within an exposure time in which the pixel data constituting the image data is obtained; and an exposure-time-ratio detection step of detecting a ratio of a time interval of the frames to the exposure time based on the amount of movement between the frames and the amount of movement within the exposure time of the foreground object. 
   In the exposure-time-ratio detection step, the exposure time may be detected based on a ratio of the amount of movement between the frames to the amount of movement within the exposure time of the foreground object and based on the time interval of the frames. 
   The mixture-ratio detection step may include a relational-expression generating step of extracting, with respect to a designated pixel of the designated frame of the image data, the pixel data of a peripheral frame around the designated frame as background pixel data corresponding to the background object, also extracting the designated pixel data of the designated pixel and proximity pixel data of a proximity pixel in a proximity of the designated pixel in the designated frame, and generating a plurality of relational expressions indicating relationships among the designated pixel data, the proximity pixel data, and the background pixel data corresponding to the designated pixel and the proximity pixel, a mixture ratio of the designated pixel and of the proximity pixel being detected based on the relational expressions in the mixture-ratio detection step. 
   In the relational-expression generating step, the relational expressions may be generated based on a first approximation in which values of components of the foreground object included in the designated pixel data and in the proximity pixel data are the same, and based on a second approximation in which the mixture ratio changes linearly in the mixed area in relation to the positions of the pixels. 
   In the amount-of-movement-within-exposure-time detection step, a reciprocal of a gradient of the linear change of the mixture ratio in the mixed area in relation to the positions of the pixels may be detected as the amount of movement of the foreground object within the exposure time in which the image data is obtained, outputting the amount of movement detected. 
   An image-capturing apparatus according to the present invention includes image-capturing means for outputting a subject image captured by an image-capturing device including a predetermined number of pixels having a time integrating function as image data consisting of a predetermined number of pixel data; amount-of-inter-frame-movement detection means for detecting an amount of movement between frames of a foreground object of a designated frame of the image data based on the designated frame and a frame preceding or subsequent to the designated frame; mixture-ratio detection means for detecting a mixture ratio indicating a ratio of mixture of a foreground object component constituting the foreground object and a background object component constituting a background object in a mixed area in the designated frame, in which the foreground object component and the background object component are mixed; amount-of-movement-within-exposure-time detection means for detecting, in accordance with the mixture ratio, an amount of movement of the foreground object within an exposure time in which the pixel data constituting the image data is obtained; and exposure-time-ratio detection means for detecting a ratio of a time interval of the frames to the exposure time based on the amount of movement between the frames and the amount of movement within the exposure time of the foreground object. 
   The exposure-time-ratio detection means may detect the exposure time based on a ratio of the amount of movement between the frames to the amount of movement within the exposure time of the foreground object and based on the time interval of the frames. 
   The mixture-ratio detection means may include relational-expression generating means for extracting, with respect to a designated pixel of the designated frame of the image data, the pixel data of a peripheral frame around the designated frame as background pixel data corresponding to the background object, also extracting the designated pixel data of the designated pixel and proximity pixel data of a proximity pixel in a proximity of the designated pixel in the designated frame, and generating a plurality of relational expressions indicating relationships among the designated pixel data, the proximity pixel data, and the background pixel data corresponding to the designated pixel and the proximity pixel, the mixture-ratio detection means detecting a mixture ratio of the designated pixel and of the proximity pixel based on the relational expressions. 
   The relational-expression generating means may generate the relational expressions based on a first approximation in which values of components of the foreground object included in the designated pixel data and in the proximity pixel data are the same, and based on a second approximation in which the mixture ratio changes linearly in the mixed area in relation to the positions of the pixels. 
   The amount-of-movement-within-exposure-time detection means may detect a reciprocal of a gradient of the linear change of the mixture ratio in the mixed area in relation to the positions of the pixels as the amount of movement of the foreground object within the exposure time in which the image data is obtained, outputting the amount of movement detected. 
   An amount of movement between frames of a foreground object of a designated frame of image data is detected based on the designated frame and a frame preceding or subsequent to the designated frame; a mixture ratio indicating a ratio of mixture of a foreground object component constituting the foreground object and a background object component constituting a background object in a mixed area in the designated frame, in which the foreground object component and the background object component are mixed, is detected; in accordance with the mixture ratio, an amount of movement of the foreground object within an exposure time in which the pixel data constituting the image data is obtained is detected; and a ratio of a time interval of the frames to the exposure time is detected based on the amount of movement between the frames and the amount of movement within the exposure time of the foreground object. 
   Accordingly, detection of an exposure time of an image that has already been captured is allowed. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a diagram illustrating an embodiment of an image processing apparatus according to the present invention. 
       FIG. 2  is a block diagram illustrating the configuration of the image processing apparatus. 
       FIG. 3  illustrates the image capturing performed by a sensor. 
       FIG. 4  illustrates the arrangement of pixels. 
       FIG. 5  illustrates the operation of a detection device. 
       FIG. 6A  illustrates an image obtained by image-capturing an object corresponding to a moving foreground and an object corresponding to a stationary background. 
       FIG. 6B  illustrates a model of an image obtained by image-capturing an object corresponding to a moving foreground and an object corresponding to a stationary background. 
       FIG. 7  illustrates a background area, a foreground area, a mixed area, a covered background area, and an uncovered background area. 
       FIG. 8  illustrates a model obtained by expanding in the time direction the pixel values of pixels aligned side-by-side in an image obtained by image-capturing an object corresponding to a stationary foreground and an the object corresponding to a stationary background. 
       FIG. 9  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 10  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 11  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 12  illustrates an example in which pixels in a foreground area, a background area, and a mixed area are extracted. 
       FIG. 13  illustrates the relationships between pixels and a model obtained by expanding the pixel values in the time direction. 
       FIG. 14  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 15  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 16  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 17  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 18  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 19  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 20  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 21  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 22  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 23  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 24A  is a diagram illustrating adjustment of the amount of motion blur. 
       FIG. 24B  is a diagram illustrating adjustment of the amount of motion blur. 
       FIG. 25A  is a diagram illustrating adjustment of the amount of motion blur. 
       FIG. 25B  is a diagram illustrating adjustment of the amount of motion blur. 
       FIG. 26  is a flowchart illustrating the processing for adjusting the amount of motion blur. 
       FIG. 27  is a block diagram illustrating an example of the configuration of an area specifying unit  103 . 
       FIG. 28  illustrates an image when an object corresponding to a foreground is moving. 
       FIG. 29  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 30  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 31  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 32  illustrates the conditions for determining the area. 
       FIG. 33A  illustrates an example of the result obtained by specifying the area by the area specifying unit  103 . 
       FIG. 33B  illustrates an example of the result obtained by specifying the area by the area specifying unit  103 . 
       FIG. 33C  illustrates an example of the result obtained by specifying the area by the area specifying unit  103 . 
       FIG. 33D  illustrates an example of the result obtained by specifying the area by the area specifying unit  103 . 
       FIG. 34  illustrates an example of the result obtained by specifying the area by the area specifying unit  103 . 
       FIG. 35  is a flowchart illustrating the area specifying processing. 
       FIG. 36  is a block diagram illustrating another example of the configuration of the area specifying unit  103 . 
       FIG. 37  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 38  illustrates an example of a background image. 
       FIG. 39  is a block diagram illustrating the configuration of a binary-object-image extracting portion  302 . 
       FIG. 40A  illustrates the calculation of a correlation value. 
       FIG. 40B  illustrates the calculation of a correlation value. 
       FIG. 41A  illustrates the calculation of a correlation value. 
       FIG. 41B  illustrates the calculation of a correlation value. 
       FIG. 42  illustrates an example of the binary object image. 
       FIG. 43  is a block diagram illustrating the configuration of a time change detector  303 . 
       FIG. 44  illustrates determinations made by an area determining portion  342 . 
       FIG. 45  illustrates an example of determinations made by the time change detector  303 . 
       FIG. 46  is a flowchart illustrating the area specifying processing performed by the area specifying unit  103 . 
       FIG. 47  is a flowchart illustrating details of the area specifying processing. 
       FIG. 48  is a block diagram illustrating the configuration of a mixture-ratio calculator  104 . 
       FIG. 49  illustrates an example of the ideal mixture ratio α. 
       FIG. 50  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 51  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 52  illustrates a straight line for approximating the mixture ratio α. 
       FIG. 53  illustrates a plane for approximating the mixture ratio α. 
       FIG. 54  illustrates the relationships of the pixels in a plurality of frames when the mixture ratio α is calculated. 
       FIG. 55  is a block diagram illustrating the configuration of the mixture-ratio estimation processor  401 . 
       FIG. 56  is a diagram for explaining a motion vector within a shutter time, output from a mixture-ratio determining portion  403 . 
       FIG. 57  is a diagram illustrating an example of an estimated mixture ratio. 
       FIG. 58  is a block diagram illustrating another configuration of the mixture-ratio calculator  104 . 
       FIG. 59  is a flowchart illustrating the processing for calculating the mixture ratio and the motion vector within the shutter time. 
       FIG. 60  is a flowchart illustrating the processing for estimating the mixture ratio and the motion vector by using a model corresponding to a covered background area. 
       FIG. 61  is a block diagram illustrating an example of the configuration of a foreground/background separator  105 . 
       FIG. 62A  illustrates an input image, a foreground component image, and a background component image. 
       FIG. 62B  illustrates a model of an input image, a foreground component image, and a background component image. 
       FIG. 63  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 64  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 65  illustrates a model in which pixel values are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 66  is a block diagram illustrating an example of the configuration of a separating portion  601 . 
       FIG. 67A  illustrates an example of a separated foreground component image. 
       FIG. 67B  illustrates an example of a separated background component image. 
       FIG. 68  is a flowchart illustrating the processing for separating a foreground and a background. 
       FIG. 69  is a block diagram illustrating an example of the configuration of a motion-blur adjusting unit  106 . 
       FIG. 70  illustrates the unit of processing. 
       FIG. 71  illustrates a model in which the pixel values of a foreground component image are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 72  illustrates a model in which the pixel values of a foreground component image are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 73  illustrates a model in which the pixel values of a foreground component image are expanded in the time direction and the period corresponding to the shutter-time is divided. 
       FIG. 74  illustrates a model in which the pixel values of a foreground component image are expanded in the time direction and the period corresponding to the shutter time is divided. 
       FIG. 75  illustrates an example of another configuration of the motion-blur adjusting unit  107 . 
       FIG. 76  is a flowchart illustrating the processing for adjusting the amount of motion blur contained in a foreground component image performed by the motion-blur adjusting unit  107 . 
       FIG. 77  is a block diagram illustrating an example of another configuration of the motion-blur adjusting unit  107 . 
       FIG. 78  illustrates an example of a model in which the relationships between pixel values and foreground components are indicated. 
       FIG. 79  illustrates the calculation of foreground components. 
       FIG. 80  illustrates the calculation of foreground components. 
       FIG. 81  is a flowchart illustrating the processing for adjusting the amount of motion blur contained in a foreground. 
       FIG. 82  illustrates the configuration of a synthesizer  108 . 
       FIG. 83  is a block diagram illustrating another configuration of the function of the image processing apparatus. 
       FIG. 84  is a block diagram illustrating the configuration of a mixture-ratio calculator  1101 . 
       FIG. 85  is a block diagram illustrating the configuration of a foreground/background separator  1102 . 
       FIG. 86  is a block diagram illustrating the configuration of a synthesizer  1103 . 
   

   BEST MODE FOR CARRYING OUT THE INVENTION 
     FIG. 1  is a diagram showing an embodiment of an image processing apparatus according to the present invention. A CPU (Central Processing Unit)  21  executes various types of processing according to programs stored in a ROM (Read Only Memory)  22  or in a storage unit  28 . Programs executed by the CPU  21  and data are stored in a RAM (Random Access Memory)  23  as required. The CPU  21 , the ROM  22 , and the RAM  23  are connected to each other by a bus  24 . 
   An input/output interface  25  is also connected to the CPU  21  via the bus  24 . An input unit  26 , which is formed of a keyboard, a mouse, a microphone, and so on, and an output unit  27 , which is formed of a display, a speaker, and so on, are connected to the input/output interface  25 . The CPU  21  executes various types of processing in response to a command input from the input unit  26 . The CPU  21  then outputs an image or sound obtained as a result of the processing to the output unit  27 . 
   The storage unit  28  connected to the input/output interface  25  is formed of, for example, a hard disk, and stores programs executed by the CPU  21  and various types of data. A communication unit  29  communicates with an external device via the Internet or another network. In this example, the communication unit  29  serves as an obtaining unit for obtaining an output of a sensor. 
   Alternatively, a program may be obtained via the communication unit  29  and stored in the storage unit  28 . 
   A drive  30  connected to the input/output interface  25  drives a magnetic disk  51 , an optical disc  52 , a magneto-optical disk  53 , a semiconductor memory  54 , or the like, when such a recording medium is attached to the drive  30 , and obtains a program or data stored in the corresponding medium. The obtained program or data is transferred to the storage unit  28  and stored therein if necessary. 
     FIG. 2  is a block diagram illustrating the image processing apparatus. 
   It does not matter whether the individual functions of the image processing apparatus are implemented by hardware or software. That is, the block diagrams of this specification may be hardware block diagrams or software functional block diagrams. 
   In this specification, an image to be captured corresponding to an object in the real world is referred to as an image object. 
   An input image supplied to the image processing apparatus is supplied to an object extracting unit  101 , an area specifying unit  103 , a mixture-ratio calculator  104 , and a foreground/background separator  105 . 
   The object extracting unit  101  extracts a rough image object corresponding to a foreground object contained in the input image, and supplies the extracted image object to a motion detector  102 . The object extracting unit  101  detects, for example, an outline of the foreground image object contained in the input image so as to extract a rough image object corresponding to the foreground object. 
   The object extracting unit  101  extracts a rough image object corresponding to a background object contained in the input image, and supplies the extracted image object to the motion detector  102 . The object extracting unit  101  extracts a rough image object corresponding to the background object from, for example, the difference between the input image and the extracted image object corresponding to the foreground object. 
   Alternatively, for example, the object extracting unit  101  may extract the rough image object corresponding to the foreground object and the rough image object corresponding to the background object from the difference between the background image stored in a built-in background memory and the input image. 
   The motion detector  102  calculates an inter-frame motion vector, indicating motion between frames, of the roughly extracted image object corresponding to the foreground object according to a technique, such as block matching, gradient, phase correlation, or pel-recursive technique, and supplies the calculated inter-frame motion vector and the inter-frame motion-vector positional information (which is information for specifying the positions of the pixels corresponding to the motion vector) to a shutter-time calculator  106 . 
   The motion detector  102  may output the inter-frame motion vector of each image object, together with the pixel positional information for specifying the pixels of the image object, to the shutter-time calculator  106 . 
   The area specifying unit  103  determines to which of a foreground area, a background area, or a mixed area each pixel of the input image belongs, and supplies information indicating to which area each pixel belongs (hereinafter referred to as “area information”) to the mixture-ratio calculator  104 , the foreground/background separator  105 , a motion-blur adjusting unit  107 , and a synthesizer  108 . 
   The mixture-ratio calculator  104  calculates the mixture ratio corresponding to the pixels contained in a mixed area (hereinafter referred to as the “mixture ratio α”) based on the input image and the area information supplied from the area specifying unit  103 , and supplies the mixture ratio α to the foreground/background separator  105  and the synthesizer  108 . 
   The mixture ratio α is a value indicating the ratio of the image components corresponding to the background object (hereinafter also be referred to as “background components”) to the pixel value as expressed by equation (3), which is shown below. 
   The mixture-ratio calculator  104  generates a motion vector within a shutter time, indicating motion of image within a frame, and positional information corresponding to the motion vector within the shutter time, indicating a pixel or an image object, based on the input image and the area information supplied from the area specifying unit  103 , and supplies the motion vector within the shutter time and the positional information thereof to the shutter-time calculator  106  and the motion-blur adjusting unit  107 . 
   The motion vector within the shutter time, output by the mixture-ratio calculator  104 , includes information corresponding to an amount of movement v within the shutter time. 
   The amount of movement v within the shutter time is a value indicating a positional change in an image corresponding to a moving object in units of the pixel pitch. For example, if a component of an object image corresponding to a foreground is moving so as to be included in four pixels in a frame, the amount of movement v within the shutter time of the object image corresponding to the foreground is 4. 
   The foreground/background separator  105  separates the input image into a foreground component image formed of only the image components corresponding to the foreground object (hereinafter also be referred to as “foreground components”) and a background component image formed of only the background components based on the area information supplied from the area specifying unit  103  and the mixture ratio α supplied from the mixture-ratio calculator  104 , and supplies the foreground component image to the motion-blur adjusting unit  107  and a selector  107 . The separated foreground component image may be set as the final output. A more precise foreground and background can be obtained compared to a known method in which only a foreground and a background are specified without considering the mixed area. 
   The shutter-time calculator  106  calculates a shutter time based on the inter-frame motion vector and the positional information thereof supplied from the motion detector  102  and the motion vector within the shutter time and the positional information thereof supplied from the mixture-ratio calculator  104 , and supplies the shutter time to the motion-blur adjusting unit  107 . 
   The motion-blur adjusting unit  107  determines the unit of processing indicating at least one pixel contained in the foreground component image based on the amount of movement v within the shutter time obtained from the motion vector within the shutter time and based on the area information. The unit of processing is data that specifies a group of pixels to be subjected to the motion-blur adjustments. 
   Based on the shutter time, the foreground component image supplied from the foreground/background separator  105 , the motion vector within the shutter time and the positional information thereof supplied from the motion detector  102 , and the unit of processing, the motion-blur adjusting unit  107  adjusts the amount of motion blur contained in the foreground component image by removing, decreasing, or increasing the motion blur contained in the foreground component image. The motion-blur adjusting unit  106  then outputs the foreground component image in which amount of motion blur is adjusted to the synthesizer  108 . 
   Motion blur is a distortion contained in an image corresponding to a moving object caused by the movement of an object to be captured in the real world and the image-capturing characteristics of the sensor. 
   Based on the area information supplied from the area specifying unit  103  and the mixture ratio supplied from the mixture-ratio calculator  104 , the synthesizer  108  combines a certain background image input to the image processing apparatus with the foreground component image supplied from the motion-blur adjusting unit  107 , in which the amount of motion blur is adjusted, outputting a synthesized image. 
   An input image supplied to the image processing apparatus is discussed below with reference to  FIGS. 3 through 25 . 
     FIG. 3  illustrates image capturing performed by a sensor. The sensor is formed of, for example, a CCD (Charge-Coupled Device) video camera provided with a CCD area sensor, which is a solid-state image-capturing device. An object  111  corresponding to a foreground in the real world moves, for example, horizontally from the left to the right, between an object  112  corresponding to a background and the sensor. 
   The sensor captures the image of the object  111  corresponding to the foreground together with the image of the object  112  corresponding to the background. The sensor outputs the captured image in units of frames. For example, the sensor outputs an image having 30 frames per second. 
   In this specification, a time interval of frames will be referred to as a frame interval time. 
   The exposure time of the sensor can be 1/30 second. The exposure time is a period from when the sensor starts converting input light into electrical charge until when the conversion from the input light to the electrical charge is finished. The exposure time is also referred to as a “shutter time”. 
     FIG. 4  illustrates the arrangement of pixels. In  FIG. 4 , A through I indicate the individual pixels. The pixels are disposed on a plane of a corresponding image. One detection device corresponding to each pixel is disposed on the sensor. When the sensor performs image capturing, each detection device outputs a pixel value of the corresponding pixel forming the image. For example, the position of the detection device in the X direction corresponds to the horizontal direction on the image, while the position of the detection device in the Y direction corresponds to the vertical direction on the image. 
   As shown in  FIG. 5 , the detection device, which is, for example, a CCD, converts input light into electrical charge during a period corresponding to a shutter time, and stores the converted electrical charge. The amount of charge is almost proportional to the intensity of the input light and the period for which the light is input. The detection device sequentially adds the electrical charge converted from the input light to the stored electrical charge during the period corresponding to the shutter time. That is, the detection device integrates the input light during the period corresponding to the shutter time and stores the electrical charge corresponding to the amount of integrated light. It can be considered that the detection device has an integrating function with respect to time. 
   The electrical charge stored in the detection device is converted into a voltage value by a circuit (not shown), and the voltage value is further converted into a pixel value, such as digital data, and is output. Accordingly, each pixel value output from the sensor is a value projected on a linear space, which is a result of integrating a certain three-dimensional portion of the object corresponding to the foreground or the background with respect to the shutter time. 
   The image processing apparatus extracts significant information embedded in the output signal, for example, the mixture ratio α, by the storage operation of the sensor. The image processing apparatus adjusts the amount of distortion, for example, the amount of motion blur, caused by the mixture of the foreground image object itself. The image processing apparatus also adjusts the amount of distortion caused by the mixture of the foreground image object and the background image object. 
     FIG. 6A  illustrates an image obtained by image-capturing an object corresponding to a moving foreground and an object corresponding to a stationary background. 
     FIG. 6B  illustrates an image obtained by image-capturing an object corresponding to a moving foreground and an object corresponding to a stationary background. 
     FIG. 6A  illustrates an image obtained by capturing a moving object corresponding to a foreground and a stationary object corresponding to a background. In the example shown in  FIG. 6A , the object corresponding to the foreground is moving horizontally from the left to the right with respect to the screen. 
     FIG. 6B  illustrates a model obtained by expanding pixel values corresponding to one line of the image shown in  FIG. 6A  in the time direction. The horizontal direction shown in  FIG. 6B  corresponds to the spatial direction X in  FIG. 6A . 
   The values of the pixels in the background area are formed only from the background components, that is, the image components corresponding to the background object. The values of the pixels in the foreground area are formed only from the foreground components, that is, the image components corresponding to the foreground object. 
   The values of the pixels of the mixed area are formed from the background components and the foreground components. Since the values of the pixels in the mixed area are formed from the background components and the foreground components, it may be referred to as a “distortion area”. The mixed area is further classified into a covered background area and an uncovered background area. 
   The covered background area is a mixed area at a position corresponding to the leading end in the direction in which the foreground object is moving, where the background components are gradually covered with the foreground over time. 
   In contrast, the uncovered background area is a mixed area corresponding to the trailing end in the direction in which the foreground object is moving, where the background components gradually appear over time. 
   As discussed above, the image containing the foreground area, the background area, or the covered background area or the uncovered background area is input into the area specifying unit  103 , the mixture-ratio calculator  104 , and the foreground/background separator  105  as the input image. 
     FIG. 7  illustrates the background area, the foreground area, the mixed area, the covered background area, and the uncovered background area discussed above. In the areas corresponding to the image shown in  FIG. 6A , the background area is a stationary portion, the foreground area is a moving portion, the covered background area of the mixed area is a portion that changes from the background to the foreground, and the uncovered background area of the mixed area is a portion that changes from the foreground to the background. 
     FIG. 8  illustrates a model obtained by expanding in the time direction the pixel values of the pixels aligned side-by-side in the image obtained by capturing the image of the object corresponding to the stationary foreground and the image of the object corresponding to the stationary background. For example, as the pixels aligned side-by-side, pixels arranged in one line on the screen can be selected. 
   The pixel values indicated by F 01  through F 04  shown in  FIG. 8  are values of the pixels corresponding to the object of the stationary foreground. The pixel values indicated by B 01  through B 04  shown in  FIG. 8  are values of the pixels corresponding to the object of the stationary background. 
   The vertical direction in  FIG. 8  corresponds to time, and time elapses from the top to the bottom as viewed in the figure. The position at the top side of the rectangle in  FIG. 8  corresponds to the time at which the sensor starts converting input light into electrical charge, and the position at the bottom side of the rectangle in  FIG. 8  corresponds to the time at which the conversion from the input light into the electrical charge is finished. That is, the distance from the top side to the bottom side of the rectangle in  FIG. 8  corresponds to the shutter time. 
   The horizontal direction in  FIG. 8  corresponds to the spatial direction X in  FIG. 6A . More specifically, in the example shown in  FIG. 8 , the distance from the left side of the rectangle indicated by “F 01 ” in  FIG. 8  to the right side of the rectangle indicated by “B 04 ” is eight times the pixel pitch, i.e., eight consecutive pixels. 
   When the foreground object and the background object are stationary, the light input into the sensor does not change during the period corresponding to the shutter time. 
   The period corresponding to the shutter time is divided into two or more portions of equal periods. For example, if the number of virtual divided portions is 4, the model shown in  FIG. 8  can be represented by the model shown in  FIG. 9 . The number of virtual divided portions can be set according to the amount of movement v of the object corresponding to the foreground within the shutter time. For example, the number of virtual divided portions is set to 4 when the amount of movement v within the shutter time is 4, and the period corresponding to the shutter time is divided into four portions. 
   The uppermost line in the figure corresponds to the first divided period from when the shutter has opened. The second line in the figure corresponds to the second divided period from when the shutter has opened. The third line in the figure corresponds to the third divided period from when the shutter has opened. The fourth line in the figure corresponds to the fourth divided period from when the shutter has opened. 
   The shutter time divided in accordance with the amount of movement v within the shutter time is also hereinafter referred to as the “shutter time/v”. 
   When the object corresponding to the foreground is stationary, the light input into the sensor does not change, and thus, the foreground component F 01 /v is equal to the value obtained by dividing the pixel value F 01  by the number of virtual divided portions. Similarly, when the object corresponding to the foreground is stationary, the foreground component F 02 /v is equal to the value obtained by dividing the pixel value F 02  by the number of virtual divided portions, the foreground component F 03 /v is equal to the value obtained by dividing the pixel value F 03  by the number of virtual divided portions, and the foreground component F 04 /v is equal to the value obtained by dividing the pixel value F 04  by the number of virtual divided portions. 
   When the object corresponding to the background is stationary, the light input into the sensor does not change, and thus, the background component B 01 /v is equal to the value obtained by dividing the pixel value B 01  by the number of virtual divided portions. Similarly, when the object corresponding to the background is stationary, the background component B 02 /v is equal to the value obtained by dividing the pixel value B 02  by the number of virtual divided portions, the background component B 03 /v is equal to the value obtained by dividing the pixel value B 03  by the number of virtual divided portions, and the background component B 04 /v is equal to the value obtained by dividing the pixel value B 04  by the number of virtual divided portions. 
   More specifically, when the object corresponding to the foreground is stationary, the light corresponding to the foreground object input into the sensor does not change during the period corresponding to the shutter time. Accordingly, the foreground component F 01 /v corresponding to the first portion of the shutter time/v from when the shutter has opened, the foreground component F 01 /v corresponding to the second portion of the shutter time/v from when the shutter has opened, the foreground component F 01 /v corresponding to the third portion of the shutter time/v from when the shutter has opened, and the foreground component F 01 /v corresponding to the fourth portion of the shutter time/v from when the shutter has opened become the same value. The same applies to F 02 /v through F 04 /v, as in the case of F 01 /v. 
   When the object corresponding to the background is stationary, the light corresponding to the background object input into the sensor does not change during the period corresponding to the shutter time. Accordingly, the background component B 01 /v corresponding to the first portion of the shutter time/v from when the shutter has opened, the background component B 01 /v corresponding to the second portion of the shutter time/v from when the shutter has opened, the background component B 01 /v corresponding to the third portion of the shutter time/v from when the shutter has opened, and the background component B 01 /v corresponding to the fourth portion of the shutter time/v from when the shutter has opened become the same value. The same applies to B 02 /v through B 04 /v. 
   A description is given of the case in which the object corresponding to the foreground is moving and the object corresponding to the background is stationary. 
     FIG. 10  illustrates a model obtained by expanding in the time direction the pixel values of the pixels in one line, including a covered background area, when the object corresponding to the foreground is moving to the right as viewed in the figure. 
   Since one frame is a short period, it can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity. In  FIG. 10 , a component of the image of the object corresponding to the foreground moves so as to be included in four pixels. 
   For example, the foreground component F 04 /v is included in the first to the fourth pixels from the left. 
   In  FIG. 10 , the amount of movement v within the shutter time is 4. 
   In  FIG. 10 , the pixels from the leftmost pixel to the fourth pixel belong to the foreground area. In  FIG. 10 , the pixels from the fifth pixel to the seventh pixel from the left belong to the mixed area, which is the covered background area. In  FIG. 10 , the rightmost pixel belongs to the background area. 
   The object corresponding to the foreground is moving such that it gradually covers the object corresponding to the background over time. Accordingly, the components contained in the pixel values of the pixels belonging to the covered background area change from the background components to the foreground components at a certain time during the period corresponding to the shutter time. 
   For example, the pixel value M surrounded by the thick frame in  FIG. 10  is expressed by equation (1) below.
 
 M=B 02/ v+B 02/ v+F 07/ v+F 06/ v   (1)
 
   For example, the fifth pixel from the left contains a background component corresponding to one portion of the shutter time/v and foreground components corresponding to three portions of the shutter time/v, and thus, the mixture ratio α of the fifth pixel from the left is 1/4. The sixth pixel from the left contains background components corresponding to two portions of the shutter time/v and foreground components corresponding to two portions of the shutter time/v, and thus, the mixture ratio α of the sixth pixel from the left is 1/2. The seventh pixel from the left contains background components corresponding to three portions of the shutter time/v and a foreground component corresponding to one portion of the shutter time/v, and thus, the mixture ratio α of the fifth pixel from the left is 3/4. 
   It can be assumed that the object corresponding to the foreground is a rigid body, and a foreground component is moving with constant velocity so as to be included in four pixels. Accordingly, for example, the foreground component F 07 /v of the fourth pixel from the left in  FIG. 10  corresponding to the first portion of the shutter time/v from when the shutter has opened is equal to the foreground component of the fifth pixel from the left in  FIG. 10  corresponding to the second portion of the shutter time/v from when the shutter has opened. Similarly, the foreground component F 07 /v is equal to the foreground component of the sixth pixel from the left in  FIG. 10  corresponding to the third portion of the shutter time/v from when the shutter has opened, and the foreground component of the seventh pixel from the left in  FIG. 10  corresponding to the fourth portion of the shutter time/v from when the shutter has opened. 
   It can be assumed that the object corresponding to the foreground is a rigid body, and a foreground component is moving with constant velocity so as to be included in four pixels. Accordingly, for example, the foreground component F 06 /v of the third pixel from the left in  FIG. 10  corresponding to the first portion of the shutter time/v from when the shutter has opened is equal to the foreground component of the fourth pixel from the left in  FIG. 10  corresponding to the second portion of the shutter time/v from when the shutter has opened. Similarly, the foreground component F 06 /v is equal to the foreground component of the fifth pixel from the left in  FIG. 10  corresponding to the third portion of the shutter time/v from when the shutter has opened, and the foreground component of the sixth pixel from the left in  FIG. 10  corresponding to the fourth portion of the shutter time/v from when the shutter has opened. 
   It can be assumed that the object corresponding to the foreground is a rigid body, and a foreground component is moving with constant velocity so as to be included in four pixels. Accordingly, for example, the foreground component F 05 /v of the second pixel from the left in  FIG. 10  corresponding to the first portion of the shutter time/v from when the shutter has opened is equal to the foreground component of the third pixel from the left in  FIG. 10  corresponding to the second portion of the shutter time/v from when the shutter has opened. Similarly, the foreground component F 05 /v is equal to the foreground component of the fourth pixel from the left in  FIG. 10  corresponding to the third portion of the shutter time/v from when the shutter has opened, and the foreground component of the fifth pixel from the left in  FIG. 10  corresponding to the fourth portion of the shutter time/v from when the shutter has opened. 
   It can be assumed that the object corresponding to the foreground is a rigid body, and a foreground component is moving with constant velocity so as to be included in four pixels. Accordingly, for example, the foreground component F 04 /v of the left most pixel in  FIG. 10  corresponding to the first portion of the shutter time/v from when the shutter has opened is equal to the foreground component of the second pixel from the left in  FIG. 10  corresponding to the second portion of the shutter time/v from when the shutter has opened. Similarly, the foreground component F 04 /v is equal to the foreground component of the third pixel from the left in  FIG. 10  corresponding to the third portion of the shutter time/v from when the shutter has opened, and the foreground component of the fourth pixel from the left in  FIG. 10  corresponding to the fourth portion of the shutter time/v from when the shutter has opened. 
   Since the foreground area corresponding to the moving object contains motion blur as discussed above, it can also be referred to as a “distortion area”. 
     FIG. 11  illustrates a model obtained by expanding in the time direction the pixel values of the pixels in one line including an uncovered background area when the object corresponding to the foreground is moving to the right as viewed in the figure. 
   Since one frame is a short period, it can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity. In  FIG. 11 , the object image component corresponding to the foreground is moving with constant velocity so as to be included in four pixels. For example, the foreground component F 01 /v moves so as to be included in the fifth to eighth pixels from the left. 
   In  FIG. 11 , the amount of movement v within the shutter time of the foreground is 4. 
   In  FIG. 11 , the pixels from the leftmost pixel to the fourth pixel belong to the background area. In  FIG. 11 , the pixels from the fifth pixel to the seventh pixels from the left belong to the mixed area, which is an uncovered background area. In  FIG. 11 , the rightmost pixel belongs to the foreground area. 
   The object corresponding to the foreground which covers the object corresponding to the background is moving such that it is gradually removed from the object corresponding to the background over time. Accordingly, the components contained in the pixel values of the pixels belonging to the uncovered background area change from the foreground components to the background components at a certain time of the period corresponding to the shutter time. 
   For example, the pixel value M′ surrounded by the thick frame in  FIG. 11  is expressed by equation (2).
 
 M′=F 02/ v+F 01/ v+B 26/ v+B 26/ v   (2)
 
   For example, the fifth pixel from the left contains background components corresponding to three portions of the shutter time/v and a foreground component corresponding to one shutter portion of the shutter time/v, and thus, the mixture ratio α of the fifth pixel from the left is 3/4. The sixth pixel from the left contains background components corresponding to two portions of the shutter time/v and foreground components corresponding to two portions of the shutter time/v, and thus, the mixture ratio α of the sixth pixel from the left is 1/2. The seventh pixel from the left contains a background component corresponding to one portion of the shutter time/v and foreground components corresponding to three portions of the shutter time/v, and thus, the mixture ratio α of the seventh pixel from the left is 1/4. 
   When equations (1) and (2) are generalized, the pixel value M can be expressed by equation (3): 
                 M   =       α   ·   B     +       ∑   i     ⁢     F   ⁢           ⁢     i   /   v                   (   3   )               
where α is the mixture ratio, B indicates a pixel value of the background, and Fi/v designates a foreground component.
 
   It can be assumed that the object corresponding to the foreground is a rigid body, which is moving with constant velocity, and the amount of movement within the shutter time is 4. Accordingly, for example, the foreground component F 01 /v of the fifth pixel from the left in  FIG. 11  corresponding to the first portion of the shutter time/v from when the shutter has opened is equal to the foreground component of the sixth pixel from the left in  FIG. 11  corresponding to the second portion of the shutter time/v from when the shutter has opened. Similarly, the foreground component F 01 /v is equal to the foreground component of the seventh pixel from the left in  FIG. 11  corresponding to the third portion of the shutter time/v from when the shutter has opened, and the foreground component of the eighth pixel from the left in  FIG. 11  corresponding to the fourth portion of the shutter time/v from when the shutter has opened. 
   It can be assumed that the object corresponding to the foreground is a rigid body, which is moving with constant velocity, and the number of virtual divided portions is 4. Accordingly, for example, the foreground component F 02 /v of the sixth pixel from the left in  FIG. 11  corresponding to the first portion of the shutter time/v from when the shutter has opened is equal to the foreground component of the seventh pixel from the left in  FIG. 11  corresponding to the second portion of the shutter time/v from when the shutter has opened. Similarly, the foreground component F 02 /v is equal to the foreground component of the eighth pixel from the left in  FIG. 11  corresponding to the third portion of the shutter time/v from when the shutter has opened. 
   It can be assumed that the object corresponding to the foreground is a rigid body, which is moving with constant velocity, and the amount of movement v within the shutter time is 4. Accordingly, for example, the foreground component F 03 /v of the seventh pixel from the left in  FIG. 11  corresponding to the first portion of the shutter time/v from when the shutter has opened is equal to the foreground component of the eighth pixel from the left in  FIG. 11  corresponding to the second portion of the shutter time/v from when the shutter has opened. 
   It has been described with reference to  FIGS. 9 through 11  that the number of virtual divided portions is 4. The number of virtual divided portions corresponds to the amount of movement v within the shutter time. Generally, the amount of movement v within the shutter time corresponds to the moving speed of the object corresponding to the foreground. For example, if a foreground component moves so as to be included in four pixels in a frame, the amount of movement v within the shutter time is set to 4. The number of virtual divided portions is set to 4 in accordance with the amount of movement v within the shutter time. 
   Similarly, if a foreground component moves so as to be included in six pixels in a frame, the amount of movement v within the shutter time is set to 6, and the number of virtual divided portions is set to 6. 
     FIGS. 12 and 13  illustrate the relationship of the foreground area, the background area, and the mixed area which consists of a covered background or an uncovered background, which are discussed above, to the foreground components and the background components corresponding to the divided periods of the shutter time. 
     FIG. 12  illustrates an example in which pixels in the foreground area, the background area, and the mixed area are extracted from an image containing a foreground corresponding to an object moving in front of a stationary background. In the example shown in  FIG. 12 , the object corresponding to the foreground, indicated by A, is horizontally moving with respect to the screen. 
   Frame #n+1 is a frame subsequent to frame #n, and frame #n+2 is a frame subsequent to frame #n+1. 
   Pixels in the foreground area, the background area, and the mixed area are extracted from one of frames #n through #n+2, and the amount of movement v within the shutter time is set to 4. A model obtained by expanding the pixel values of the extracted pixels in the time direction is shown in  FIG. 13 . 
   Since the object corresponding to the foreground is moving, the pixel values in the foreground area are formed of four different foreground components corresponding to the shutter time/v. For example, the leftmost pixel of the pixels in the foreground area shown in  FIG. 13  consists of F 01 /v, F 02 /v, F 03 /v, and F 04 /v. That is, the pixels in the foreground contain motion blur. 
   Since the object corresponding to the background is stationary, light input into the sensor corresponding to the background during the shutter time does not change. In this case, the pixel values in the background area do not contain motion blur. 
   The pixel values in the mixed area consisting of a covered background area or an uncovered background area are formed of foreground components and background components. 
   A description is given below of a model obtained by expanding in the time direction the pixel values of the pixels which are aligned side-by-side in a plurality of frames and which are located at the same positions when the frames are overlapped when the image corresponding to the object is moving. For example, when the image corresponding to the object is moving horizontally with respect to the screen, pixels aligned on the screen can be selected as the pixels aligned side-by-side. 
     FIG. 14  illustrates a model obtained by expanding in the time direction the pixels which are aligned side-by-side in three frames of an image obtained by capturing an object corresponding to a stationary background and which are located at the same positions when the frames are overlapped. In  FIG. 14 , the shutter time is as long as the frame interval time. 
   Frame #n is the frame subsequent to frame #n−1, and frame #n+1 is the frame subsequent to frame #n. The same applies to the other frames. 
   The pixel values B 01  through B 12  shown in  FIG. 14  are pixel values corresponding to the stationary background object. Since the object corresponding to the background is stationary, the pixel values of the corresponding pixels in frame #n−1 through frame #n+1 do not change. For example, the pixel in frame #n and the pixel in frame #n+1 located at the corresponding position of the pixel having the pixel value B 05  in frame #n−1 have the pixel value B 05 . 
   With reference to  FIGS. 15 and 16 , an image including a covered background area, for which the shutter time and the frame interval time are the same, will be described. 
     FIG. 15  illustrates a model obtained by expanding in the time direction the pixels which are aligned side-by-side in three frames of an image obtained by capturing an object corresponding to a foreground that is moving to the right as viewed in the figure together with an object corresponding to a stationary background and which are located at the same positions when the frames are overlapped. 
   In  FIG. 15 , it can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity, and a foreground component moves so as to be included in four pixels in a frame. Accordingly, the amount of movement v within the shutter time is 4, and the number of virtual divided portions is 4. 
   For example, the foreground component of the leftmost pixel of frame #n−1 in  FIG. 15  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 12 /v, and the foreground component of the second pixel from the left in  FIG. 15  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 12 /v. The foreground component of the third pixel from the left in  FIG. 15  corresponding to the third portion of the shutter time/v from when the shutter has opened and the foreground component of the fourth pixel from the left in  FIG. 15  corresponding to the fourth portion of the shutter time/v from when the shutter has opened are F 12 /v. 
   The foreground component of the leftmost pixel of frame #n−1 in  FIG. 15  corresponding to the second portion of the shutter time/v from when the shutter has opened is F 11 /v. The foreground component of the second pixel from the left in  FIG. 15  corresponding to the third portion of the shutter time/v from when the shutter has opened is also F 11 /v. The foreground component of the third pixel from the left in  FIG. 15  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is F 11 /v. 
   The foreground component of the leftmost pixel of frame #n−1 in  FIG. 15  corresponding to the third portion of the shutter time/v from when the shutter has opened is F 10 /v. The foreground component of the second pixel from the left in  FIG. 15  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is also F 10 /v. The foreground component of the leftmost pixel of frame #n−1 in  FIG. 15  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is F 09 /v. 
   Since the object corresponding to the background is stationary, the background component of the second pixel from the left of frame #n−1 in  FIG. 15  corresponding to the first portion of the shutter time/v from when the shutter has opened is B 01 /v. The background components of the third pixel from the left of frame #n−1 in  FIG. 15  corresponding to the first and second portions of the shutter time/v from when the shutter has opened are B 02 /v. The background components of the fourth pixel from the left of frame #n−1 in  FIG. 15  corresponding to the first through third portions of the shutter time/v from when the shutter has opened are B 03 /v. 
   In frame #n−1 in  FIG. 15 , the leftmost pixel from the left belongs to the foreground area, and the second through fourth pixels from the left belong to the mixed area, which is a covered background area. 
   The fifth through twelfth pixels from the left of frame #n−1 in  FIG. 15  belong to the background area, and the pixel values thereof are B 04  through B 11 , respectively. 
   The first through fifth pixels from the left in frame #n in  FIG. 15  belong to the foreground area. The foreground component in the shutter time/v in the foreground area of frame #n is any one of F 05 /v through F 12 /v. 
   It can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity, the amount of movement v within the shutter time is 4, and the shutter time and the frame interval time are the same, the foreground image moves so as to be displayed four pixels to the right in the subsequent frame. 
   The foreground component of the fifth pixel from the left of frame #n in  FIG. 15  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 12 /v, and the foreground component of the sixth pixel from the left in  FIG. 15  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 12 /v. The foreground component of the seventh pixel from the left in  FIG. 15  corresponding to the third portion of the shutter time/v from when the shutter has opened and the foreground component of the eighth pixel from the left in  FIG. 15  corresponding to the fourth portion of the shutter time/v from when the shutter has opened are F 12 /v. 
   The foreground component of the fifth pixel from the left of frame #n in  FIG. 15  corresponding to the second portion of the shutter time/v from when the shutter has opened is F 11 /v. The foreground component of the sixth pixel from the left in  FIG. 15  corresponding to the third portion of the shutter time/v from when the shutter has opened is also F 11 /v. The foreground component of the seventh pixel from the left in  FIG. 15  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is F 11 /v. 
   The foreground component of the fifth pixel from the left of frame #n in  FIG. 15  corresponding to the third portion of the shutter time/v from when the shutter has opened is F 10 /v. The foreground component of the sixth pixel from the left in  FIG. 15  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is also F 10 /v. The foreground component of the fifth pixel from the left of frame #n in  FIG. 15  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is F 09 /v. 
   Since the object corresponding to the background is stationary, the background component of the sixth pixel from the left of frame #n in  FIG. 15  corresponding to the first portion of the shutter time/v from when the shutter has opened is B 05 /v. The background components of the seventh pixel from the left of frame #n in  FIG. 15  corresponding to the first and second portions of the shutter time/v from when the shutter has opened are B 06 /v. The background components of the eighth pixel from the left of frame #n in  FIG. 15  corresponding to the first through third portions of the shutter time/v from when the shutter has opened are B 07 /v. 
   In frame #n in  FIG. 15 , the sixth through eighth pixels from the left belong to the mixed area, which is a covered background area. 
   The ninth through twelfth pixels from the left of frame #n in  FIG. 15  belong to the background area, and the pixel values thereof are B 08  through B 11 , respectively. 
   The first through ninth pixels from the left in frame #n+1 in  FIG. 15  belong to the foreground area. The foreground component in the shutter time/v in the foreground area of frame #n+1 is any one of F 01 /v through F 12 /v. 
   It can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity, the amount of movement v within the shutter time is 4, and the shutter time and the frame interval time are the same. Accordingly, the foreground component of the ninth pixel from the left of frame #n+1 in  FIG. 15  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 12 /v, and the foreground component of the tenth pixel from the left in  FIG. 15  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 12 /v. The foreground component of the eleventh pixel from the left in  FIG. 15  corresponding to the third portion of the shutter time/v from when the shutter has opened and the foreground component of the twelfth pixel from the left in  FIG. 15  corresponding to the fourth portion of the shutter time/v from when the shutter has opened are F 12 /v. 
   The foreground component of the ninth pixel from the left of frame #n+1 in  FIG. 15  corresponding to the second portion of the shutter time/v from when the shutter has opened is F 11 /v. The foreground component of the tenth pixel from the left in  FIG. 15  corresponding to the third portion of the shutter time/v from when the shutter has opened is also F 11 /v. The foreground component of the eleventh pixel from the left in  FIG. 15  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is F 11 /v. 
   The foreground component of the ninth pixel from the left of frame #n+1 in  FIG. 15  corresponding to the third portion of the shutter time/v from when the shutter has opened is F 10 /v. The foreground component of the tenth pixel from the left in  FIG. 15  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is also F 10 /v. The foreground component of the ninth pixel from the left of frame #n+1 in  FIG. 15  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is F 09 /v. 
   Since the object corresponding to the background is stationary, the background component of the tenth pixel from the left of frame #n+1 in  FIG. 15  corresponding to the first portion of the shutter time/v from when the shutter has opened is B 09 /v. The background components of the eleventh pixel from the left of frame #n+1 in  FIG. 15  corresponding to the first and second portions of the shutter time/v from when the shutter has opened are B 10 /v. The background components of the twelfth pixel from the left of frame #n+1 in  FIG. 15  corresponding to the first through third portions of the shutter time/v from when the shutter has opened are B 11 /v. 
   In frame #n+1 in  FIG. 15 , the tenth through twelfth pixels from the left belong to the mixed area, which is a covered background area. 
     FIG. 16  is a model of an image obtained by extracting the foreground components from the pixel values shown in  FIG. 15 . 
     FIG. 17  illustrates a model obtained by expanding in the time direction the pixels which are aligned side-by-side in three frames of an image obtained by capturing an object corresponding to a stationary background and which are located at the same positions when the frames are overlapped. In  FIG. 17 , the frame interval time is twice as long as the shutter time. 
   The pixel values B 01  through B 12  shown in  FIG. 17  are pixel values corresponding to the stationary background object. Since the object corresponding to the background is stationary, the pixel values of the corresponding pixels in frame #n−1 through frame #n+1 do not change. For example, the pixel in frame #n and the pixel in frame #n+1 located at the corresponding position of the pixel having the pixel value B 05  in frame #n−1 have the pixel value B 05 . 
   As described above, components of an image obtained by capturing only a stationary object do not change even if relationship between the shutter time and the frame interval time changes. 
   With reference to  FIGS. 18 and 19 , an image including a covered background area, for which the shutter time is half as long as the frame interval time, will be described. 
     FIG. 18  illustrates a model obtained by expanding in the time direction the pixels which are aligned side-by-side in three frames of an image obtained by capturing an object corresponding to a foreground that is moving to the right as viewed in the figure together with an object corresponding to a stationary background and which are located at the same positions when the frames are overlapped. 
   In  FIG. 18 , it can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity, and a foreground component moves so as to be included in four pixels in a frame. Accordingly, the amount of movement v within the shutter time is 4, and the number of virtual divided portions is 4. 
   For example, the foreground component of the leftmost pixel of frame #n−1 in  FIG. 18  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 20 /v, and the foreground component of the second pixel from the left in  FIG. 18  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 20 /v. The foreground component of the third pixel from the left in  FIG. 18  corresponding to the third portion of the shutter time/v from when the shutter has opened and the foreground component of the fourth pixel from the left in  FIG. 18  corresponding to the fourth portion of the shutter time/v from when the shutter has opened are F 20 /v. 
   The foreground component of the leftmost pixel of frame #n−1 in  FIG. 18  corresponding to the second portion of the shutter time/v from when the shutter has opened is F 19 /v. The foreground component of the second pixel from the left in  FIG. 18  corresponding to the third portion of the shutter time/v from when the shutter has opened is also F 19 /v. The foreground component of the third pixel from the left in  FIG. 18  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is F 19 /v. 
   The foreground component of the leftmost pixel of frame #n−1 in  FIG. 18  corresponding to the third portion of the shutter time/v from when the shutter has opened is F 18 /v. The foreground component of the second pixel from the left in  FIG. 18  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is also F 18 /v. The foreground component of the leftmost pixel of frame #n−1 in  FIG. 18  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is F 17 /v. 
   Since the object corresponding to the background is stationary, the background component of the second pixel from the left of frame #n−1 in  FIG. 18  corresponding to the first portion of the shutter time/v from when the shutter has opened is B 01 /v. The background components of the third pixel from the left of frame #n−1 in  FIG. 18  corresponding to the first and second portions of the shutter time/v from when the shutter has opened are B 02 /v. The background components of the fourth pixel from the left of frame #n−1 in  FIG. 18  corresponding to the first through third portions of the shutter time/v from when the shutter has opened are B 03 /v. 
   In frame #n−1 in  FIG. 18 , the leftmost pixel from the left belongs to the foreground area, and the second through fourth pixels from the left belong to the mixed area, which is a covered background area. 
   The fifth through twentieth pixels from the left of frame #n−1 in  FIG. 18  belong to the background area, and the pixel values thereof are B 04  through B 19 , respectively. 
   The first through ninth pixels from the left in frame #n in  FIG. 18  belong to the foreground area. The foreground component in the shutter time/v in the foreground area of frame #n is any one of F 09 /v through F 20 /v. 
   It can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity, and the frame interval time is twice as long as the shutter time, so that the foreground image moves so as to be displayed eight pixels to the right in the subsequent frame. 
   The foreground component of the ninth pixel from the left of frame #n in  FIG. 18  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 20 /v, and the foreground component of the tenth pixel from the left in  FIG. 18  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 20 /v. The foreground component of the eleventh pixel from the left in  FIG. 18  corresponding to the third portion of the shutter time/v from when the shutter has opened and the foreground component of the twelfth pixel from the left in  FIG. 18  corresponding to the fourth portion of the shutter time/v from when the shutter has opened are F 20 /v. 
   The foreground component of the ninth pixel from the left of frame #n in  FIG. 18  corresponding to the second portion of the shutter time/v from when the shutter has opened is F 19 /v. The foreground component of the tenth pixel from the left in  FIG. 18  corresponding to the third portion of the shutter time/v from when the shutter has opened is also F 19 /v. The foreground component of the eleventh pixel from the left in  FIG. 18  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is F 19 /v. 
   The foreground component of the ninth pixel from the left of frame #n in  FIG. 18  corresponding to the third portion of the shutter time/v from when the shutter has opened is F 18 /v. The foreground component of the tenth pixel from the left in  FIG. 18  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is also F 18 /v. The foreground component of the ninth pixel from the left of frame #n in  FIG. 18  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is F 17 /v. 
   Since the object corresponding to the background is stationary, the background component of the tenth pixel from the left of frame #n in  FIG. 18  corresponding to the first portion of the shutter time/v from when the shutter has opened is B 09 /v. The background components of the eleventh pixel from the left of frame #n in  FIG. 18  corresponding to the first and second portions of the shutter time/v from when the shutter has opened are B 10 /v. The background components of the twelfth pixel from the left of frame #n in  FIG. 18  corresponding to the first through third portions of the shutter time/v from when the shutter has opened are B 11 /v. 
   In frame #n in  FIG. 18 , the tenth through twelfth pixels from the left belong to the mixed area, which is a covered background area. 
   The thirteenth through twentieth pixels from the left of frame #n in  FIG. 18  belong to the background area, and the pixel values thereof are B 12  through B 19 , respectively. 
   The first through seventeenth pixels from the left in frame #n+1 in  FIG. 18  belong to the foreground area. The foreground component in the shutter time/v in the foreground area of frame #n+1 is any one of F 01 /v through F 20 /v. 
   It can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity, the amount of movement v within the shutter time is 4, and the frame interval time is twice as long as the shutter time. Accordingly, the foreground component of the seventeenth pixel from the left of frame #n+1 in  FIG. 18  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 20 /v, and the foreground component of the eighteenth pixel from the left in  FIG. 18  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 20 /v. The foreground component of the nineteenth pixel from the left in  FIG. 18  corresponding to the third portion of the shutter time/v from when the shutter has opened and the foreground component of the twentieth pixel from the left in  FIG. 18  corresponding to the fourth portion of the shutter time/v from when the shutter has opened are F 20 /v. 
   The foreground component of the seventeenth pixel from the left of frame #n+1 in  FIG. 18  corresponding to the second portion of the shutter time/v from when the shutter has opened is F 19 /v. The foreground component of the eighteenth pixel from the left in  FIG. 18  corresponding to the third portion of the shutter time/v from when the shutter has opened is also F 19 /v. The foreground component of the nineteenth pixel from the left in  FIG. 18  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is F 19 /v. 
   The foreground component of the seventeenth pixel from the left of frame #n+1 in  FIG. 18  corresponding to the third portion of the shutter time/v from when the shutter has opened is F 18 /v. The foreground component of the eighteenth pixel from the left in  FIG. 18  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is also F 18 /v. The foreground component of the seventeenth pixel from the left of frame #n+1 in  FIG. 18  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is F 17 /v. 
   Since the object corresponding to the background is stationary, the background component of the eighteenth pixel from the left of frame #n+1 in  FIG. 18  corresponding to the first portion of the shutter time/v from when the shutter has opened is B 17 /v. The background components of the nineteenth pixel from the left of frame #n+1 in  FIG. 18  corresponding to the first and second portions of the shutter time/v from when the shutter has opened are B 18 /v. The background components of the twentieth pixel from the left of frame #n+1 in  FIG. 18  corresponding to the first through third portions of the shutter time/v from when the shutter has opened are B 19 /v. 
   In frame #n+1 in  FIG. 18 , the eighth through twentieth pixels from the left belong to the mixed area, which is a covered background area. 
     FIG. 19  illustrates a model of an image obtained by extracting the foreground components from the pixel values shown in  FIG. 18 . 
   Next, with reference to  FIGS. 20 and 21 , an image including an uncovered background area, for which the shutter time and the frame interval time are the same, will be described. 
     FIG. 20  illustrates a model obtained by expanding in the time direction the pixels which are aligned side-by-side in three frames of an image obtained by capturing an object corresponding to a foreground that is moving to the right as viewed in the figure together with an object corresponding to a stationary background and which are located at the same positions when the frames are overlapped. 
   In  FIG. 20 , it can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity. Since a foreground component moves so as to be included in four pixels in a frame, the amount of movement v within the shutter time is 4. 
   For example, the foreground component of the leftmost pixel of frame #n−1 in  FIG. 20  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 13 /v, and the foreground component of the second pixel from the left in  FIG. 20  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 13 /v. The foreground component of the third pixel from the left in  FIG. 20  corresponding to the third portion of the shutter time/v from when the shutter has opened and the foreground component of the fourth pixel from the left in  FIG. 20  corresponding to the fourth portion of the shutter time/v from when the shutter has opened are F 13 /v. 
   The foreground component of the second pixel from the left of frame #n−1 in  FIG. 20  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 14 /v. The foreground component of the third pixel from the left in  FIG. 20  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 14 /v. The foreground component of the third pixel from the left in  FIG. 20  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 15 /v. 
   Since the object corresponding to the background is stationary, the background components of the leftmost pixel of frame #n−1 in  FIG. 20  corresponding to the second through fourth portions of the shutter time/v from when the shutter has opened are B 25 /v. The background components of the second pixel from the left of frame #n−1 in  FIG. 20  corresponding to the third and fourth portions of the shutter time/v from when the shutter has opened are B 26 /v. The background component of the third pixel from the left of frame #n−1 in  FIG. 20  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is B 27 /v. 
   In frame #n−1 in  FIG. 20 , the leftmost pixel through the third pixel belong to the mixed area, which is an uncovered background area. 
   The fourth through twelfth pixels from the left of frame #n−1 in  FIG. 20  belong to the foreground area. The foreground component of the frame is any one of F 13 /v through F 24 /v. 
   The leftmost pixel through the fourth pixel from the left of frame #n in  FIG. 20  belong to the background area, and the pixel values thereof are B 25  through B 28 , respectively. 
   It can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity, and a foreground component moves so as to be included in four pixels in a frame. Accordingly, the foreground component of the fifth pixel from the left of frame #n in  FIG. 20  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 13 /v, and the foreground component of the sixth pixel from the left in  FIG. 20  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 13 /v. The foreground component of the seventh pixel from the left in  FIG. 20  corresponding to the third portion of the shutter time/v from when the shutter has opened and the foreground component of the eighth pixel from the left in  FIG. 20  corresponding to the fourth portion of the shutter time/v from when the shutter has opened are F 13 /v. 
   The foreground component of the sixth pixel from the left of frame #n in  FIG. 20  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 14 /v. The foreground component of the seventh pixel from the left in  FIG. 20  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 14 /v. The foreground component of the eighth pixel from the left in  FIG. 20  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 15 /v. 
   Since the object corresponding to the background is stationary, the background components of the fifth pixel from the left of frame #n in  FIG. 20  corresponding to the second through fourth portions of the shutter time/v from when the shutter has opened are B 29 /v. The background components of the sixth pixel from the left of frame #n in  FIG. 20  corresponding to the third and fourth portions of the shutter time/v from when the shutter has opened are B 30 /v. The background component of the seventh pixel from the left of frame #n in  FIG. 20  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is B 31 /v. 
   In frame #n in  FIG. 20 , the fifth pixel through the seventh pixel from the left belong to the mixed area, which is an uncovered background area. 
   The eighth through twelfth pixels from the left of frame #n in  FIG. 20  belong to the foreground area. The value in the foreground area of frame #n corresponding to the period of the shutter time/v is any one of F 13 /v through F 20 /v. 
   The leftmost pixel through the eighth pixel from the left of frame #n+1 in  FIG. 20  belong to the background area, and the pixel values thereof are B 25  through B 32 , respectively. 
   It can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity, and a foreground component moves so as to be included in four pixels in a frame. Accordingly, the foreground component of the ninth pixel from the left of frame #n+1 in  FIG. 20  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 13 /v, and the foreground component of the tenth pixel from the left in  FIG. 20  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 13 /v. The foreground component of the eleventh pixel from the left in  FIG. 20  corresponding to the third portion of the shutter time/v from when the shutter has opened and the foreground component of the twelfth pixel from the left in  FIG. 20  corresponding to the fourth portion of the shutter time/v from when the shutter has opened are F 13 /v. 
   The foreground component of the tenth pixel from the left of frame #n+1 in  FIG. 20  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 14 /v. The foreground component of the eleventh pixel from the left in  FIG. 20  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 14 /v. The foreground component of the twelfth pixel from the left in  FIG. 20  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 15 /v. 
   Since the object corresponding to the background is stationary, the background components of the ninth pixel from the left of frame #n+1 in  FIG. 20  corresponding to the second through fourth portions of the shutter time/v from when the shutter has opened are B 33 /v. The background components of the tenth pixel from the left of frame #n+1 in  FIG. 20  corresponding to the third and fourth portions of the shutter time/v from when the shutter has opened are B 34 /v. The background component of the eleventh pixel from the left of frame #n+1 in  FIG. 20  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is B 35 /v. 
   In frame #n+1 in  FIG. 20 , the ninth through eleventh pixels from the left as viewed in the figure belong to the mixed area, which is an uncovered background area. 
   The twelfth pixel from the left of frame #n+1 in  FIG. 20  belongs to the foreground area. The foreground component in the shutter time/v in the foreground area of frame #n+1 is any one of F 13 /v through F 16 /v, respectively. 
     FIG. 21  illustrates a model of an image obtained by extracting the foreground components from the pixel values shown in  FIG. 20 . 
   Next, with reference to  FIGS. 22 and 23 , an image including an uncovered background area, for which the frame interval time is twice as long as the shutter time, will be described. 
     FIG. 22  illustrates a model obtained by expanding in the time direction the pixels which are aligned side-by-side in three frames of an image obtained by capturing an object corresponding to a foreground that is moving to the right as viewed in the figure together with an object corresponding to a stationary background and which are located at the same positions when the frames are overlapped. 
   In  FIG. 22 , it can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity. Since a foreground component moves so as to be included in four pixels in a frame, the amount of movement v within the shutter time is 4. 
   For example, the foreground component of the leftmost pixel of frame #n−1 in  FIG. 22  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 13 /v, and the foreground component of the second pixel from the left in  FIG. 22  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 13 /v. The foreground component of the third pixel from the left in  FIG. 22  corresponding to the third portion of the shutter time/v from when the shutter has opened and the foreground component of the fourth pixel from the left in  FIG. 22  corresponding to the fourth portion of the shutter time/v from when the shutter has opened are F 13 /v. 
   The foreground component of the second pixel from the left of frame #n−1 in  FIG. 22  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 14 /v. The foreground component of the third pixel from the left in  FIG. 22  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 14 /v. The foreground component of the third pixel from the left in  FIG. 22  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 15 /v. 
   Since the object corresponding to the background is stationary, the background components of the leftmost pixel of frame #n−1 in  FIG. 22  corresponding to the second through fourth portions of the shutter time/v from when the shutter has opened are B 25 /v. The background components of the second pixel from the left of frame #n−1 in  FIG. 22  corresponding to the third and fourth portions of the shutter time/v from when the shutter has opened are B 26 /v. The background component of the third pixel from the left of frame #n−1 in  FIG. 22  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is B 27 /v. 
   In frame #n−1 in  FIG. 22 , the leftmost pixel through the third pixel belong to the mixed area, which is an uncovered background area. 
   The fourth through twentieth pixels from the left of frame #n−1 in  FIG. 22  belong to the foreground area. The foreground component of the frame is any one of F 13 /v through F 32 /v. 
   The leftmost pixel through the eighth pixel from the left of frame #n in  FIG. 22  belong to the background area, and the pixel values thereof are B 25  through B 32 , respectively. 
   It can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity, the amount of movement v within the shutter time is 4, and the frame interval time is twice as long as the shutter time. Accordingly, the foreground image moves so as to be displayed eight pixels to the right in the subsequent frame. 
   The foreground component of the ninth pixel from the left of frame #n in  FIG. 22  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 13 /v, and the foreground component of the tenth pixel from the left in  FIG. 22  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 13 /v. The foreground component of the eleventh pixel from the left in  FIG. 22  corresponding to the third portion of the shutter time/v from when the shutter has opened and the foreground component of the twelfth pixel from the left in  FIG. 22  corresponding to the fourth portion of the shutter time/v from when the shutter has opened are F 13 /v. 
   The foreground component of the tenth pixel from the left of frame #n in  FIG. 22  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 14 /v. The foreground component of the eleventh pixel from the left in  FIG. 22  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 14 /v. The foreground component of the twelfth pixel from the left in  FIG. 22  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 15 /v. 
   Since the object corresponding to the background is stationary, the background components of the ninth pixel from the left of frame #n in  FIG. 22  corresponding to the second through fourth portions of the shutter time/v from when the shutter has opened are B 33 /v. The background components of the tenth pixel from the left of frame #n in  FIG. 22  corresponding to the third and fourth portions of the shutter time/v from when the shutter has opened are B 34 /v. The background component of the eleventh pixel from the left of frame #n in  FIG. 22  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is B 35 /v. 
   In frame #n in  FIG. 22 , the ninth pixel through the eleventh pixel from the left belong to the mixed area, which is an uncovered background area. 
   The twelfth through twentieth pixels from the left of frame #n in  FIG. 22  belong to the foreground area. The value in the foreground area of frame #n corresponding to the period of the shutter time/v is any one of F 13 /v through F 24 /v. 
   The leftmost pixel through the sixteenth pixel from the left of frame #n+1 in  FIG. 22  belong to the background area, and the pixel values thereof are B 25  through B 40 , respectively. 
   It can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity, and a foreground component moves so as to be included in four pixels in a frame. Accordingly, the foreground component of the seventeenth pixel from the left of frame #n+1 in  FIG. 22  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 13 /v, and the foreground component of the eighteenth pixel from the left in  FIG. 22  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 13 /v. The foreground component of the nineteenth pixel from the left in  FIG. 22  corresponding to the third portion of the shutter time/v from when the shutter has opened and the foreground component of the twentieth pixel from the left in  FIG. 22  corresponding to the fourth portion of the shutter time/v from when the shutter has opened are F 13 /v. 
   The foreground component of the eighteenth pixel from the left of frame #n+1 in  FIG. 22  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 14 /v. The foreground component of the nineteenth pixel from the left in  FIG. 22  corresponding to the second portion of the shutter time/v from when the shutter has opened is also F 14 /v. The foreground component of the nineteenth pixel from the left in  FIG. 22  corresponding to the first portion of the shutter time/v from when the shutter has opened is F 15 /v. 
   Since the object corresponding to the background is stationary, the background components of the seventeenth pixel from the left of frame #n+1 in  FIG. 22  corresponding to the second through fourth portions of the shutter time/v from when the shutter has opened are B 41 /v. The background components of the eighteenth pixel from the left of frame #n+1 in  FIG. 22  corresponding to the third and fourth portions of the shutter time/v from when the shutter has opened are B 42 /v. The background component of the nineteenth pixel from the left of frame #n+1 in  FIG. 22  corresponding to the fourth portion of the shutter time/v from when the shutter has opened is B 43 /v. 
   In frame #n+1 in  FIG. 22 , the seventeenth through nineteenth pixels from the left belong to the mixed area, which is an uncovered background area. 
   The twentieth pixel from the left of frame #n+1 in  FIG. 22  belongs to the foreground area. The foreground component in the shutter time/v in the foreground area of frame #n+1 is any one of F 13  through F 16 , respectively. 
     FIG. 23  illustrates a model of an image obtained by extracting the foreground components from the pixel values shown in  FIG. 22 . 
   Referring back to  FIG. 2 , the area specifying unit  103  specifies flags indicating to which of a foreground area, a background area, a covered background area, or an uncovered background area the individual pixels of the input image belong by using the pixel values of a plurality of frames, and supplies the flags to the mixture-ratio calculator  104  and the motion-blur adjusting unit  107  as the area information. 
   The mixture-ratio calculator  104  calculates the mixture ratio α for each pixel contained in the mixed area based on the pixel values of a plurality of frames and the area information, and supplies the calculated mixture ratio α to the foreground/background separator  105 . 
   Based on the input image and the area information supplied from the area specifying unit  103 , the mixture-ratio calculator  104  generates a motion vector within the shutter time and positional information indicating a pixel or an image object corresponding to the motion vector within the shutter time, and supplies the motion vector within the shutter time and the positional information thereof to the motion-blur adjusting unit  107 . The magnitude of the motion vector within the shutter time, obtained by the mixture-ratio calculator  104 , represents the amount of movement v within the shutter time. 
   The foreground/background separator  105  extracts the foreground component image consisting of only the foreground components based on the pixel values of a plurality of frames, the area information, and the mixture ratio α, and supplies the foreground component image to the motion-blur adjusting unit  107 . 
   The motion-blur adjusting unit  107  adjusts the amount of motion blur contained in the foreground component image based on the foreground component image supplied from the foreground/background separator  105 , the motion vector within the shutter time supplied from the mixture-ratio calculator  104 , and the area information supplied from the area specifying unit  103 , and then outputs the foreground component image in which motion blur is adjusted. 
   Processing for adjusting motion blur in a foreground component image by the motion-blur adjusting unit  107  will be described with reference to  FIGS. 24A through 25B . 
     FIGS. 24A and 24B  are diagrams showing a foreground component image before adjusting motion blur and a background image to be combined. In  FIG. 24A , A denotes the foreground component image, and B denotes the amount of movement within the shutter time. In  FIG. 24B , C denotes the background image to be combined. 
   The description will be made in the context of an example where the frame interval time of the foreground component image is as long as the frame interval time of the background image to be combined, the shutter time of the foreground component image is half as long as the frame interval time, and the shutter time of the background image is as long as the frame interval time. 
   As shown in  FIG. 24A , since the shutter time is shorter, the amount of movement within the shutter time of the foreground component image is smaller compared with a case where the shutter time is as long as the frame interval time. Thus, if the foreground component image is combined with the background image without adjusting the amount of motion blur, the synthesized image looks unnatural to a viewer. 
   Accordingly, as shown in  FIGS. 25A and 25B , the motion-blur adjusting unit  107  adjusts the amount of movement within the shutter time, i.e., the amount of motion blur, of the foreground component image in accordance with the length of the shutter time and the length of the frame interval time. In  FIG. 25A , A denotes the foreground component image, and B denotes the amount of movement within the shutter time. In  FIG. 25B , C denotes the background image to be combined. 
   Thus, the amounts of motion blur included in the foreground component image and in the background component image become equal, so that the synthesized image obtained by combining the foreground component image and the background image does not look unnatural to a viewer. 
   The processing for adjusting the amount of motion blur performed by the image processing apparatus is described below with reference to the flowchart of  FIG. 26 . In step S 11 , the area specifying unit  103  executes area specifying processing, based on an input image, for generating area information indicating to which of a foreground area, a background area, a covered background area, or an uncovered background area each pixel of the input image belongs. Details of the area specifying processing are given below. The area specifying unit  103  supplies the generated area information to the mixture-ratio calculator  104 . 
   In step S 11 , the area specifying unit  103  may generate, based on the input image, area information indicating to which of the foreground area, the background area, or the mixed area (regardless of whether each pixel belongs to a covered background area or an uncovered background area) each pixel of the input image belongs. In this case, the foreground/background separator  105  and the motion-blur adjusting unit  107  determine whether the mixed area is a covered background area or an uncovered background area based on the direction of the motion vector within the shutter time or the inter-frame motion vector. For example, if the input image is disposed in the order of the foreground area, the mixed area, and the background area in the direction of the motion vector within the shutter time or the inter-frame motion vector, it is determined that the mixed area is a covered background area. If the input image is disposed in the order of the background area, the mixed area, and the foreground area in the direction of the motion vector within the shutter time or the inter-frame motion vector, it is determined that the mixed area is an uncovered background area. 
   In step S 12 , the mixture-ratio calculator  104  calculates the mixture ratio α and the motion vector within the shutter time for each pixel contained in the mixed area based on the input image and the area information. Details of the processing for calculating the mixture ratio α and the motion vector within the shutter time are given below. The mixture-ratio calculator  104  supplies the calculated mixture ratio α to the foreground/background separator  105  and the synthesizer  108 , and supplies the motion vector within the shutter time to the shutter-time calculator  106  and the motion-blur adjusting unit  107 . 
   In step S 13 , the shutter-time calculator  106  calculates the shutter time based on the inter-frame motion vector and the positional information thereof supplied from the motion detector  102  and the motion vector within the shutter time and the positional information thereof supplied from the mixture-ratio calculator  104 . The shutter-time calculator  106  supplies the calculated shutter time to the motion-blur adjusting unit  107 . 
   In step S 14 , the foreground/background separator  105  extracts the foreground components from the input image based on the area information and the mixture ratio α, and supplies the foreground components to the motion-blur adjusting unit  107  as the foreground component image. 
   In step S 15 , the motion-blur adjusting unit  107  generates, based on the motion vector within the shutter time and the area information, the unit of processing that indicates the positions of consecutive pixels disposed in the moving direction and belonging to any of the uncovered background area, the foreground area, and the covered background area. Based on the shutter time, the motion-blur adjusting unit  107  adjusts the amount of motion blur contained in the foreground components corresponding to the unit of processing. The motion-blur adjusting unit  107  supplies the foreground component image in which motion blur is adjusted to the synthesizer  108 . Details of the processing for adjusting the amount of motion blur are given below. 
   In step S 16 , the image processing apparatus determines whether the processing is finished for the whole screen. If it is determined that the processing is not finished for the whole screen, the process returns to step S 15 , and the processing for adjusting the amount of motion blur for the foreground components corresponding to the unit of processing is repeated. 
   If it is determined in step S 16  that the processing is finished for the whole screen, the processing proceeds to step S 17 , in which the synthesizer  108  combines a certain background image input to the image processing apparatus with the foreground component image supplied from the motion-blur adjusting unit  107 , in which motion blur is adjusted, based on the area information supplied from the area specifying unit  103  and the mixture ratio supplied from the mixture-ratio calculator  104 , and the processing is then exited. 
   In this manner, the image processing apparatus is capable of adjusting the amount of motion blur contained in the foreground and combining the resulting image with a desired background image. That is, the image processing apparatus is capable of adjusting the amount of motion blur contained in sampled data indicating the pixel values of the foreground pixels. 
   The configuration of each of the area specifying unit  103 , the mixture-ratio calculator  104 , the foreground/background separator  105 , and the motion-blur adjusting unit  107  is described below. 
     FIG. 27  is a block diagram illustrating an example of the configuration of the area specifying unit  103 . A frame memory  201  stores an input image in units of frames. When the image to be processed is frame #n, the frame memory  201  stores frame #n−2, which is the frame two frames before frame #n, frame #n−1, which is the frame one frame before frame #n, frame #n, frame #n+1, which is the frame one frame after frame #n, frame #n+2, which is the frame two frames after frame #n. 
   A stationary/moving determining portion  202 - 1  reads the pixel value of the pixel of frame #n+2 located at the same position as a designated pixel of frame #n in which the area to which the pixel belongs is determined, and reads the pixel value of the pixel of frame #n+1 located at the same position of the designated pixel of frame #n from the frame memory  201 , and calculates the absolute value of the difference between the read pixel values. The stationary/moving determining portion  202 - 1  determines whether the absolute value of the difference between the pixel value of frame #n+2 and the pixel value of frame #n+1 is greater than a preset threshold Th. If it is determined that the difference is greater than the threshold Th, a stationary/moving determination indicating “moving” is supplied to an area determining portion  203 - 1 . If it is determined that the absolute value of the difference between the pixel value of the pixel of frame #n+2 and the pixel value of the pixel of frame #n+1 is smaller than or equal to the threshold Th, the stationary/moving determining portion  202 - 1  supplies a stationary/moving determination indicating “stationary” to the area determining portion  203 - 1 . 
   A stationary/moving determining portion  202 - 2  reads the pixel value of a designated pixel of frame #n in which the area to which the pixel belongs is determined, and reads the pixel value of the pixel of frame #n+1 located at the same position as the designated pixel of frame #n from the frame memory  201 , and calculates the absolute value of the difference between the pixel values. The stationary/moving determining portion  202 - 2  determines whether the absolute value of the difference between the pixel value of frame #n+1 and the pixel value of frame #n is greater than a preset threshold Th. If it is determined that the absolute value of the difference between the pixel values is greater than the threshold Th, a stationary/moving determination indicating “moving” is supplied to the area determining portion  203 - 1  and an area determining portion  203 - 2 . If it is determined that the absolute value of the difference between the pixel value of the pixel of frame #n+1 and the pixel value of the pixel of frame #n is smaller than or equal to the threshold Th, the stationary/moving determining portion  202 - 2  supplies a stationary/moving determination indicating “stationary” to the area determining portion  203 - 1  and the area determining portion  203 - 2 . 
   A stationary/moving determining portion  202 - 3  reads the pixel value of a designated pixel of frame #n in which the area to which the pixel belongs is determined, and reads the pixel value of the pixel of frame #n−1 located at the same position as the designated pixel of frame #n from the frame memory  201 , and calculates the absolute value of the difference between the pixel values. The stationary/moving determining portion  202 - 3  determines whether the absolute value of the difference between the pixel value of frame #n and the pixel value of frame #n−1 is greater than a preset threshold Th. If it is determined that the absolute value of the difference between the pixel values is greater than the threshold Th, a stationary/moving determination indicating “moving” is supplied to the area determining portion  203 - 2  and an area determining portion  203 - 3 . If it is determined that the absolute value of the difference between the pixel value of the pixel of frame #n and the pixel value of the pixel of frame #n−1 is smaller than or equal to the threshold Th, the stationary/moving determining portion  202 - 3  supplies a stationary/moving determination indicating “stationary” to the area determining portion  203 - 2  and the area determining portion  203 - 3 . 
   A stationary/moving determining portion  202 - 4  reads the pixel value of the pixel of frame #n−1 located at the same position as a designated pixel of frame #n in which the area to which the pixel belongs is determined, and reads the pixel value of the pixel of frame #n−2 located at the same position as the designated pixel of frame #n from the frame memory  201 , and calculates the absolute value of the difference between the pixel values. The stationary/moving determining portion  202 - 4  determines whether the absolute value of the difference between the pixel value of frame #n−1 and the pixel value of frame #n−2 is greater than a preset threshold Th. If it is determined that the absolute value of the difference between the pixel values is greater than the threshold Th, a stationary/moving determination indicating “moving” is supplied to the area determining portion  203 - 3 . If it is determined that the absolute value of the difference between the pixel value of the pixel of frame #n−1 and the pixel value of the pixel of frame #n−2 is smaller than or equal to the threshold Th, the stationary/moving determining portion  202 - 4  supplies a stationary/moving determination indicating “stationary” to the area determining portion  203 - 3 . 
   When the stationary/moving determination supplied from the stationary/moving determining portion  202 - 1  indicates “stationary” and when the stationary/moving determination supplied from the stationary/moving determining portion  202 - 2  indicates “moving”, the area determining portion  203 - 1  determines that the designated pixel of frame #n belongs to an uncovered background area, and sets “1”, which indicates that the designated pixel belongs to an uncovered background area, in an uncovered-background-area determining flag associated with the designated pixel. 
   When the stationary/moving determination supplied from the stationary/moving determining portion  202 - 1  indicates “moving” or when the stationary/moving determination supplied from the stationary/moving determining portion  202 - 2  indicates “stationary”, the area specifying unit  203 - 1  determines that the designated pixel of frame #n does not belong to an uncovered background area, and sets “0”, which indicates that the designated pixel does not belong to an uncovered background area, in the uncovered-background-area determining flag associated with the designated pixel. 
   The area determining portion  203 - 1  supplies the uncovered-background-area determining flag in which “1” or “0” is set as discussed above to a determining-flag-storing frame memory  204 . 
   When the stationary/moving determination supplied from the stationary/moving determining portion  202 - 2  indicates “stationary” and when the stationary/moving determination supplied from the stationary/moving determining portion  202 - 3  indicate “stationary”, the area determining portion  203 - 2  determines that the designated pixel of frame #n belongs to the stationary area, and sets “1”, which indicates that the pixel belongs to the stationary area, in a stationary-area determining flag associated with the designated pixel. 
   When the stationary/moving determination supplied from the stationary/moving determining portion  202 - 2  indicates “moving” or when the stationary/moving determination supplied from the stationary/moving determining portion  202 - 3  indicate “moving”, the area determining portion  203 - 2  determines that the designated pixel of frame #n does not belong to the stationary area, and sets “0”, which indicates that the pixel does not belong to the stationary area, in the stationary-area determining flag associated with the designated pixel. 
   The area determining portion  203 - 2  supplies the stationary-area determining flag in which “1” or “0” is set as discussed above to the determining-flag-storing frame memory  204 . 
   When the stationary/moving determination supplied from the stationary/moving determining portion  202 - 2  indicates “moving” and when the stationary/moving determination supplied from the stationary/moving determining portion  202 - 3  indicate “moving”, the area determining portion  203 - 2  determines that the designated pixel of frame #n belongs to the moving area, and sets “1”, which indicates that the designated pixel belongs to the moving area, in a moving-area determining flag associated with the designated pixel. 
   When the stationary/moving determination supplied from the stationary/moving determining portion  202 - 2  indicates “stationary” or when the stationary/moving determination supplied from the stationary/moving determining portion  202 - 3  indicate “stationary”, the area determining portion  203 - 2  determines that the designated pixel of frame #n does not belong to the moving area, and sets “0”, which indicates that the pixel does not belong to the moving area, in the moving-area determining flag associated with the designated pixel. 
   The area determining portion  203 - 2  supplies the moving-area determining flag in which “1” or “0” is set as discussed above to the determining-flag-storing frame memory  204 . 
   When the stationary/moving determination supplied from the stationary/moving determining portion  202 - 3  indicates “moving” and when the stationary/moving determination supplied from the stationary/moving determining portion  202 - 4  indicate “stationary”, the area determining portion  203 - 3  determines that the designated pixel of frame #n belongs to a covered background area, and sets “1”, which indicates that the designated pixel belongs to the covered background area, in a covered-background-area determining flag associated with the designated pixel. 
   When the stationary/moving determination supplied from the stationary/moving determining portion  202 - 3  indicates “stationary” or when the stationary/moving determination supplied from the stationary/moving determining portion  202 - 4  indicate “moving”, the area determining portion  203 - 3  determines that the designated pixel of frame #n does not belong to a covered background area, and sets “0”, which indicates that the designated pixel does not belong to a covered background area, in the covered-background-area determining flag associated with the designated pixel. 
   The area determining portion  203 - 3  supplies the covered-background-area determining flag in which “1” or “0” is set as discussed above to the determining-flag-storing frame memory  204 . 
   The determining-flag-storing frame memory  204  thus stores the uncovered-background-area determining flag supplied from the area determining portion  203 - 1 , the stationary-area determining flag supplied from the area determining portion  203 - 2 , the moving-area determining flag supplied from the area determining portion  203 - 2 , and the covered-background-area determining flag supplied from the area determining portion  203 - 3 . 
   The determining-flag-storing frame memory  204  supplies the uncovered-background-area determining flag, the stationary-area determining flag, the moving-area determining flag, and the covered-background-area determining flag stored therein to a synthesizer  205 . The synthesizer  205  generates area information indicating to which of the uncovered background area, the stationary area, the moving area, or the covered background area each pixel belongs based on the uncovered-background-area determining flag, the stationary-area determining flag, the moving-area determining flag, and the covered-background-area determining flag supplied from the determining-flag-storing frame memory  204 , and supplies the area information to a determining-flag-storing frame memory  206 . 
   The determining-flag-storing frame memory  206  stores the area information supplied from the synthesizer  205 , and also outputs the area information stored therein. 
   An example of the processing performed by the area specifying unit  103  in a case where the frame interval time is as long as the shutter time is described below with reference to  FIGS. 28 through 32 . 
   When the object corresponding to the foreground is moving, the position of the image corresponding to the object on the screen changes in every frame. As shown in  FIG. 28 , the image corresponding to the object located at the position indicated by Yn(x,y) in frame #n is positioned at Yn+1(x,y) in frame #n+1, which is subsequent to frame #n. 
   A model obtained by expanding in the time direction the pixel values of the pixels aligned side-by-side in the moving direction of the image corresponding to the foreground object is shown in  FIG. 22 . For example, if the moving direction of the image corresponding to the foreground object is horizontal with respect to the screen, the model shown in  FIG. 29  is a model obtained by expanding in the time direction the pixel values of the pixels disposed on a line side-by-side. 
   In  FIG. 29 , the line in frame #n is equal to the line in frame #n+1. 
   The foreground components corresponding to the object contained in the second pixel to the thirteenth pixel from the left in frame #n are contained in the sixth pixel through the seventeenth pixel from the left in frame #n+1. 
   In frame #n, the pixels belonging to the covered background area are the eleventh through thirteenth pixels from the left, and the pixels belonging to the uncovered background area are the second through fourth pixels from the left. In frame #n+1, the pixels belonging to the covered background area are the fifteenth through seventeenth pixels from the left, and the pixels belonging to the uncovered background area are the sixth through eighth pixels from the left. 
   In the example shown in  FIG. 29 , since the foreground components contained in frame #n moves so as to be included in four pixels, the amount of movement v within the shutter time is 4. The number of virtual divided portions is 4 in accordance with the amount of movement v within the shutter time. 
   A description is now given of a change in pixel values of the pixels belonging to the mixed area in the frames before and after a designated frame. 
   In  FIG. 30 , the pixels belonging to a covered background area in frame #n in which the background is stationary and the amount of movement v within the shutter time in the foreground is 4 are the fifteenth through seventeenth pixels from the left. Since the amount of movement v within the shutter time is 4, the fifteenth through seventeenth frames from the left in the previous frame #n−1 contain only background components and belong to the background area. The fifteenth through seventeenth pixels from the left in frame #n−2, which is one before frame #n−1, contain only background components and belong to the background area. 
   Since the object corresponding to the background is stationary, the pixel value of the fifteenth pixel from the left in frame #n−1 does not change from the pixel value of the fifteenth pixel from the left in frame #n−2. Similarly, the pixel value of the sixteenth pixel from the left in frame #n−1 does not change from the pixel value of the sixteenth pixel from the left in frame #n−2, and the pixel value of the seventeenth pixel from the left in frame #n−1 does not change from the pixel value of the seventeenth pixel from the left in frame #n−2. 
   That is, the pixels in frame #n−1 and frame #n−2 corresponding to the pixels belonging to the covered background area in frame #n consist of only background components, and the pixel values thereof do not change. Accordingly, the absolute value of the difference between the pixel values is almost 0. Thus, the stationary/moving determination made for the pixels in frame #n−1 and frame #n−2 corresponding to the pixels belonging to the mixed area in frame #n by the stationary/moving determining portion  202 - 4  is “stationary”. 
   Since the pixels belonging to the covered background area in frame #n contain foreground components, the pixel values thereof are different from those of frame #n−1 consisting of only background components. Accordingly, the stationary/moving determination made for the pixels belonging to the mixed area in frame #n and the corresponding pixels in frame #n−1 by the stationary/moving determining portion  202 - 3  is “moving”. 
   When the stationary/moving determination result indicating “moving” is supplied from the stationary/moving determining portion  202 - 3 , and when the stationary/moving determination result indicating “stationary” is supplied from the stationary/moving determining portion  202 - 4 , as discussed above, the area determining portion  203 - 3  determines that the corresponding pixels belong to a covered background area. 
   In  FIG. 31 , in frame #n in which the background is stationary and the amount of movement v within the shutter time in the foreground is 4, the pixels contained in an uncovered background area are the second through fourth pixels from the left. Since the frame interval time is as long as the shutter time and the amount of movement v within the shutter time is 4, the second through fourth pixels from the left in the subsequent frame #n+1 contain only background components and belong to the background area. In frame #n+2, which is subsequent to frame #n+1, the second through fourth pixels from the left contain only background components and belong to the background area. 
   Since the object corresponding to the background is stationary, the pixel value of the second pixel from the left in frame #n+2 does not change from the pixel value of the second pixel from the left in frame #n+1. Similarly, the pixel value of the third pixel from the left in frame #n+2 does not change from the pixel value of the third pixel from the left in frame #n+1, and the pixel value of the fourth pixel from the left in frame #n+2 does not change from the pixel value of the fourth pixel from the left in frame #n+1. 
   That is, the pixels in frame #n+1 and frame #n+2 corresponding to the pixels belonging to the uncovered background area in frame #n consist of only background components, and the pixel values thereof do not change. Accordingly, the absolute value of the difference between the pixel values is almost 0. Thus, the stationary/moving determination made for the pixels in frame #n+1 and frame #n+2 corresponding to the pixels belonging to the mixed area in frame #n by the stationary/moving determining portion  202 - 1  is “stationary”. 
   Since the pixels belonging to the uncovered background area in frame #n contain foreground components, the pixel values thereof are different from those of frame #n+1 consisting of only background components. Accordingly, the stationary/moving determination made for the pixels belonging to the mixed area in frame #n and the corresponding pixels in frame #n+1 by the stationary/moving determining portion  202 - 2  is “moving”. 
   When the stationary/moving determination result indicating “moving” is supplied from the stationary/moving determining portion  202 - 2 , and when the stationary/moving determination result indicating “stationary” is supplied from the stationary/moving determining portion  202 - 1 , as discussed above, the area determining portion  203 - 1  determines that the corresponding pixels belong to an uncovered background area. 
     FIG. 32  illustrates determination conditions for frame #n made by the area specifying unit  103 . When the determination result for the pixel in frame #n−2 located at the same image position as a pixel in frame #n to be processed and for the pixel in frame #n−1 located at the same position as the pixel in frame #n is stationary, and when the determination result for the pixel in frame #n and the pixel in frame #n−1 located at the same image position as the pixel in frame #n is moving, the area specifying unit  103  determines that the pixel in frame #n belongs to a covered background area. 
   When the determination result for the pixel in frame #n and the pixel in frame #n−1 located at the same image position as the pixel in frame #n is stationary, and when the determination result for the pixel in frame #n and the pixel in frame #n+1 located at the same image position as the pixel in frame #n is stationary, the area specifying unit  103  determines that the pixel in frame #n belongs to the stationary area. 
   When the determination result for the pixel in frame #n and the pixel in frame #n−1 located at the same image position as the pixel in frame #n is moving, and when the determination result for the pixel in frame #n and the pixel in frame #n+1 located at the same image position as the pixel in frame #n is moving, the area specifying unit  103  determines that the pixel in frame #n belongs to the moving area. 
   When the determination result for the pixel in frame #n and the pixel in frame #n+1 located at the same image position as the pixel in frame #n is moving, and when the determination result for the pixel in frame #n+1 located at the same image position as the pixel in frame #n and the pixel in frame #n+2 located at the same image position as the pixel in frame #n is stationary, the area specifying unit  103  determines that the pixel in frame #n belongs to an uncovered background area. 
     FIGS. 33A through 33D  illustrate examples of the area determination results obtained by the area specifying unit  103 . In  FIG. 33A , the pixels which are determined to belong to a covered background area are indicated in white. In  FIG. 33B , the pixels which are determined to belong to an uncovered background area are indicated in white. 
   In  FIG. 33C , the pixels which are determined to belong to a moving area are indicated in white. In  FIG. 33D , the pixels which are determined to belong to a stationary area are indicated in white. 
     FIG. 34  illustrates the area information indicating the mixed area, in the form of an image, selected from the area information output from the determining-flag-storing frame memory  206 . In  FIG. 34 , the pixels which are determined to belong to the covered background area or the uncovered background area, i.e., the pixels which are determined to belong to the mixed area, are indicated in white. The area information indicating the mixed area output from the determining-flag-storing frame memory  206  designates the mixed area and the portions having a texture surrounded by the portions without a texture in the foreground area. 
   The area specifying processing performed by the area specifying unit  103  is described below with reference to the flowchart of  FIG. 35 . In step S 201 , the frame memory  201  obtains an image of frame #n−2 through frame #n+2 including frame #n. 
   In step S 202 , the stationary/moving determining portion  202 - 3  determines whether the determination result for the pixel in frame #n−1 and the pixel in frame #n located at the same position is stationary. If it is determined that the determination result is stationary, the process proceeds to step S 203  in which the stationary/moving determining portion  202 - 2  determines whether the determination result for the pixel in frame #n and the pixel in frame #n+1 located at the same position is stationary. 
   If it is determined in step S 203  that the determination result for the pixel in frame #n and the pixel in frame #n+1 located at the same position is stationary, the process proceeds to step S 204 . In step S 204 , the area determining portion  203 - 2  sets “1”, which indicates that the pixel to be processed belongs to the stationary area, in the stationary-area determining flag associated with the pixel to be processed. The area determining portion  203 - 2  supplies the stationary-area determining flag to the determining-flag-storing frame memory  204 , and the process proceeds to step S 205 . 
   If it is determined in step S 202  that the determination result for the pixel in frame #n−1 and the pixel in frame #n located at the same position is moving, or if it is determined in step S 203  that the determination result for the pixel in frame #n and the pixel in frame #n+1 located at the same position is moving, the pixel to be processed does not belong to a stationary area. Accordingly, the processing of step S 204  is skipped, and the process proceeds to step S 205 . 
   In step S 205 , the stationary/moving determining portion  202 - 3  determines whether the determination result for the pixel in frame #n−1 and the pixel in frame #n located at the same position is moving. If it is determined that the determination result is moving, the process proceeds to step S 206  in which the stationary/moving determining portion  202 - 2  determines whether the determination result for the pixel in frame #n and the pixel in frame #n+1 located at the same position is moving. 
   If it is determined in step S 206  that the determination result for the pixel in frame #n and the pixel in frame #n+1 located at the same position is moving, the process proceeds to step S 207 . In step S 207 , the area determining portion  203 - 2  sets “1”, which indicates that the pixel to be processed belongs to a moving area, in the moving-area determining flag associated with the pixel to be processed. The area determining area  203 - 2  supplies the moving-area determining flag to the determining-flag-storing frame memory  204 , and the process proceeds to step S 208 . 
   If it is determined in step S 205  that the determination result for the pixel in frame #n−1 and the pixel in frame #n located at the same position is stationary, or if it is determined in step S 206  that the determination result for the pixel in frame #n and the pixel in frame #n+1 located at the same position is stationary, the pixel in frame #n does not belong to a moving area. Accordingly, the processing of step S 207  is skipped, and the process proceeds to step S 208 . 
   In step S 208 , the stationary/moving determining portion  202 - 4  determines whether the determination result for the pixel in frame #n−2 and the pixel in frame #n−1 located at the same position is stationary. If it is determined that the determination result is stationary, the process proceeds to step S 209  in which the stationary/moving determining portion  202 - 3  determines whether the determination result for the pixel in frame #n−1 and the pixel in frame #n located at the same position is moving. 
   If it is determined in step S 209  that the determination result for the pixel in frame #n−1 and the pixel in frame #n located at the same position is moving, the process proceeds to step S 210 . In step S 210 , the area determining portion  203 - 3  sets “1”, which indicates that the pixel to be processed belongs to a covered background area, in the covered-background-area determining flag associated with the pixel to be processed. The area determining portion  203 - 3  supplies the covered-background-area determining flag to the determining-flag-storing frame memory  204 , and the process proceeds to step S 211 . The area determining portion  203 - 3  supplies the covered-background-area determining flag to the determining-flag-storing frame memory  204 , and the process proceeds to step S 211 . 
   If it is determined in step S 208  that the determination result for the pixel in frame #n−2 and the pixel in frame #n−1 located at the same position is moving, or if it is determined in step S 209  that the pixel in frame #n−1 and the pixel in frame #n located at the same position is stationary, the pixel in frame #n does not belong to a covered background area. Accordingly, the processing of step S 210  is skipped, and the process proceeds to step S 211 . 
   In step S 211 , the stationary/moving determining portion  202 - 2  determines whether the determination result for the pixel in frame #n and the pixel in frame #n+1 located at the same position is moving. If it is determined in step S 211  that the determination result is moving, the process proceeds to step S 212  in which the stationary/moving determining portion  202 - 1  determines whether the determination result for the pixel in frame #n+1 and the pixel in frame #n+2 located at the same position is stationary. 
   If it is determined in step S 212  that the determination result for the pixel in frame #n+1 and the pixel in frame #n+2 located at the same position is stationary, the process proceeds to step S 213 . In step S 213 , the area determining portion  203 - 1  sets “1”, which indicates that the pixel to be processed belongs to an uncovered background area, in the uncovered-background-area determining flag associated with the pixel to be processed. The area determining portion  203 - 1  supplies the uncovered-background-flag determining flag to the determining-flag-storing frame memory  204 , and the process proceeds to step S 214 . 
   If it is determined in step S 211  that the determination result for the pixel in frame #n and the pixel in frame #n+1 located at the same position is stationary, or if it is determined in step S 212  that the determination result for the pixel in frame #n+1 and the pixel in frame #n+2 is moving, the pixel in frame #n does not belong to an uncovered background area. Accordingly, the processing of step S 213  is skipped, and the process proceeds to step S 214 . 
   In step S 214 , the area specifying unit  103  determines whether the areas of all the pixels in frame #n are specified. If it is determined that the areas of all the pixels in frame #n are not yet specified, the process returns to step S 202 , and the area specifying processing is repeated for the remaining pixels. 
   If it is determined in step S 214  that the areas of all the pixels in frame #n are specified, the process proceeds to step S 215 . In step S 215 , the synthesizer  215  generates area information indicating the mixed area based on the uncovered-background-area determining flag and the covered-background-area determining flag stored in the determining-flag-storing frame memory  204 , and also generates area information indicating to which of the uncovered background area, the stationary area, the moving area, or the covered background area each pixel belongs, and sets the generated area information in the determining-flag-storing frame memory  206 . The processing is then completed. 
   As discussed above, the area specifying unit  103  is capable of generating area information indicating to which of the moving area, the stationary area, the uncovered background area, or the covered background area each of the pixels contained in a frame belongs. 
   The area specifying unit  103  may apply logical OR to the area information corresponding to the uncovered background area and the area information corresponding to the covered background area so as to generate area information corresponding to the mixed area, and then may generate area information consisting of flags indicating to which of the moving area, the stationary area, or the mixed area the individual pixels contained in the frame belong. 
   When the object corresponding to the foreground has a texture, the area specifying unit  103  is able to specify the moving area more precisely. 
   The area specifying unit  103  is able to output the area information indicating the moving area as the area information indicating the foreground area, and outputs the area information indicating the stationary area as the area information indicating the background area. 
   The embodiment has been described, assuming that the object corresponding to the background is stationary. However, the above-described area specifying processing can be applied even if the image corresponding to the background area contains motion. For example, if the image corresponding to the background area is uniformly moving, the area specifying unit  103  shifts the overall image in accordance with this motion, and performs processing in a manner similar to the case in which the object corresponding to the background is stationary. If the image corresponding to the background area contains locally different motions, the area specifying unit  103  selects the pixels corresponding to the motions, and executes the above-described processing. 
     FIG. 36  is a block diagram illustrating another example of the configuration of the area specifying unit  103 . A background image generator  301  generates a background image corresponding to an input image, and supplies the generated background image to a binary-object-image extracting portion  302 . The background image generator  301  extracts, for example, an image object corresponding to a background object contained in the input image, and generates the background image. 
   An example of a model obtained by expanding in the time direction the pixel values of pixels aligned side-by-side in the moving direction of an image corresponding to a foreground object is shown in  FIG. 37 . For example, if the moving direction of the image corresponding to the foreground object is horizontal with respect to the screen, the model shown in  FIG. 37  is a model obtained by expanding the pixel values of pixels disposed side-by-side on a single line in the time domain. 
   In  FIG. 37 , the line in frame #n is the same as the line in frame #n−1 and the line in frame #n+1. 
   In frame #n, the foreground components corresponding to the object contained in the sixth through seventeenth pixels from the left are contained in the second through thirteenth pixels from the left in frame #n−1 and are also contained in the tenth through twenty-first pixel from the left in frame #n+1. 
   In frame #n−1, the pixels belonging to the covered background area are the eleventh through thirteenth pixels from the left, and the pixels belonging to the uncovered background area are the second through fourth pixels from the left. In frame #n, the pixels belonging to the covered background area are the fifteenth through seventeenth pixels from the left, and the pixels belonging to the uncovered background area are the sixth through eighth pixels from the left. In frame #n+1, the pixels belonging to the covered background area are the nineteenth through twenty-first pixels from the left, and the pixels belonging to the uncovered background area are the tenth through twelfth pixels from the left. 
   In frame #n−1, the pixels belonging to the background area are the first pixel from the left, and the fourteenth through twenty-first pixels from the left. In frame #n, the pixels belonging to the background area are the first through fifth pixels from the left, and the eighteenth through twenty-first pixels from the left. In frame #n+1, the pixels belonging to the background area are the first through ninth pixels from the left. 
   An example of the background image corresponding to the example shown in  FIG. 37  generated by the background image generator  301  is shown in  FIG. 38 . The background image consists of the pixels corresponding to the background object, and does not contain image components corresponding to the foreground object. 
   The binary-object-image extracting portion  302  generates a binary object image based on the correlation between the background image and the input image, and supplies the generated binary object image to a time change detector  303 . 
     FIG. 39  is a block diagram illustrating the configuration of the binary-object-image extracting portion  302 . A correlation-value calculator  321  calculates the correlation between the background image supplied from the background image generator  301  and the input image so as to generate a correlation value, and supplies the generated correlation value to a threshold-value processor  322 . 
   The correlation-value calculator  321  applies equation (4) to, for example, 3×3-background image blocks having X 4  at the center, as shown in  FIG. 40A , and to, for example, 3×3-background image blocks having Y 4  at the center which corresponds to the background image blocks, as shown in  FIG. 40B , thereby calculating a correlation value corresponding to Y 4 . 
   
     
       
         
           
             
               
                 
                   Correlation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   value 
                 
                 = 
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       8 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             X 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             i 
                           
                           - 
                           
                             X 
                             _ 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             0 
                           
                           8 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               Y 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               i 
                             
                             - 
                             
                               Y 
                               _ 
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         8 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 i 
                               
                               - 
                               
                                 X 
                                 _ 
                               
                             
                             ) 
                           
                           2 
                         
                         · 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               0 
                             
                             8 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   Y 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   i 
                                 
                                 - 
                                 
                                   Y 
                                   _ 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 4 
                 ) 
               
             
           
           
             
               
                 
                   X 
                   _ 
                 
                 = 
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       8 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       X 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       i 
                     
                   
                   9 
                 
               
             
             
               
                 ( 
                 5 
                 ) 
               
             
           
           
             
               
                 
                   Y 
                   _ 
                 
                 = 
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       8 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Y 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       i 
                     
                   
                   9 
                 
               
             
             
               
                 ( 
                 6 
                 ) 
               
             
           
         
       
     
   
   The correlation-value calculator  321  supplies the correlation value calculated for each pixel as discussed above to the threshold-value processor  322 . 
   Alternatively, the correlation-value calculator  321  may apply equation (7) to, for example, 3×3-background image blocks having X 4  at the center, as shown in  FIG. 41A , and to, for example, 3×3-background image blocks having Y 4  at the center which corresponds to the background image blocks, as shown in  FIG. 41B , thereby calculating the absolute values of differences corresponding to Y 4 . 
   
     
       
         
           
             
               
                 
                   
                     Sum 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     of 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     absolute 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     values 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     of 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     differences 
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       8 
                     
                     ⁢ 
                     
                        
                       
                         ( 
                         
                           
                             X 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             i 
                           
                           - 
                           
                             Y 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             i 
                           
                         
                         ) 
                       
                        
                     
                   
                 
                 ⁢ 
                 
                     
                 
               
             
             
               
                 ( 
                 7 
                 ) 
               
             
           
         
       
     
   
   The correlation-value calculator  321  supplies the absolute values of the differences calculated as described above to the threshold-value processor  322  as the correlation value. 
   The threshold-value processor  322  compares the pixel value of the correlation image with a threshold value th 0 . If the correlation value is smaller than or equal to the threshold value th 0 , 1 is set in the pixel value of the binary object image. If the correlation value is greater than the threshold value th 0 , 0 is set in the pixel value of the binary object image. The threshold-value processor  322  then outputs the binary object image whose pixel value is set to 0 or 1. The threshold-value processor  322  may store the threshold value th 0  therein in advance, or may use the threshold value th 0  input from an external source. 
     FIG. 42  illustrates the binary object image corresponding to the model of the input image shown in  FIG. 37 . In the binary object image, 0 is set in the pixel values of the pixels having a higher correlation with the background image. 
     FIG. 43  is a block diagram illustrating the configuration of the time change detector  303 . When determining the area of a pixel in frame #n, a frame memory  341  stores a binary object image of frame #n−1, frame #n, and frame #n+1 supplied from the binary-object-image extracting portion  302 . 
   An area determining portion  342  determines the area of each pixel of frame #n based on the binary object image of frame #n−1, frame #n, and frame #n+1 so as to generate area information, and outputs the generated area information. 
     FIG. 44  illustrates the determinations made by the area determining portion  342 . When the designated pixel of the binary object image in frame #n is 0, the area determining portion  342  determines that the designated pixel in frame #n belongs to the background area. 
   When the designated pixel of the binary object image in frame #n is 1, and when the corresponding pixel of the binary object image in frame #n−1 is 1, and when the corresponding pixel of the binary object image in frame #n+1 is 1, the area determining portion  342  determines that the designated pixel in frame #n belongs to the foreground area. 
   When the designated pixel of the binary object image in frame #n is 1, and when the corresponding pixel of the binary object image in frame #n−1 is 0, the area determining portion  342  determines that the designated pixel in frame #n belongs to a covered background area. 
   When the designated pixel of the binary object image in frame #n is 1, and when the corresponding pixel of the binary object image in frame #n+1 is 0, the area determining portion  342  determines that the designated pixel in frame #n belongs to an uncovered background area. 
     FIG. 45  illustrates an example of the determinations made by the time change detector  303  on the binary object image corresponding to the model of the input image shown in  FIG. 37 . The time change detector  303  determines that the first through fifth pixels from the left in frame #n belong to the background area since the corresponding pixels of the binary object image in frame #n are 0. 
   The time change detector  303  determines that the sixth through ninth pixels from the left belong to the uncovered background area since the pixels of the binary object image in frame #n are 1, and the corresponding pixels in frame #n+1 are 0. 
   The time change detector  303  determines that the tenth through thirteenth pixels from the left belong to the foreground area since the pixels of the binary object image in frame #n are 1, the corresponding pixels in frame #n−1 are 1, and the corresponding pixels in frame #n+1 are 1. 
   The time change detector  303  determines that the fourteenth through seventeenth pixels from the left belong to the covered background area since the pixels of the binary object image in frame #n are 1, and the corresponding pixels in frame #n−1 are 0. 
   The time change detector  303  determines that the eighteenth through twenty-first pixels from the left belong to the background area since the corresponding pixels of the binary object image in frame #n are 0. 
   The area specifying processing performed by the area specifying unit  103  is described below with reference to the flowchart of  FIG. 46 . In step S 301 , the background image generator  301  of the area specifying unit  103  extracts, for example, an image object corresponding to a background object contained in an input image based on the input image so as to generate a background image, and supplies the generated background image to the binary-object-image extracting portion  302 . 
   In step S 302 , the binary-object-image extracting portion  302  calculates a correlation value between the input image and the background image supplied from the background image generator  301  according to, for example, calculation discussed with reference to  FIGS. 40A and 40B . In step S 303 , the binary-object-image extracting portion  302  computes a binary object image from the correlation value and the threshold value th 0  by, for example, comparing the correlation value with the threshold value th 0 . 
   In step S 304 , the time change detector  303  executes the area determining processing, and the processing is completed. 
   Details of the area determining processing in step S 304  are described below with reference to the flowchart of  FIG. 47 . In step S 321 , the area determining portion  342  of the time change detector  303  determines whether the designated pixel in frame #n stored in the frame memory  341  is 0. If it is determined that the designated pixel in frame #n is 0, the process proceeds to step S 322 . In step S 322 , it is determined that the designated pixel in frame #n belongs to the background area, and the processing is completed. 
   If it is determined in step S 321  that the designated pixel in frame #n is 1, the process proceeds to step S 323 . In step S 323 , the area determining portion  342  of the time change detector  303  determines whether the designated pixel in frame #n stored in the frame memory  341  is 1, and whether the corresponding pixel in frame #n−1 is 0. If it is determined that the designated pixel in frame #n is 1 and the corresponding pixel in frame #n−1 is 0, the process proceeds to step S 324 . In step S 324 , it is determined that the designated pixel in frame #n belongs to the covered background area, and the processing is completed. 
   If it is determined in step S 323  that the designated pixel in frame #n is 0, or that the corresponding pixel in frame #n−1 is 1, the process proceeds to step S 325 . In step S 325 , the area determining portion  342  of the time change detector  303  determines whether the designated pixel in frame #n stored in the frame memory  341  is 1, and whether the corresponding pixel in frame #n+1 is 0. If it is determined that the designated pixel in frame #n is 1 and the corresponding pixel in frame #n+1 is 0, the process proceeds to step S 326 . In step S 326 , it is determined that the designated pixel in frame #n belongs to the uncovered background area, and the processing is completed. 
   If it is determined in step S 325  that the designated pixel in frame #n is 0, or that the corresponding pixel in frame #n+1 is 1, the process proceeds to step S 327 . In step S 327 , the area determining portion  342  of the time change detector  303  determines that the designated pixel in frame #n belongs to the foreground area, and the processing is completed. 
   As discussed above, the area specifying unit  103  is able to specify, based on the correlation value between the input image and the corresponding background image, to which of the foreground area, the background area, the covered background area, or the uncovered background area each pixel of the input image belongs, and generates area information corresponding to the specified result. 
     FIG. 48  is a block diagram illustrating the configuration of the mixture-ratio calculator  104 . An estimated-mixture-ratio processor  401  calculates an estimated mixture ratio for each pixel by calculating a model of a covered background area based on the input image, and supplies the calculated estimated mixture ratio to a mixture-ratio determining portion  403 . The estimated-mixture-ratio processor  401  calculates an estimated motion vector based on the estimated mixture ratio calculated for each pixel by calculating the model of a covered background area, and supplies the calculated estimated motion vector to a mixture-ratio determining portion  403 . 
   An estimated-mixture-ratio processor  402  calculates an estimated mixture ratio for each pixel by calculating a model of an uncovered background area based on the input image, and supplies the calculated estimated mixture ratio to the mixture-ratio determining portion  403 . The estimated-mixture-ratio processor  402  calculates an estimated motion vector based on the estimated mixture ratio calculated for each pixel by calculating the model of an uncovered background area, and supplies the calculated estimated motion vector to the mixture-ratio determining portion  403 . 
   The mixture-ratio determining portion  403  sets the mixture ratio α based on the area information supplied from the area specifying unit  103  and indicating to which of the foreground area, the background area, the covered background area, or the uncovered background area the pixel for which the mixture ratio α is to be calculated belongs. The mixture-ratio determining portion  403  sets the mixture ratio α to 0 when the corresponding pixel belongs to the foreground area, and sets the mixture ratio α to 1 when the corresponding pixel belongs to the background area. When the corresponding pixel belongs to the covered background area, the mixture-ratio determining portion  403  sets the estimated mixture ratio supplied from the estimated-mixture-ratio processor  401  as the mixture ratio α. When the corresponding pixel belongs to the uncovered background area, the mixture-ratio determining portion  403  sets the estimated mixture ratio supplied from the estimated-mixture-ratio processor  402  as the mixture ratio α. The mixture-ratio determining portion  403  outputs the mixture ratio α which has been set based on the area information. 
   Based on the area information supplied from the area specifying unit  103 , the mixture-ratio determining portion  403  sets the estimated motion vector supplied from the estimated-mixture-ratio processor  401  as the motion vector within the shutter time if the corresponding pixel belongs to the covered background area, whereas it supplies the estimated motion vector supplied from the estimated-mixture-ratio processor  402  as the motion vector within the shutter time if the corresponding pixel belongs to the uncovered background area. 
   The mixture-ratio determining portion  403  outputs the motion vector within the shutter time and the positional information thereof that have been set based on the area information. 
   Since it can be assumed that the object corresponding to the foreground is moving with constant velocity within the shutter time, the mixture ratio α of the pixels belonging to a mixed area exhibits the following characteristics. That is, the mixture ratio α linearly changes according to the positional change in the pixels. If the positional change in the pixels is one-dimensional, a change in the mixture ratio α can be represented linearly. If the positional change in the pixels is two-dimensional, a change in the mixture ratio α can be represented on a plane. 
   Since the period of one frame is short, it can be assumed that the object corresponding to the foreground is a rigid body moving with constant velocity. 
   The gradient of the mixture ratio α is inversely proportional to the amount of movement v within the shutter time of the foreground. 
   An example of the ideal mixture ratio α is shown in  FIG. 49 . The gradient l of the ideal mixture ratio α in the mixed area can be represented by the reciprocal of the amount of movement v within the shutter time. 
   As shown in  FIG. 49 , the ideal mixture ratio α has the value of 1 in the background area, the value of 0 in the foreground area, and the value of greater than 0 and smaller than 1 in the mixed area. 
   In the example shown in  FIG. 50 , the pixel value C 06  of the seventh pixel from the left in frame #n can be indicated by equation (8) by using the pixel value P 06  of the seventh pixel from the left in frame #n−1. 
   
     
       
         
           
             
               
                 
                   
                     
                       C06 
                       = 
                       
                         
                           B06 
                           / 
                           v 
                         
                         + 
                         
                           B06 
                           / 
                           v 
                         
                         + 
                         
                           F01 
                           / 
                           v 
                         
                         + 
                         
                           F02 
                           / 
                           v 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           P06 
                           / 
                           v 
                         
                         + 
                         
                           P06 
                           / 
                           v 
                         
                         + 
                         
                           F01 
                           / 
                           v 
                         
                         + 
                         
                           F02 
                           / 
                           v 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           
                             2 
                             / 
                             v 
                           
                           · 
                           P06 
                         
                         + 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             2 
                           
                           ⁢ 
                           
                             F 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               i 
                               / 
                               v 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 8 
                 ) 
               
             
           
         
       
     
   
   In equation (8), the pixel value C 06  is indicated by a pixel value M of the pixel in the mixed area, while the pixel value P 06  is indicated by a pixel value B of the pixel in the background area. That is, the pixel value M of the pixel in the mixed area and the pixel value B of the pixel in the background area can be represented by equations (9) and (10), respectively.
 
M=C06  (9)
 
B=P06  (10)
 
   In equation (8), 2/v corresponds to the mixture ratio α. Since the amount of movement v within the shutter time is 4, the mixture ratio α of the seventh pixel from the left in frame #n is 0.5. 
   As discussed above, the pixel value C in the designated frame #n is considered as the pixel value in the mixed area, while the pixel value P of frame #n−1 prior to frame #n is considered as the pixel value in the background area. Accordingly, equation (3) indicating the mixture ratio α can be represented by equation (11):
 
 C=α·P+f   (11)
 
where f in equation (11) indicates the sum of the foreground components Σ i Fi/v contained in the designated pixel. The variables contained in equation (11) are two factors, i.e., the mixture ratio α and the sum f of the foreground components.
 
   Similarly, a model obtained by expanding in the time direction the pixel values in which the amount of movement v within the shutter time is 4 and the number of virtual divided portions is 4 in an uncovered background area is shown in  FIG. 51 . 
   As in the representation of the covered background area, in the uncovered background area, the pixel value C of the designated frame #n is considered as the pixel value in the mixed area, while the pixel value N of frame #n+1 subsequent to frame #n is considered as the background area. Accordingly, equation (3) indicating the mixture ratio α can be represented by equation (12).
 
 C=α·N+f   (12)
 
   The embodiment has been described, assuming that the background object is stationary. However, equations (8) through (12) can be applied to the case in which the background object is moving by using the pixel value of a pixel located corresponding to the amount of movement v within the shutter time of the background. It is now assumed, for example, in  FIG. 50  that the amount of movement v within the shutter time of the object corresponding to the background is 2, and the number of virtual divided portions is 2. In this case, when the object corresponding to the background is moving to the right in  FIG. 49 , the pixel value B of the pixel in the background area in equation (10) is represented by a pixel value P 04 . 
   Since equations (11) and (12) each contain two variables, the mixture ratio α cannot be determined without modifying the equations. 
   The mixture ratio α linearly changes in accordance with a change in the position of the pixels associated with the object corresponding to the foreground moving with constant velocity. By utilizing this characteristic, an equation in which the mixture ratio α and the sum f of the foreground components are approximated in the spatial direction can hold true. By utilizing a plurality of sets of the pixel values of the pixels belonging to the mixed area and the pixel values of the pixels belonging to the background area, the equations in which the mixture ratio α and the sum f of the foreground components are approximated are solved. 
   When a change in the mixture ratio α is approximated as a straight line, the mixture ratio α can be expressed by equation (13).
 
α= il+p   (13)
 
In equation (13), i indicates the spatial index when the position of the designated pixel is set to 0, 1 designates the gradient of the straight line of the mixture ratio α, and p designates the intercept of the straight line of the mixture ratio α and also indicates the mixture ratio α of the designated pixel. In equation (13), the index i is known, and the gradient l and the intercept p are unknown.
 
   The relationship among the index i, the gradient l, and the intercept p is shown in  FIG. 52 . In  FIG. 52 , the white dot indicates the designated pixel, and the black dots indicate the pixels located in close proximity with the designated pixel. 
   By approximating the mixture ratio α as equation (13), a plurality of different mixture ratios a for a plurality of pixels can be expressed by two variables. In the example shown in  FIG. 52 , the five mixture ratios for five pixels are expressed by the two variables, i.e., the gradient l and the intercept p. 
   When the mixture ratio α is approximated in the plane shown in  FIG. 53 , equation (13) is expanded into the plane by considering the movement v corresponding to the two directions, i.e., the horizontal direction and the vertical direction of the image, and the mixture ratio α can be expressed by equation (14). In  FIG. 53 , the white dot indicates the designated pixel.
 
α= jm+kq+p   (14)
 
In equation (14), j is the index in the horizontal direction and k is the index in the vertical direction when the position of the designated pixel is 0. m designates the horizontal gradient of the mixture ratio α in the plane, and q indicates the vertical gradient of the mixture ratio α in the plane. p indicates the intercept of the mixture ratio α in the plane.
 
   For example, in frame #n shown in  FIG. 50 , equations (15) through (17) can hold true for C 05  through C 07 , respectively.
 
 C 05=α05· B 05/ v+f 05  (15)
 
 C 06=α06· B 06/ v+f 06  (16)
 
 C 07=α07· B 07/ v+f 07  (17)
 
   Assuming that the foreground components positioned in close proximity with each other are equal to each other, i.e., that F 01  through F 03  are equal, equation (18) holds true by replacing F 01  through F 03  by fc.
 
 f ( x )=(1−α( x ))· Fc   (18)
 
In equation (18), x indicates the position in the spatial direction.
 
   When α(x) is replaced by equation (14), equation (18) can be expressed by equation (19). 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         f 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               
                                 ( 
                                 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     m 
                                   
                                   + 
                                   
                                     k 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     q 
                                   
                                   + 
                                   p 
                                 
                                 ) 
                               
                             
                             ) 
                           
                           · 
                           F 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         c 
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           j 
                           · 
                           
                             ( 
                             
                               
                                 
                                   - 
                                   m 
                                 
                                 · 
                                 F 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                             
                             ) 
                           
                         
                         + 
                         
                           k 
                           · 
                           
                             ( 
                             
                               
                                 
                                   - 
                                   q 
                                 
                                 · 
                                 F 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               c 
                             
                             ) 
                           
                         
                         + 
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   p 
                                 
                                 ) 
                               
                               · 
                               F 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             c 
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           s 
                         
                         + 
                         
                           k 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                         + 
                         u 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 19 
                 ) 
               
             
           
         
       
     
   
   In equation (19), (−m·Fc), (−q·Fc), and (1−p)·Fc are replaced, as expressed by equations (20) through (22), respectively.
 
 s=−m·Fc   (20)
 
 t=−q·Fc   (21)
 
 u =(1− p )· Fc   (22)
 
   In equation (19), j is the index in the horizontal direction and k is the index in the vertical direction when the position of the designated pixel is 0. 
   As discussed above, since it can be assumed that the object corresponding to the foreground is moving with constant velocity within the shutter period, and that the foreground components positioned in close proximity with each other are uniform, the sum of the foreground components can be approximated by equation (19). 
   When the mixture ratio α is approximated by a straight line, the sum of the foreground components can be expressed by equation (23).
 
 f ( x )= is+u   (23)
 
   By replacing the mixture ratio α and the sum of the foreground components in equation (13) by using equations (14) and (19), the pixel value M can be expressed by equation (24). 
   
     
       
         
           
             
               
                 
                   
                     
                       M 
                       = 
                         
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 m 
                               
                               + 
                               
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 q 
                               
                               + 
                               p 
                             
                             ) 
                           
                           · 
                           B 
                         
                         + 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           s 
                         
                         + 
                         
                           k 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                         + 
                         u 
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             B 
                             · 
                             m 
                           
                         
                         + 
                         
                           k 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             B 
                             · 
                             q 
                           
                         
                         + 
                         
                           B 
                           · 
                           p 
                         
                         + 
                         
                           j 
                           · 
                           s 
                         
                         + 
                         
                           k 
                           · 
                           t 
                         
                         + 
                         u 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 24 
                 ) 
               
             
           
         
       
     
   
   In equation (24), unknown variables are six factors, such as the horizontal gradient m of the mixture ratio α in the plane, the vertical gradient q of the mixture ratio α in the plane, and the intercepts of the mixture ratio α in the plane, p, s, t, and u. 
   More specifically, according to the pixels in close proximity with the designated pixel, the pixel value M or the pixel value B is set in the normal equation expressed in equation (24). Then, a plurality of normal equations in which the pixel value M or the pixel value B is set are solved by the method of least squares, thereby calculating the mixture ratio α. 
   For example, the horizontal index j of the designated pixel is set to 0, and the vertical index k is set to 0. Then, the pixel value M or the pixel value B is set in the normal equation expressed in equation (24) for 3×3 pixels located close to the designated pixel, thereby obtaining equations (25) through (33).
 
 M   −1,−1 =(−1)· B   −1,−1   ·m +(−1)· B   −1,−1   ·q+B   −1,−1   p +(−1)· s +(−1)· t+u   (25)
 
 M   0,−1 =(0)· B   0,−1   ·m +(−1)· B   0,−1   ·q+B   0,−1   ·p +(0)· s +(−1)· t+u   (26)
 
 M   +1,−1 =(+1)· B   +1,−1   ·m +(−1)· B   +1,−1   ·q+B   +1,−1   ·p +(+1)· s +(−1)· t+u   (27)
 
 M   −1,0 =(−1)· B   −1,0   ·m +(0)· B   −1,0   ·q+B   −1,0   ·p +(−1)· s +(0)· t+u   (28)
 
 M   0,0 =(0)· B   0,0   ·m +(0)· B   0,0   ·q+B   0,0   ·p +(0)· s +(0)· t+u   (29)
 
 M   +1,0 =(+1)· B   +1,0   ·m +(0)· B   +1,0   ·q+B   +1,0   ·p +(+1)· s +(0)· t+u   (30)
 
 M   −1,+1 =(−1)· B   −1,+1   ·m +(+1)· B   −1,+1   ·q+B   −1,30 1   ·p +(−1)· s +(+1)· t+u   (31)
 
 M   0,+1 =(0)· B   0,+1   ·m +(+1)· B   0,+1   ·q+B   0,+1   ·p +(0)· s +(+1)· t+u   (32)
 
 M   +1,+1 =(+1)· B   +1,+1   ·m +(+1)· B   +1,+1   ·q+B   +1,+1   ·p +(+1)· s +(+1)· t+u   (33)
 
   Since the horizontal index j of the designated pixel is 0, and the vertical index k of the designated pixel is 0, the mixture ratio α of the designated pixel is equal to the value when j is 0 and k is 0 in equation (14), i.e., the mixture ratio α is equal to the intercept p in equation (14). 
   Accordingly, based on nine equations, i.e., equations (25) through (33), the horizontal gradient m, the vertical gradient q, and the intercepts p, s, t, and u are calculated by the method of least squares, and the intercept p is output as the mixture ratio α. 
   A specific process for calculating the mixture ratio α by applying the method of least squares is as follows. 
   When the index i and the index k are expressed by a single index x, the relationship among the index i, the index k, and the index x can be expressed by equation (34).
 
 x =( j+ 1)·3+( k+ 1)  (34)
 
   It is now assumed that the horizontal gradient m, the vertical gradient q, and the intercepts p, s, t, and u are expressed by variables w 0 , w 1 , w 2 , w 3 , w 4 , and w 5 , respectively, and jB, kB, B, j, k and  1  are expressed by a 0 , a 1 , a 2 , a 3 , a 4 , and a 5 , respectively. In consideration of the error ex, equations (25) through (33) can be modified into equation (35). 
   
     
       
         
           
             
               
                 
                   M 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   x 
                 
                 = 
                 
                   
                     
                       ∑ 
                       
                         y 
                         = 
                         0 
                       
                       5 
                     
                     ⁢ 
                     
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         y 
                         · 
                         w 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       y 
                     
                   
                   + 
                   
                     e 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     x 
                   
                 
               
             
             
               
                 ( 
                 35 
                 ) 
               
             
           
         
       
     
   
   In equation (35), x is any one of the integers from 0 to 8. 
   Equation (36) can be found from equation (35). 
   
     
       
         
           
             
               
                 
                   e 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   x 
                 
                 = 
                 
                   
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     x 
                   
                   - 
                   
                     
                       ∑ 
                       
                         y 
                         = 
                         0 
                       
                       5 
                     
                     ⁢ 
                     
                       a 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         y 
                         · 
                         w 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       y 
                     
                   
                 
               
             
             
               
                 ( 
                 36 
                 ) 
               
             
           
         
       
     
   
   Since the method of least squares is applied, the square sum E of the error is defined as follows, as expressed by equation (37). 
   
     
       
         
           
             
               
                 E 
                 = 
                 
                   
                     ∑ 
                     
                       x 
                       = 
                       0 
                     
                     8 
                   
                   ⁢ 
                   
                     e 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       x 
                       2 
                     
                   
                 
               
             
             
               
                 ( 
                 37 
                 ) 
               
             
           
         
       
     
   
   In order to minimize the error, the partial differential value of the variable Wv with respect to the square sum E of the error should be 0. v is any one of the integers from 0 to 5. Thus, wy is determined so that equation (38) is satisfied. 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ∂ 
                           E 
                         
                         
                           
                             ∂ 
                             W 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           v 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         2 
                         · 
                         
                           
                             ∑ 
                             
                               x 
                               = 
                               0 
                             
                             8 
                           
                           ⁢ 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               x 
                               · 
                               
                                 
                                   
                                     ∂ 
                                     e 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   x 
                                 
                                 
                                   
                                     ∂ 
                                     W 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   v 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           2 
                           · 
                           
                             
                               ∑ 
                               
                                 x 
                                 = 
                                 0 
                               
                               8 
                             
                             ⁢ 
                             
                               e 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 x 
                                 · 
                                 a 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               v 
                             
                           
                         
                         = 
                         0 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 38 
                 ) 
               
             
           
         
       
     
   
   By substituting equation (36) into equation (38), equation (39) is obtained. 
   
     
       
         
           
             
               
                 
                   
                     ∑ 
                     
                       x 
                       = 
                       0 
                     
                     8 
                   
                   ⁢ 
                   
                     ( 
                     
                       av 
                       · 
                       
                         
                           ∑ 
                           
                             y 
                             = 
                             0 
                           
                           5 
                         
                         ⁢ 
                         
                           ay 
                           · 
                           Wy 
                         
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ∑ 
                     
                       x 
                       = 
                       0 
                     
                     8 
                   
                   ⁢ 
                   
                     
                       av 
                       · 
                       M 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     x 
                   
                 
               
             
             
               
                 ( 
                 39 
                 ) 
               
             
           
         
       
     
   
   For example, the sweep-out method (Gauss-Jordan elimination) is applied to six equations obtained by substituting one of the integers from 0 to 5 into v in equation (39), thereby obtaining wy. As stated above, w 0  is the horizontal gradient m, w 1  is the vertical gradient q, w 2  is the intercept p, w 3  is s, w 4  is t, and w 5  is u. 
   As discussed above, by applying the method of least squares to the equations in which the pixel value. M and the pixel value B are set, the horizontal gradient m, the vertical gradient q, and the intercepts p, s, t, and u can be determined. 
   A description has been given with reference to equations (25) through (33), by assuming that the pixel value of the pixel contained in the mixed area is M, and the pixel value of the pixel contained in the background area is B. In this case, it is necessary to set normal equations for each of the cases where the designated pixel is contained in the covered background area, or the designated pixel is contained in the uncovered background area. 
   For example, when the mixture ratio α of the pixel contained in the covered background area in frame #n shown in  FIG. 50  is determined, C 04  through C 08  of the pixels in frame #n and the pixel values P 04  through P 08  of the pixels in frame #n−1 are set in the normal equations. 
   For determining the mixture ratio α of the pixel contained in the uncovered background area in frame #n shown in  FIG. 51 , the pixels C 28  through C 32  of frame #n and the pixel values N 28  through N 32  of the pixels in frame #n+1 are set in the normal equations. 
   Moreover, if, for example, the mixture ratio α of the pixel contained in the covered background area shown in FIG.  54  is calculated, the following equations (40) through (48) are set. In  FIG. 54 , the white dots indicate pixels to belong to the background, and the black dots indicate pixels to belong to the mixed area. The pixel value of the pixel for which the mixture ratio α is calculated is Mc 5 .
 
 Mc 1=(−1)· Bc 1· m +(−1)· Bc 1· q+Bc 1· p +(−1)· s +(−1)· t+u   (40)
 
 Mc 2=(0)· Bc 2· m +(−1)· Bc 2· q+Bc 2· p +(0)· s +(−1)· t+u   (41)
 
 Mc 3=(+1)· Bc 3· m +(−1)· Bc 3· q+Bc 3· p +(+1)· s +(−1)· t+u   (42)
 
 Mc 4=(−1)· Bc 4· m +(0)· Bc 4· q+Bc 4· p +(−1)· s +(0)· t+u   (43)
 
 Mc 5=(0)· Bc 5· m +(0)· Bc 5· q+Bc 5· p +(0)· s +(0)· t+u   (44)
 
 Mc 6=(+1)· Bc 6· m +(0)· Bc 6· q+Bc 6· p +(+1)· s +(0)· t+u   (45)
 
 Mc 7=(−1)· Bc 7· m +(+1)· Bc 7· q+Bc 7· p +(−1)· s +(+1)· t+u   (46)
 
 Mc 8=(0)· Bc 8· m +(+1)· Bc 8· q+Bc 8· p +(0)· s +(+1)· t+u   (47)
 
 Mc 9=(+1)· Bc 9· m +(+1)· Bc 9· q+Bc 9· p +(+1)· s +(+1)· t+u   (48)
 
   When calculating the mixture ratio α of the pixel contained in the covered background area in frame #n, the pixel values Bc 1  through Bc 9  of the pixels in the background area in frame #n−1 corresponding to the pixels in frame #n are used in equations (40) through (48). 
   When calculating the mixture ratio α of the pixel contained in the uncovered background area shown in  FIG. 54 , the following equations (49) through (57) can hold true. The pixel value of the pixel for which the mixture ratio α is calculated is Mu 5 .
 
 Mu 1=(−1)· Bu 1· m +(−1)· Bu 1· q+Bu 1· p +(−1)· s +(−1)· t+u   (49)
 
 Mu 2=(0)· Bu 2· m +(−1)· Bu 2· q+Bu 2· p +(0)· s +(−1)· t+u   (50)
 
 Mu 3=(+1)· Bu 3· m +(−1)· Bu 3· q+Bu 3· p +(+1)· s +(−1)· t+u   (51)
 
 Mu 4=(−1)· Bu 4· m +(0)· Bu 4· q+Bu 4· p +(−1)· s +(0)· t+u   (52)
 
 Mu 5=(0)· Bu 5· m +(0)· Bu 5· q+Bu 5· p +(0)· s +(0)· t+u   (53)
 
 Mu 6=(+1)· Bu 6· m +(0)· Bu 6· q+Bu 6· p +(+1)· s +(0)· t+u   (54)
 
 Mu 7=(−1)· Bu 7· m +(+1)· Bu 7· q+Bu 7· p +(−1)· s +(+1)· t+u   (55)
 
 Mu 8=(0)· Bu 8· m +(+1)· Bu 8· q+Bu 8· p +(0)· s +(+1)· t+u   (56)
 
 Mu 9=(+1)· Bu 9· m +(+1)· Bu 9· q+Bu 9· p +(+1)· s +(+1)· t+u   (57)
 
   When calculating the mixture ratio α of the pixel contained in the uncovered background area in frame #n, the pixel values Bu 1  through Bu 9  of the pixels of the background area in frame #n+1 corresponding to the pixels of frame #n are used in equations (49) through (57). 
     FIG. 55  is a block diagram illustrating the configuration of the estimated-mixture-ratio processor  401 . An image input into the estimated-mixture-ratio processor  401  is supplied to a delay circuit  421  and an adder  422 . 
   The delay circuit  421  delays the input image for one frame, and supplies the image to the adder  422 . When frame #n is supplied as the input image to the adder  422 , the delay circuit  421  supplies frame #n−1 to the adder  422 . 
   The adder  422  sets the pixel value of the pixel adjacent to the pixel for which the mixture ratio α is calculated, and the pixel value of frame #n−1 in the normal equation. For example, the adder  422  sets the pixel values Mc 1  through Mc 9  and the pixel values Bc 1  through Bc 9  in the normal equations based on equations (40) through (48), respectively. The adder  422  supplies the normal equations in which the pixel values are set to a calculator  423 . 
   The calculator  423  determines the estimated mixture ratio by solving the normal equations supplied from the adder  422 , for example, by the sweep-out method, and outputs the determined estimated mixture ratio. 
   The calculator  423  calculates the motion v within the shutter time by equation (58) based on the gradient a of the mixture ratio α.
 
 a= 1/ v   (58)
 
   More specifically, the calculator  423  calculates the motion vix within the shutter time in the x direction and the motion viy within the shutter time in the y direction based on the gradient m of the plane of the mixture ratio α in the horizontal direction and the gradient q of the plane of the mixture ratio α in the vertical direction in equation (24).
 
 vix= 1/ m   (59)
 
 viy= 1/ q   (60)
 
   The calculator  423  outputs an estimated motion vector represented by the motion vix within the shutter time in the x direction and the motion viy within the shutter time in the y direction. 
   As shown in  FIG. 56 , the magnitude of the estimated motion vector output by the calculator  423  corresponds to the amount of movement v within the shutter time. 
   The amount of inter-frame movement vf is a value representing motion of an object between two adjacent frames. For example, if an object image corresponding to a foreground is moving such that it is displayed at a position eight pixels away from a reference frame when it is positioned in the subsequent frame, the amount of inter-frame movement vf of the object image corresponding to the foreground is 8. In  FIG. 56 , A denotes a background object. 
   In this manner, the estimated-mixture-ratio processor  401  is able to calculate the estimated mixture ratio and the estimated motion vector based on the input image, and supplies them to the mixture-ratio determining portion  403 . 
   The estimated-mixture-ratio processor  402  is configured similar to the estimated-mixture-ratio processor  401 , and an explanation thereof is thus omitted. 
     FIG. 57  is a diagram illustrating an example of the estimated mixture ratio calculated by the estimated-mixture-ratio processor  401 .  FIG. 57  shows an estimated mixture ratio for one line calculated in a case where the amount of movement v of a foreground corresponding to an object moving with constant velocity is 11 and equations are generated for units of 7×7 pixel block. 
   It is understood from  FIG. 57  that the estimated mixture ratio is changing substantially linearly in the mixed area. 
     FIG. 58  is a block diagram illustrating another configuration of the mixture-ratio calculator  104 . The same elements as those shown in  FIG. 48  are designated with like reference numerals, and an explanation thereof is thus omitted. 
   A selector  441  supplies a pixel belonging to the covered background area and the corresponding pixels in the previous and subsequent frames to the estimated-mixture-ratio processor  401  based on the area information supplied from the area specifying unit  103 . The selector  441  supplies a pixel belonging to the uncovered background area and the corresponding pixels in the previous and subsequent frames to the estimated-mixture-ratio processor  402  based on the area information supplied from the area specifying unit  103 . 
   The estimated-mixture-ratio processor  401  calculates an estimated mixture ratio of the designated pixel belonging to the covered background area based on the pixel value input from the selector  441 , and supplies the estimated mixture ratio to a selector  442 . The estimated-mixture-ratio processor  401  calculates an estimated motion vector based on the estimated mixture ratio, and supplies the estimated motion vector to the selector  442 . 
   The estimated-mixture-ratio processor  402  calculates an estimated mixture ratio of the designated pixel belonging to the uncovered background area based on the pixel value input from the selector  441 , and supplies the estimated mixture ratio to the selector  442 . The estimated-mixture-ratio processor  402  calculates an estimated motion vector based on the estimated mixture ratio, and supplies the estimated motion vector to the selector  442 . 
   Based on the area information supplied from the area specifying unit  103 , the selector  442  sets the mixture ratio α to 0 when the designated pixel belongs to the foreground area, and sets the mixture ratio α to 1 when the designated pixel belongs to the background area. When the designated pixel belongs to the covered background area, the selector  442  selects the estimated mixture ratio supplied from the estimated-mixture-ratio processor  442  and sets it as the mixture ratio α. When the designated pixel belongs to the uncovered background area, the selector  442  selects the estimated mixture ratio supplied from the estimated-mixture-ratio processor  443  and sets it as the mixture ratio α. The selector  442  then outputs the mixture ratio α which has been selected and set based on the area information. 
   Based on the area information supplied from the area specifying unit  103 , the selector  442  selects the estimated motion vector supplied from the estimated-mixture-ratio processor  401  and sets it as the motion vector within the shutter time when the designated pixel belongs to the covered background area. When the designated pixel belongs to the uncovered background area, the selector  442  selects the estimated motion vector supplied from the estimated-mixture-ratio processor  402  and sets it as the motion vector within the shutter time. The selector  442  then outputs the motion vector within the shutter time which has been selected and set based on the area information. 
   As discussed above, the mixture-ratio calculator  104  is able to calculate the mixture ratio α for each pixel contained in the image, and outputs the calculated mixture ratio α and motion vector within the shutter time. 
   The processing for calculating the mixture ratio α and the motion vector within the shutter time, performed by the mixture-ratio calculator  104 , is discussed below with reference to the flowchart of  FIG. 59 . In step S 401 , the mixture-ratio calculator  104  obtains area information supplied from the area specifying unit  103 . In step S 402 , the estimated-mixture-ratio processor  401  executes the processing for estimating the mixture ratio and the motion vector by using a model corresponding to a covered background area, and supplies the estimated mixture ratio and the estimated motion vector to the mixture-ratio determining portion  403 . Details of the processing for estimating the mixture ratio are discussed below with reference to the flowchart of  FIG. 60 . 
   In step S 403 , the estimated-mixture-ratio processor  402  executes the processing for estimating the mixture ratio and the motion vector by using a model corresponding to an uncovered background area, and supplies the estimated mixture ratio and the estimated motion vector to the mixture-ratio determining portion  403 . 
   In step S 404 , the mixture-ratio calculator  104  determines whether the mixture ratios have been estimated for the whole frame. If it is determined that the mixture ratios have not yet been estimated for the whole frame, the process returns to step S 402 , and the processing for estimating the mixture ratio for the subsequent pixel is executed. 
   If it is determined in step S 404  that the mixture ratios have been estimated for the whole frame, the process proceeds to step S 405 . In step S 405 , the mixture-ratio determining portion  403  determines the mixture ratio α and the motion vector within the shutter time based on the area information supplied from the area specifying unit  103  and indicating to which of the foreground area, the background area, the covered background area, or the uncovered background area the pixel for which the mixture ratio α is to be calculated belongs. The mixture-ratio determining portion  403  sets the mixture ratio α to 0 when the corresponding pixel belongs to the foreground area, and sets the mixture ratio α to 1 when the corresponding pixel belongs to the background area. When the corresponding pixel belongs to the covered background area, the mixture-ratio determining portion  403  sets the estimated mixture ratio supplied from the estimated-mixture-ratio processor  401  as the mixture ratio α. When the corresponding pixel belongs to the uncovered background area, the mixture-ratio determining portion  403  sets the estimated mixture ratio supplied from the estimated-mixture-ratio processor  402  as the mixture ratio α. 
   Based on the area information supplied from the area specifying unit  103 , the mixture-ratio determining portion  403  selects the estimated motion vector supplied from the estimated-mixture-ratio processor  401  and sets it as the motion vector within the shutter time if the designated pixel belongs to the covered background area, whereas the mixture-ratio determining portion  403  selects the estimated motion vector supplied from the estimated-mixture ratio processor  402  and sets it as the motion vector within the shutter time if the designated pixel belongs to the uncovered background area. The processing is then exited. 
   As discussed above, the mixture-ratio calculator  104  is able to calculate the mixture ratio α and the motion vector within the shutter time, which indicate feature quantities corresponding to each pixel, based on the area information supplied from the area specifying unit  103 , and the input image. 
   The processing for calculating the mixture ratio α performed by the mixture-ratio calculator  104  configured as shown in  FIG. 58  is similar to that discussed with reference to the flowchart of  FIG. 59 , and an explanation thereof is thus omitted. 
   A description is now given, with reference to the flowchart of  FIG. 60 , of the processing for estimating the mixture ratio and the motion vector by using a model of the covered background area, performed by the estimated-mixture-ratio processor  401 , which corresponds to the processing in step S 402 . 
   In step S 421 , the adder  422  sets the pixel value contained in the input image and the pixel value contained in the image supplied from the delay circuit  421  in a normal equation corresponding to a model of the covered background area. 
   In step S 422 , the estimated-mixture-ratio processor  401  determines whether the setting of the target pixels is finished. If it is determined that the setting of the target pixels is not finished, the process returns to step S 521 , and the processing for setting the pixel values in the normal equation is repeated. 
   If it is determined in step S 422  that the setting for the target pixels is finished, the process proceeds to step S 423 . In step S 423 , a calculator  423  calculates the estimated mixture ratio based on the normal equations in which the pixels values are set, and outputs the calculated mixture ratio. 
   In step S 424 , the calculator  423  calculates the estimated motion vector based on the gradient of the estimated mixture ratio, and the processing is then exited. 
   As discussed above, the estimated-mixture-ratio processor  401  having the configuration shown in  FIG. 55  is able to calculate the estimated mixture ratio and the estimated motion vector based on the input image. 
   The processing for estimating the mixture ratio and the motion vector by using a model corresponding to the uncovered background area is similar to the processing indicated by the flowchart of  FIG. 60  by using the normal equations corresponding to a model of the uncovered background area, and an explanation thereof is thus omitted. 
   The embodiment has been described, assuming that the object corresponding to the background is stationary. However, the above-described mixture-ratio calculation processing can be applied even if the image corresponding to the background area contains motion. For example, if the image corresponding to the background area is uniformly moving, the estimated-mixture-ratio processor  401  shifts the overall image in accordance with this motion, and performs processing in a manner similar to the case in which the object corresponding to the background is stationary. If the image corresponding to the background area contains locally different motions, the estimated-mixture-ratio processor  401  selects the pixels corresponding to the motions as the pixels belonging to the mixed area, and executes the above-described processing. 
   The mixture-ratio calculator  104  may execute the mixture-ratio estimating processing on all the pixels only by using a model corresponding to the covered background area, and outputs the calculated estimated mixture ratio as the mixture ratio α. In this case, the mixture ratio α indicates the ratio of the background components for the pixels belonging to the covered background area, and indicates the ratio of the foreground components for the pixels belonging to the uncovered background area. Concerning the pixels belonging to the uncovered background area, the absolute value of the difference between the calculated mixture ratio α and 1 is determined, and the calculated absolute value is set as the mixture ratio α. Then, the image processing apparatus is able to determine the mixture ratio α indicating the ratio of the background components for the pixels belonging to the uncovered background area. 
   Similarly, the mixture-ratio processor  104  may execute the mixture-ratio estimating processing on all the pixels only by using a model corresponding to the uncovered background area, and outputs the calculated estimated mixture ratio as the mixture ratio α. 
   As discussed above, the mixture-ratio calculator  104  is able to calculate the mixture ratio α and the motion vector, which indicate feature quantities corresponding to each pixel, based on the area information supplied from the area specifying unit  103 , and the input image. 
   By utilizing the mixture ratio α calculated by the mixture ratio calculator  104 , it is possible to separate the foreground components and the background components contained in the pixel values while maintaining the information of motion blur contained in the image corresponding to the moving object. 
   By combining the images based on the mixture ratio α, it is also possible to create an image which contains correct motion blur that coincides with the speed of a moving object and which faithfully reflects the real world. 
   The motion vector calculated by the mixture-ratio calculator  104  represents the amount of movement v within the shutter time, detection of which has not hitherto been allowed. 
   By utilizing the motion vector calculated by the mixture-ratio calculator  104 , adjustment of the amount of motion blur included in an image corresponding to a moving object is allowed. 
   The mixture-ratio calculator  104  may set an estimated motion vector corresponding to an estimated mixture ratio exceeding 0 and below 1 as the motion vector within the shutter time. In this case, the mixture-ratio calculator  104  is allowed to generate the motion vector within the shutter time without using area information. 
   The foreground/background separator  105  is discussed below.  FIG. 61  is a block diagram illustrating an example of the configuration of the foreground/background separator  105 . The input image supplied to the foreground/background separator  105  is supplied to a separating portion  601 , a switch  602 , and a switch  604 . The area information supplied from the area specifying unit  103  and indicating the information of the covered background area and the uncovered background area is supplied to the separating portion  601 . The area information indicating the foreground area is supplied to the switch  602 . The area information indicating the background area supplied to the switch  604 . 
   The mixture ratio α supplied from the mixture-ratio calculator  104  is supplied to the separating portion  601 . 
   The separating portion  601  separates the foreground components from the input image based on the area information indicating the covered background area, the area information indicating the uncovered background area, and the mixture ratio α, and supplies the separated foreground components to a synthesizer  603 . The separating portion  601  also separates the background components from the input image, and supplies the separated background components to a synthesizer  605 . 
   The switch  602  is closed when a pixel corresponding to the foreground is input based on the area information indicating the foreground area, and supplies only the pixels corresponding to the foreground contained in the input image to the synthesizer  603 . 
   The switch  604  is closed when a pixel corresponding to the background is input based on the area information indicating the background area, and supplies only the pixels corresponding to the background contained in the input image to the synthesizer  605 . 
   The synthesizer  603  synthesizes a foreground component image based on the foreground components supplied from the separating portion  601  and the pixels corresponding to the foreground supplied from the switch  602 , and outputs the synthesized foreground component image. Since the foreground area and the mixed area do not overlap, the synthesizer  603  applies, for example, logical OR to the foreground components and the foreground pixels, thereby synthesizing the foreground component image. 
   In the initializing processing executed at the start of the synthesizing processing for the foreground component image, the synthesizer  603  stores an image whose pixel values are all 0 in a built-in frame memory. Then, in the synthesizing processing for the foreground component image, the synthesizer  603  stores the foreground component image (overwrites the previous image by the foreground component image). Accordingly, 0 is stored in the pixels corresponding to the background area in the foreground component image output from the synthesizer  603 . 
   The synthesizer  605  synthesizes a background component image based on the background components supplied from the separating portion  601  and the pixels corresponding to the background supplied from the switch  604 , and outputs the synthesized background component image. Since the background area and the mixed area do not overlap, the synthesizer  605  applies, for example, logical OR to the background components and the background pixels, thereby synthesizing the background component image. 
   In the initializing processing executed at the start of the synthesizing processing for the background component image, the synthesizer  605  stores an image whose pixel values are all 0 in a built-in frame memory. Then, in the synthesizing processing for the background component image, the synthesizer  605  stores the background component image (overwrites the previous image by the background component image). Accordingly, 0 is stored in the pixels corresponding to the foreground area in the background component image output from the synthesizer  605 . 
     FIG. 62A  illustrates the input image input into the foreground/background separator  105  and the foreground component image and the background component image output from the foreground/background separator  105 .  FIG. 62B  illustrates a model corresponding to the input image input into the foreground/background separator  105  and the foreground component image and the background component image output from the foreground/background separator  105 . 
     FIG. 62A  is a schematic diagram illustrating the image to be displayed, and  FIG. 62B  is a model obtained by expanding in the time direction the pixels disposed in one line including the pixels belonging to the foreground area, the pixels belonging to the background area, and the pixels belonging to the mixed area corresponding to  FIG. 62A . 
   As shown in  FIGS. 62A and 62B , the background component image output from the foreground/background separator  105  consists of the pixels belonging to the background area and the background components contained in the pixels of the mixed area. 
   As shown in  FIGS. 62A and 62B , the foreground component image output from the foreground/background separator  105  consists of the pixel belonging to the foreground area and the foreground components contained in the pixels of the mixed area. 
   The pixel values of the pixels in the mixed area are separated into the background components and the foreground components by the foreground/background separator  105 . The separated background components form the background component image together with the pixels belonging to the background area. The separated foreground components form the foreground component image together with the pixels belonging to the foreground area. 
   As discussed above, in the foreground component image, the pixel values of the pixels corresponding to the background area are set to 0, and significant pixel values are set in the pixels corresponding to the foreground area and the pixels corresponding to the mixed area. Similarly, in the background component image, the pixel values of the pixels corresponding to the foreground area are set to 0, and significant pixel values are set in the pixels corresponding to the background area and the pixels corresponding to the mixed area. 
   A description is given below of the processing executed by the separating portion  601  for separating the foreground components and the background components from the pixels belonging to the mixed area, in the context of an example where the frame interval time is as long as the shutter time. 
     FIG. 63  illustrates a model of an image indicating foreground components and background components in two frames including a foreground object moving from the left to the right as viewed in the figure. In the model of the image shown in  FIG. 63 , the amount of movement v within the shutter time is 4, and the number of virtual divided portions is 4. 
   In frame #n, the leftmost pixel and the fourteenth through eighteenth pixels from the left consist of only the background components and belong to the background area. In frame #n, the second through fourth pixels from the left contain the background components and the foreground components, and belong to the uncovered background area. In frame #n, the eleventh through thirteenth pixels from the left contain background components and foreground components, and belong to the covered background area. In frame #n, the fifth through tenth pixels from the left consist of only the foreground components, and belong to the foreground area. 
   In frame #n+1, the first through fifth pixels from the left and the eighteenth pixel from the left consist of only the background components, and belong to the background area. In frame #n+1, the sixth through eighth pixels from the left contain background components and foreground components, and belong to the uncovered background area. In frame #n+1, the fifteenth through seventeenth pixels from the left contain background components and foreground components, and belong to the covered background area. In frame #n+1, the ninth through fourteenth pixels from the left consist of only the foreground components, and belong to the foreground area. 
     FIG. 64  illustrates the processing for separating the foreground components from the pixels belonging to the covered background area. In  FIG. 64 , α 1  through α 18  indicate mixture ratios of the individual pixels of frame #n. In  FIG. 64 , the fifteenth through seventeenth pixels from the left belong to the covered background area. 
   The pixel value C 15  of the fifteenth pixel from the left in frame #n can be expressed by equation (61): 
                       C15   =       ⁢       B15   /   v     +     F09   /   v     +     F08   /   v     +     F07   /   v                   =       ⁢       α15   ·   B15     +     F09   /   v     +     F08   /   v     +     F07   /   v                   =       ⁢       α15   ·   P15     +     F09   /   v     +     F08   /   v     +     F07   /   v                     (   61   )               
where α 15  indicates the mixture ratio of the fifteenth pixel from the left in frame #n, and P 15  designates the pixel value of the fifteenth pixel from the left in frame #n−1.
 
   The sum f 15  of the foreground components of the fifteenth pixel from the left in frame #n can be expressed by equation (62) based on equation (61). 
   
     
       
         
           
             
               
                 
                   
                     
                       f15 
                       = 
                         
                       ⁢ 
                       
                         
                           F09 
                           / 
                           v 
                         
                         + 
                         
                           F08 
                           / 
                           v 
                         
                         + 
                         
                           F07 
                           / 
                           v 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         C15 
                         - 
                         
                           α15 
                           · 
                           P15 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 62 
                 ) 
               
             
           
         
       
     
   
   Similarly, the sum f 16  of the foreground components of the sixteenth pixel from the left in frame #n can be expressed by equation (63), and the sum f 17  of the foreground components of the seventeenth pixel from the left in frame #n can be expressed by equation (64).
 
 f 16= C 16−α16· P 16  (63)
 
 f 17= C 17−α17· P 17  (64)
 
   In this manner, the foreground components fc contained in the pixel value C of the pixel belonging to the covered background area can be expressed by equation (65):
 
 fc=C−α·P   (65)
 
where P designates the pixel value of the corresponding pixel in the previous frame.
 
     FIG. 65  illustrates the processing for separating the foreground components from the pixels belonging to the uncovered background area. In  FIG. 65 , α 1  through α 18  indicate mixture ratios of the individual pixels of frame #n. In  FIG. 65 , the second through fourth pixels from the left belong to the uncovered background area. 
   The pixel value C 02  of the second pixel from the left in frame #n can be expressed by equation (66): 
                       C02   =       ⁢       B02   /   v     +     B02   /   v     +     B02   /   v     +     F01   /   v                   =       ⁢       α2   ·   B02     +     F01   /   v                   =       ⁢       α2   ·   N02     +     F01   /   v                     (   66   )               
where α 2  indicates the mixture ratio of the second pixel from the left in frame #n, and N 02  designates the pixel value of the second pixel from the left in frame #n+1.
 
   The sum f 02  of the foreground components of the second pixel from the left in frame #n can be expressed by equation (67) based on equation (66). 
   
     
       
         
           
             
               
                 
                   
                     
                       f02 
                       = 
                         
                       ⁢ 
                       
                         F01 
                         / 
                         v 
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         C02 
                         - 
                         
                           α2 
                           · 
                           N02 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 67 
                 ) 
               
             
           
         
       
     
   
   Similarly, the sum f 03  of the foreground components of the third pixel from the left in frame #n can be expressed by equation (68), and the sum f 04  of the foreground components of the fourth pixel from the left in frame #n can be expressed by equation (69).
 
 f 03= C 03−α3· N 03  (68)
 
 f 04= C 04−α4· N 04  (69)
 
   In this manner, the foreground components fu contained in the pixel value C of the pixel belonging to the uncovered background area can be expressed by equation (70):
 
 fu=C−α·N   (70)
 
where N designates the pixel value of the corresponding pixel in the subsequent frame.
 
   As discussed above, the separating portion  601  is able to separate the foreground components from the pixels belonging to the mixed area and the background components from the pixels belonging to the mixed area based on the information indicating the covered background area and the information indicating the uncovered background area contained in the area information, and the mixture ratio α for each pixel. 
     FIG. 66  is a block diagram illustrating an example of the configuration of the separating portion  601  for executing the above-described processing. An image input into the separating portion  601  is supplied to a frame memory  621 , and the area information indicating the covered background area and the uncovered background area supplied from the mixture-ratio calculator  104  and the mixture ratio α are supplied to a separation processing block  622 . 
   The frame memory  621  stores the input images in units of frames. When a frame to be processed is frame #n, the frame memory  621  stores frame #n−1, which is the frame one frame before frame #n, frame #n, and frame #n+1, which is the frame one frame after frame #n. 
   The frame memory  621  supplies the corresponding pixels in frame #n−1, frame #n, and frame #n+1 to the separation processing block  622 . 
   The separation processing block  622  applies the calculations discussed with reference to  FIGS. 64 and 65  to the pixel values of the corresponding pixels in frame #n−1, frame #n, and frame #n+1 supplied from the frame memory  621  based on the area information indicating the covered background area and the uncovered background area and the mixture ratio α so as to separate the foreground components and the background components from the pixels belonging to the mixed area in frame #n, and supplies them to a frame memory  623 . 
   The separation processing block  622  is formed of an uncovered area processor  631 , a covered area processor  632 , a synthesizer  633 , and a synthesizer  634 . 
   A multiplier  641  of the uncovered area processor  631  multiplies the pixel value of the pixel in frame #n+1 supplied from the frame memory  621  by the mixture ratio α, and outputs the resulting pixel value to a switch  642 . The switch  642  is closed when the pixel of frame #n (corresponding to the pixel in frame #n+1) supplied from the frame memory  621  belongs to the uncovered background area, and supplies the pixel value multiplied by the mixture ratio α supplied from the multiplier  641  to a calculator  643  and the synthesizer  634 . The value obtained by multiplying the pixel value of the pixel in frame #n+1 by the mixture ratio α output from the switch  642  is equivalent to the background components of the pixel value of the corresponding pixel in frame #n. 
   The calculator  643  subtracts the background components supplied from the switch  642  from the pixel value of the pixel in frame #n supplied from the frame memory  621  so as to obtain the foreground components. The calculator  643  supplies the foreground components of the pixel in frame #n belonging to the uncovered background area to the synthesizer  633 . 
   A multiplier  651  of the covered area processor  632  multiplies the pixel value of the pixel in frame #n−1 supplied from the frame memory  621  by the mixture ratio α, and outputs the resulting pixel value to a switch  652 . The switch  652  is closed when the pixel of frame #n (corresponding to the pixel in frame #n−1) supplied from the frame memory  621  belongs to the covered background area, and supplies the pixel value multiplied by the mixture ratio α supplied from the multiplier  651  to a calculator  653  and the synthesizer  634 . The value obtained by multiplying the pixel value of the pixel in frame #n−1 by the mixture ratio α output from the switch  652  is equivalent to the background components of the pixel value of the corresponding pixel in frame #n. 
   The calculator  653  subtracts the background components supplied from the switch  652  from the pixel value of the pixel in frame #n supplied from the frame memory  621  so as to obtain the foreground components. The calculator  653  supplies the foreground components of the pixel in frame #n belonging to the covered background area to the synthesizer  633 . 
   The synthesizer  633  combines the foreground components of the pixels belonging to the uncovered background area and supplied from the calculator  643  with the foreground components of the pixels belonging to the covered background area and supplied from the calculator  653 , and supplies the synthesized foreground components to the frame memory  623 . 
   The synthesizer  634  combines the background components of the pixels belonging to the uncovered background area and supplied from the switch  642  with the background components of the pixels belonging to the covered background area and supplied from the switch  652 , and supplies the synthesized background components to the frame memory  623 . 
   The frame memory  623  stores the foreground components and the background components of the pixels in the mixed area of frame #n supplied from the separation processing block  622 . 
   The frame memory  623  outputs the stored foreground components of the pixels in the mixed area in frame #n and the stored background components of the pixels in the mixed area in frame #n. 
   By utilizing the mixture ratio α, which indicates the feature quantity, the foreground components and the background components contained in the pixel values can be completely separated. 
   The synthesizer  603  combines the foreground components of the pixels in the mixed area in frame #n output from the separating portion  601  with the pixels belonging to the foreground area so as to generate a foreground component image. The synthesizer  605  combines the background components of the pixels in the mixed area in frame #n output from the separating portion  601  with the pixels belonging to the background area so as to generate a background component image. 
     FIG. 67A  illustrates an example of the foreground component image corresponding to frame #n in  FIG. 63 . The leftmost pixel and the fourteenth pixel from the left consist of only the background components before the foreground and the background are separated, and thus, the pixel values are set to 0. 
   The second through fourth pixels from the left belong to the uncovered background area before the foreground and the background are separated. Accordingly, the background components are set to 0, and the foreground components are maintained. The eleventh through thirteenth pixels from the left belong to the covered background area before the foreground and the background are separated. Accordingly, the background components are set to 0, and the foreground components are maintained. The fifth through tenth pixels from the left consist of only the foreground components, which are thus maintained. 
     FIG. 67B  illustrates an example of the background component image corresponding to frame #n in  FIG. 63 . The leftmost pixel and the fourteenth pixel from the left consist of only the background components before the foreground and the background are separated, and thus, the background components are maintained. 
   The second through fourth pixels from the left belong to the uncovered background area before the foreground and the background are separated. Accordingly, the foreground components are set to 0, and the background components are maintained. The eleventh through thirteenth pixels from the left belong to the covered background area before the foreground and the background are separated. Accordingly, the foreground components are set to 0, and the background components are maintained. The fifth through tenth pixels from the left consist of only the foreground components, and thus, the pixel values are set to 0. 
   The processing for separating the foreground and the background executed by the foreground/background separator  105  is described below with reference to the flowchart of  FIG. 68 . In step S 601 , the frame memory  621  of the separating portion  601  obtains an input image, and stores frame #n for which the foreground and the background are separated together with the previous frame #n−1 and the subsequent frame #n+1. 
   In step S 602 , the separation processing block  622  of the separating portion  601  obtains area information supplied from the mixture-ratio calculator  104 . In step S 603 , the separation processing block  622  of the separating portion  601  obtains the mixture ratio α supplied from the mixture-ratio calculator  104 . 
   In step S 604 , the uncovered area processor  631  extracts the background components from the pixel values of the pixels belonging to the uncovered background area supplied from the frame memory  621  based on the area information and the mixture ratio α. 
   In step S 605 , the uncovered area processor  631  extracts the foreground components from the pixel values of the pixels belonging to the uncovered background area supplied from the frame memory  621  based on the area information and the mixture ratio α. 
   In step S 606 , the covered area processor  632  extracts the background components from the pixel values of the pixels belonging to the covered background area supplied from the frame memory  621  based on the area information and the mixture ratio α. 
   In step S 607 , the covered area processor  632  extracts the foreground components from the pixel values of the pixels belonging to the covered background area supplied from the frame memory  621  based on the area information and the mixture ratio α. 
   In step S 608 , the synthesizer  633  combines the foreground components of the pixels belonging to the uncovered background area extracted in the processing of step S 605  with the foreground components of the pixels belonging to the covered background area extracted in the processing of step S 607 . The synthesized foreground components are supplied to the synthesizer  603 . The synthesizer  603  further combines the pixels belonging to the foreground area supplied via the switch  602  with the foreground components supplied from the separating portion  601  so as to generate a foreground component image. 
   In step S 609 , the synthesizer  634  combines the background components of the pixels belonging to the uncovered background area extracted in the processing of step S 604  with the background components of the pixels belonging to the covered background area extracted in the processing of step S 606 . The synthesized background components are supplied to the synthesizer  605 . The synthesizer  605  further combines the pixels belonging to the background area supplied via the switch  604  with the background components supplied from the separating portion  601  so as to generate a background component image. 
   In step S 610 , the synthesizer  603  outputs the foreground component image. In step S 611 , the synthesizer  605  outputs the background component image. The processing is then completed. 
   As discussed above, the foreground/background separator  105  is able to separate the foreground components and the background components from the input image based on the area information and the mixture ratio α, and outputs the foreground component image consisting of only the foreground components and the background component image consisting of only the background components. 
   Next, calculation of the shutter time by the shutter-time calculator  106  will be described. 
   The shutter-time calculator  106  calculates the shutter time based on the inter-frame motion vfx in the x direction and the inter-frame motion vfy in the y direction included in the inter-frame motion vector supplied from the motion detector  102  and the motion vix within the shutter time in the x direction and the motion viy within the shutter time in the y direction included in the motion vector within the shutter time supplied from the mixture-ratio calculator  104 . 
   As shown in  FIG. 56 , since it can be assumed that the foreground object moves with constant velocity, the ratio of the amount of movement within the shutter time to the amount of inter-frame movement is equal to the ratio of the shutter time to the frame interval time. 
   The shutter-time calculator  106  calculates a ratio S 1  of the shutter time to the frame interval time based on the motion vix within the shutter time in the x direction included in the motion within the shutter time and the inter-frame motion vfx in the x direction included in the inter-frame motion vector, for example, by equation (71).
 
 S 1= vix/vfx   (71)
 
   For example, if the motion vix within the shutter time in the x direction is 5 and if the inter-frame motion vfx in the x direction is 10, the shutter-time calculator  106  calculates the ratio S 1  of the shutter time to the frame interval time as 0.5. 
   Alternatively, the shutter-time calculator  106  calculates a ratio S 2  of the shutter time to the frame interval time based on the inter-frame motion vfx in the x direction and the inter-frame motion vfy in the y direction included in the inter-frame motion vector and the motion vix within the shutter time in the x direction and the motion viy within the shutter time in the y direction included in the motion vector within the shutter time, for example, by equation (72).
 
 S 2=(( vix/vfx )+( viy/vfy ))/2  (72)
 
   The shutter-time calculator  106  calculates the ratio S 2  of the shutter time to the frame interval time based on the inter-frame motion vfx in the x direction and the inter-frame motion vfy in the y direction included in the inter-frame motion vector and the motion vix within the shutter time in the x direction and the motion viy within the shutter time in the y direction included in the motion vector within the shutter time, for example, by equation (73). 
   
     
       
         
           
             
               
                 S3 
                 = 
                 
                   
                     
                       
                         vix 
                         2 
                       
                       + 
                       
                         viy 
                         2 
                       
                     
                   
                   
                     
                       
                         vfx 
                         2 
                       
                       + 
                       
                         vfy 
                         2 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 73 
                 ) 
               
             
           
         
       
     
   
   The shutter-time calculator  106  is able to calculate the ratio of the shutter time to the frame interval time more simply by equation (71) than by equation (72) or equation (73). 
   The shutter-time calculator is able to calculate the ratio of the shutter time to the frame interval time more precisely by equation (72) or equation (73) than by equation (71). 
   The shutter-time calculator  106  calculates the shutter time based on the frame interval time and the calculated ratio of the shutter time to the frame interval time. 
   As described above, the shutter-time calculator  106  is able to calculate the shutter time based on the inter-frame motion vfx in the x direction and the inter-frame motion vfy in the y direction included in the inter-frame motion vector supplied from the motion detector  102  and the motion vix within the shutter time in the x direction and the motion viy within the shutter time in the y direction included in the motion vector within the shutter time supplied from the mixture-ratio calculator  104 . 
   Adjustments of the amount of motion blur in a foreground component image are described below. 
     FIG. 69  is a block diagram illustrating an example of the configuration of the motion-blur adjusting unit  107 . The motion vector within the shutter time and the positional information thereof supplied from the mixture-ratio calculator  104  are supplied to a unit-of-processing determining portion  801 , a model-forming portion  802 , and a calculator  805 . The area information supplied from the area specifying unit  103  is supplied to the unit-of-processing determining portion  801 . The foreground component image supplied from the foreground/background separator  105  is supplied to an adder  804 . 
   The unit-of-processing determining portion  801  supplies the unit of processing that is generated based on the motion vector within the shutter time and the positional information thereof and the area information to the model-forming portion  802 . The unit-of-processing determining portion  801  supplies the generated unit of processing to the adder  804 . 
   As indicated by A in an example shown in  FIG. 70 , the unit of processing generated by the unit-of-processing determining portion  801  indicates consecutive pixels disposed in the moving direction starting from the pixel corresponding to the covered background area of the foreground component image until the pixel corresponding to the uncovered background area, or indicates consecutive pixels disposed in the moving direction starting from the pixel corresponding to the uncovered background area until the pixel corresponding to the covered background area. The unit of processing is formed of two pieces of data which indicate, for example, the upper left point (which is the position of the leftmost or the topmost pixel in the image designated by the unit of processing) and the lower right point. 
   The model-forming portion  802  forms a model based on the motion vector within the shutter time and the unit of processing. More specifically, for example, the model-forming portion  802  stores in advance a plurality of models in accordance with the number of pixels contained in the unit of processing, the number of virtual divided portions of the pixel value in the time direction, and the number of foreground components for each pixel. The model-forming portion  902  then selects the model in which the correlation between the pixel values and the foreground components is designated, such as that in  FIG. 71 , based on the unit of processing and the number of virtual divided portions of the pixel value in the time direction. 
   It is now assumed, for example, that the number of pixels corresponding to the unit of processing is 12, and that the amount of movement v within the shutter time is 5. Then, the model-forming portion  802  sets the number of virtual divided portions to 5, and selects a model formed of eight types of foreground components so that the leftmost pixel contains one foreground component, the second pixel from the left contains two foreground components, the third pixel from the left contains three foreground components, the fourth pixel from the left contains four pixel components, the fifth pixel from the left contains five foreground components, the sixth pixel from the left contains five foreground components, the seventh pixel from the left contains five foreground components, the eighth pixel from the left contains five foreground components, the ninth pixel from the left contains four foreground components, the tenth pixel from the left contains three foreground components, the eleventh pixel from the left contains two foreground components, and the twelfth pixel from the left contains one foreground component. 
   Instead of selecting a model from the prestored models, the model-forming portion  802  may generate a model based on the motion vector within the shutter time and the unit of processing when the motion vector within the shutter time and the unit of processing are supplied. 
   The model-forming portion  802  supplies the selected model to an equation generator  803 . 
   The equation generator  803  generates an equation based on the model supplied from the model-forming portion  802 . A description is given below, with reference to the model of the foreground component image shown in  FIG. 71 , of equations generated by the equation generator  803  when the number of foreground components is 8, the number of pixels corresponding to the unit of processing is 12, the amount of movement v within the shutter time is 5, and the number of virtual divided portions is 5. 
   When the foreground components contained in the foreground component image corresponding to the shutter time/v are F 01 /v through F 08 /v, the relationships between F 01 /v through F 08 /v and the pixel values C 01  through C 12  can be expressed by equations (74) through (85).
 
 C 01= F 01/ v   (74)
 
 C 02= F 02/ v+F 01/ v   (75)
 
 C 03= F 03/ v+F 02/ v+F 01 v   (76)
 
 C 04= F 04/ v+F 03/ v+F 02/ v+F 01 v   (77)
 
 C 05= F 05/ v+F 04/ v+F 03/ v+F 02/ v+F 01 v   (78)
 
 C 06= F 06/ v+F 05/ v+F 04/ v+F 03/ v+F 02/ v   (79)
 
 C 07= F 07/ v+F 06/ v+F 05/ v+F 04/ v+F 03/ v   (80)
 
 C 08= F 08/ v+F 07/ v+F 06/ v+F 05/ v+F 04/ v   (81)
 
 C 09= F 08/ v+F 07/ v+F 06/ v+F 05/ v   (82)
 
 C 10= F 08/ v+F 07/ v+F 06/ v   (83)
 
 C 11= F 08/ v+F 07/ v   (84)
 
 C 12= F 08/ v   (85)
 
   The equation generator  803  generates an equation by modifying the generated equations. The equations generated by the equation generator  803  are indicated by equations (86) though (97).
 
 C 01=1· F 01/ v+ 0· F 02/ v+ 0· F 03/ v+ 0· F 04/ v+ 0· F 05/ v+ 0· F 06/ v+ 0· F 07/ v+ 0· F 08/ v   (86)
 
 C 02=1· F 01/ v+ 1· F 02/ v+ 0· F 03/ v+ 0· F 04/ v+ 0· F 05/ v+ 0· F 06/ v+ 0· F 07/ v+ 0· F 08/ v   (87)
 
 C 03=1· F 01/ v+ 1· F 02/ v+ 1· F 03/ v+ 0· F 04/ v+ 0· F 05/ v+ 0· F 06/ v+ 0· F 07/ v+ 0· F 08/ v   (88)
 
 C 04=1· F 01/ v+ 1· F 02/ v+ 1· F 03/ v+ 1· F 04/ v+ 0· F 05/ v+ 0· F 06/ v+ 0· F 07/ v+ 0· F 08/ v   (89)
 
 C 05=1· F 01/ v+ 1· F 02/ v+ 1· F 03/ v+ 1· F 04/ v+ 1· F 05/ v+ 0· F 06/ v+ 0· F 07/ v+ 0· F 08/ v   (90)
 
 C 06=0· F 01/ v+ 1· F 02/ v+ 1· F 03/ v+ 1· F 04/ v+ 1· F 05/ v+ 1· F 06/ v+ 0· F 07/ v+ 0· F 08/ v   (91)
 
 C 07=0· F 01/ v+ 0· F 02/ v+ 1· F 03/ v+ 1· F 04/ v+ 1· F 05/ v+ 1· F 06/ v+ 1· F 07/ v+ 0· F 08/ v   (92)
 
 C 08=0· F 01/ v+ 0· F 02/ v+ 0· F 03/ v+ 1· F 04/ v+ 1· F 05/ v+ 1· F 06/ v+ 1· F 07/ v+ 1· F 08/ v   (93)
 
 C 09=0· F 01/ v+ 0· F 02/ v+ 0· F 03/ v+ 0· F 04/ v+ 1· F 05/ v+ 1· F 06/ v+ 1· F 07/ v+ 1· F 08/ v   (94)
 
 C 10=0· F 01/ v+ 0· F 02/ v+ 0· F 03/ v+ 0· F 04/ v+ 0· F 05/ v+ 1· F 06/ v+ 1· F 07/ v+ 1· F 08/ v   (95)
 
 C 11=0· F 01/ v+ 0· F 02/ v+ 0· F 03/ v+ 0· F 04/ v+ 0· F 05/ v+ 0· F 06/ v+ 1· F 07/ v+ 1· F 08/ v   (96)
 
 C 12=0· F 01/ v+ 0· F 02/ v+ 0· F 03/ v+ 0· F 04/ v+ 0· F 05/ v+ 0· F 06/ v+ 0· F 07/ v+ 1· F 08/ v   (97)
 
   Equations (86) through (97) can be expressed by equation (98). 
   
     
       
         
           
             
               
                 Cj 
                 = 
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       01 
                     
                     08 
                   
                   ⁢ 
                   
                     aij 
                     · 
                     
                       Fi 
                       / 
                       v 
                     
                   
                 
               
             
             
               
                 ( 
                 98 
                 ) 
               
             
           
         
       
     
   
   In equation (98), j designates the position of the pixel. In this example, j has one of the values from 1 to 12. In equation (98), i designates the position of the foreground value. In this example, i has one of the values from 1 to 8. In equation (98), aij has the value 0 or 1 according to the values of i and j. 
   Equation (98) can be expressed by equation (99) in consideration of the error. 
   
     
       
         
           
             
               
                 Cj 
                 = 
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         01 
                       
                       08 
                     
                     ⁢ 
                     
                       aij 
                       · 
                       
                         Fi 
                         / 
                         v 
                       
                     
                   
                   + 
                   ej 
                 
               
             
             
               
                 ( 
                 99 
                 ) 
               
             
           
         
       
     
   
   In equation (99), ej designates the error contained in the designated pixel Cj. 
   Equation (99) can be modified into equation (100). 
   
     
       
         
           
             
               
                 ej 
                 = 
                 
                   Cj 
                   - 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         01 
                       
                       08 
                     
                     ⁢ 
                     
                       aij 
                       · 
                       
                         Fi 
                         / 
                         v 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 100 
                 ) 
               
             
           
         
       
     
   
   In order to apply the method of least squares, the square sum E of the error is defined as equation (101). 
   
     
       
         
           
             
               
                 E 
                 = 
                 
                   
                     ∑ 
                     
                       j 
                       = 
                       01 
                     
                     12 
                   
                   ⁢ 
                   
                     ej 
                     2 
                   
                 
               
             
             
               
                 ( 
                 101 
                 ) 
               
             
           
         
       
     
   
   In order to minimize the error, the partial differential value using the variable Fk with respect to the square sum E of the error should be 0. Fk is determined so that equation (102) is satisfied. 
   
     
       
         
           
             
               
                 
                   
                     ∂ 
                     E 
                   
                   
                     ∂ 
                     Fk 
                   
                 
                 = 
                 
                   
                     2 
                     · 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           01 
                         
                         12 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ej 
                         · 
                         
                           
                             ∂ 
                             ej 
                           
                           
                             ∂ 
                             Fk 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   = 
                   
                     2 
                     · 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           01 
                         
                         12 
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 Cj 
                                 - 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       01 
                                     
                                     08 
                                   
                                   ⁢ 
                                   
                                     aij 
                                     · 
                                     
                                       Fi 
                                       / 
                                       v 
                                     
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               ( 
                               
                                 
                                   - 
                                   akj 
                                 
                                 / 
                                 v 
                               
                               ) 
                             
                           
                           = 
                           0 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 102 
                 ) 
               
             
           
         
       
     
   
   In equation (102), since the amount of movement v within the shutter time is a fixed value, equation (103) can be deduced. 
   
     
       
         
           
             
               
                 
                   
                     ∑ 
                     
                       j 
                       = 
                       01 
                     
                     12 
                   
                   ⁢ 
                   
                     akj 
                     · 
                     
                       ( 
                       
                         Cj 
                         - 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               01 
                             
                             08 
                           
                           ⁢ 
                           
                             aij 
                             · 
                             
                               Fi 
                               / 
                               v 
                             
                           
                         
                       
                       ) 
                     
                   
                 
                 = 
                 0 
               
             
             
               
                 ( 
                 103 
                 ) 
               
             
           
         
       
     
   
   To expand equation (103) and transpose the terms, equation (104) can be obtained. 
   
     
       
         
           
             
               
                 
                   
                     ∑ 
                     
                       j 
                       = 
                       01 
                     
                     12 
                   
                   ⁢ 
                   
                     ( 
                     
                       akj 
                       · 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             01 
                           
                           08 
                         
                         ⁢ 
                         
                           aij 
                           · 
                           Fi 
                         
                       
                     
                     ) 
                   
                 
                 = 
                 
                   v 
                   ⁢ 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         01 
                       
                       12 
                     
                     ⁢ 
                     
                       akj 
                       · 
                       Cj 
                     
                   
                 
               
             
             
               
                 ( 
                 104 
                 ) 
               
             
           
         
       
     
   
   Equation (104) is expanded into eight equations by substituting the individual integers from 1 to 8 into k in equation (104). The obtained eight equations can be expressed by one matrix equation. This equation is referred to as a “normal equation”. 
   An example of the normal equation generated by the equation generator  803  based on the method of least squares is indicated by equation (105). 
   
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           5 
                         
                         
                           4 
                         
                         
                           3 
                         
                         
                           2 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           4 
                         
                         
                           5 
                         
                         
                           4 
                         
                         
                           3 
                         
                         
                           2 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           3 
                         
                         
                           4 
                         
                         
                           5 
                         
                         
                           4 
                         
                         
                           3 
                         
                         
                           2 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           2 
                         
                         
                           3 
                         
                         
                           4 
                         
                         
                           5 
                         
                         
                           4 
                         
                         
                           3 
                         
                         
                           2 
                         
                         
                           1 
                         
                       
                       
                         
                           1 
                         
                         
                           2 
                         
                         
                           3 
                         
                         
                           4 
                         
                         
                           5 
                         
                         
                           4 
                         
                         
                           3 
                         
                         
                           2 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           2 
                         
                         
                           3 
                         
                         
                           4 
                         
                         
                           5 
                         
                         
                           4 
                         
                         
                           3 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           2 
                         
                         
                           3 
                         
                         
                           4 
                         
                         
                           5 
                         
                         
                           4 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           2 
                         
                         
                           3 
                         
                         
                           4 
                         
                         
                           5 
                         
                       
                     
                     ] 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           F01 
                         
                       
                       
                         
                           F02 
                         
                       
                       
                         
                           F03 
                         
                       
                       
                         
                           F04 
                         
                       
                       
                         
                           F05 
                         
                       
                       
                         
                           F06 
                         
                       
                       
                         
                           F07 
                         
                       
                       
                         
                           F08 
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   v 
                   · 
                   
                     [ 
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 08 
                               
                               12 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 07 
                               
                               11 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 06 
                               
                               10 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 05 
                               
                               09 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 04 
                               
                               08 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 03 
                               
                               07 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 02 
                               
                               06 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 01 
                               
                               05 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 ( 
                 105 
                 ) 
               
             
           
         
       
     
   
   When equation (105) is expressed by A·F=v·C, C, A, and v are known, and F is unknown. A and v are known when the model is formed, while C becomes known when the pixel value is input in the addition processing. 
   By calculating the foreground components according to the normal equation based on the method of least squares, the error contained in the pixel C can be distributed. 
   The equation generator  803  supplies the normal equation generated as discussed above to the adder  804 . 
   The adder  804  sets, based on the unit of processing supplied from the unit-of-processing determining portion  801 , the pixel value C contained in the foreground component image in the matrix equation supplied from the equation generator  803 . The adder  804  supplies the matrix in which the pixel value C is set to a calculator  805 . 
   The calculator  805  calculates the foreground component Fi/v from which motion blur is eliminated by the processing based on a solution, such as a sweep-out method (Gauss-Jordan elimination), so as to obtain Fi corresponding to i indicating one of the integers from 1 to 8, which is the pixel value from which motion blur is eliminated. The calculator  805  then outputs the foreground component image consisting of the pixel values Fi without motion blur, such as that in  FIG. 72 , to a motion-blur adder  806  and a selector  807 . 
   In the foreground component image without motion blur shown in  FIG. 72 , the reason for setting F 01  through F 08  in C 03  through C 10 , respectively, is not to change the position of the foreground component image with respect to the screen. However, F 01  through F 08  may be set in any desired positions. 
   The motion-blur adder  806  generates an amount v′ by which motion blur is adjusted based on the shutter time supplied from the shutter-time calculator  106 . For example, the motion-blur adder  806  divides the prestored frame interval time by the shutter time supplied from the shutter-time calculator  106  and multiplies the motion within the shutter time supplied from the mixture-ratio calculator  104  by the result of the division, thereby generating the amount v′ by which motion blur is adjusted. 
   The motion-blur adder  806  is able to adjust the amount of motion blur by adding the amount v′ by which motion blur is adjusted, which is different from the amount of movement v within the shutter time, for example, the amount v′ by which motion blur is adjusted, which is one half the value of the amount of movement v within the shutter time, or the amount v′ by which motion blur is adjusted, which is irrelevant to the amount of movement v within the shutter time. For example, as shown in  FIG. 73 , the motion-blur adder  806  divides the foreground pixel value Fi without motion blur by the amount v′ by which motion blur is adjusted so as to obtain the foreground component Fi/v′. The motion-blur adder  806  then calculates the sum of the foreground components Fi/v′, thereby generating the pixel value in which the amount of motion blur is adjusted. For example, when the amount v′ by which motion blur is adjusted is 3, the pixel value C 02  is set to (F 01 )/v′, the pixel value C 03  is set to (F 0 +F 02 )/v′, the pixel value C 04  is set to (F 01 +F 02 +F 03 )/v′, and the pixel value C 05  is set to (F 02 +F 03 +F 04 )/v′. 
   The motion-blur adder  806  supplies the foreground component image in which the amount of motion blur is adjusted to a selector  807 . 
   The selector  807  selects one of the foreground component image without motion blur supplied from the calculator  805  and the foreground component image in which the amount of motion blur is adjusted supplied from the motion-blur adder  806  based on a selection signal reflecting a user&#39;s selection, and outputs the selected foreground component image. 
   As discussed above, the motion-blur adjusting unit  107  is able to adjust the amount of motion blur based on the selection signal and the amount v′ by which motion blur is adjusted. 
   Also, for example, when the number of pixels corresponding to the unit of processing is 8, and the amount of movement v within the shutter time is 4, as shown in  FIG. 74 , the motion-blur adjusting unit  107  generates a matrix equation expressed by equation (106). 
   
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           4 
                         
                         
                           3 
                         
                         
                           2 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           3 
                         
                         
                           4 
                         
                         
                           3 
                         
                         
                           2 
                         
                         
                           1 
                         
                       
                       
                         
                           2 
                         
                         
                           3 
                         
                         
                           4 
                         
                         
                           3 
                         
                         
                           2 
                         
                       
                       
                         
                           1 
                         
                         
                           2 
                         
                         
                           3 
                         
                         
                           4 
                         
                         
                           3 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           2 
                         
                         
                           3 
                         
                         
                           4 
                         
                       
                     
                     ] 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           F01 
                         
                       
                       
                         
                           F02 
                         
                       
                       
                         
                           F03 
                         
                       
                       
                         
                           F04 
                         
                       
                       
                         
                           F05 
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   v 
                   · 
                   
                     [ 
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 05 
                               
                               08 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 04 
                               
                               07 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 03 
                               
                               06 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 02 
                               
                               05 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 01 
                               
                               04 
                             
                             ⁢ 
                             Ci 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 ( 
                 106 
                 ) 
               
             
           
         
       
     
   
   In this manner, the motion-blur adjusting unit  107  calculates Fi, which is the pixel value in which the amount of motion blur is adjusted, by setting up the equation in accordance with the length of the unit of processing. Similarly, for example, when the number of pixels contained in the unit of processing is 100, the equation corresponding to 100 pixels is generated so as to calculate Fi. 
     FIG. 75  illustrates an example of another configuration of the motion-blur adjusting unit  107 . The same elements as those shown in  FIG. 69  are designated with like reference numerals, and an explanation thereof is thus omitted. 
   Based on the shutter time supplied from the shutter-time calculator  106 , a selector  821  generates the amount v′ by which motion blur is adjusted. 
   Based on a selection signal, the selector  821  directly supplies an input motion vector within the shutter time and a positional signal thereof to the unit-of-processing determining portion  801  and the model-forming portion  802 . Alternatively, the selector  821  may substitute the magnitude of the motion vector within the shutter time by the amount v′ by which motion blur is adjusted, and then supplies the motion vector and the positional signal thereof to the unit-of-processing determining portion  801  and the model-forming unit  802 . 
   With this arrangement, the unit-of-processing determining portion  801  through the calculator  805  of the motion-blur adjusting unit  107  shown in  FIG. 75  are able to adjust the amount of motion blur in accordance with the amount of movement v and the amount v′ by which motion blur is adjusted. For example, when the amount of movement v within the shutter time is 5, and the amount v′ by which motion blur is adjusted is 3, the unit-of-processing determining portion  801  through the calculator  805  of the motion-blur adjusting unit  107  shown in  FIG. 75  execute computation on the foreground component image in which the amount of movement v within the shutter time is 5 shown in  FIG. 71  according to the model shown in  FIG. 73  in which the amount v′ by which motion blur is adjusted is 3. As a result, the image containing motion blur having the amount of movement v within the shutter time of (amount of movement v within the shutter time)/(amount v′ by which motion blur is adjusted)=5/3, i.e., about 1.7 is obtained. In this case, the calculated image does not contain motion blur corresponding to the amount of movement v within the shutter time of 3. Accordingly, it should be noted that the relationship between the amount of movement v within the shutter time and the amount v′ by which motion blur is adjusted is different from the result of the motion-blur adder  806 . 
   As discussed above, the motion-blur adjusting unit  107  generates the equation in accordance with the amount of movement v within the shutter time and the unit of processing, and sets the pixel values of the foreground component image in the generated equation, thereby calculating the foreground component image in which the amount of motion blur is adjusted. 
   The processing for adjusting the amount of motion blur contained in the foreground component image executed by the motion-blur adjusting unit  107  is described below with reference to the flowchart of  FIG. 76 . 
   In step S 801 , the unit-of-processing determining portion  801  of the motion-blur adjusting unit  107  generates the unit of processing based on the motion vector within the shutter time and the area information, and supplies the generated unit of processing to the model-forming portion  802 . 
   In step S 802 , the model-forming portion  802  of the motion-blur adjusting unit  107  selects or generates the model in accordance with the amount of movement v within the shutter time and the unit of processing. In step S 803 , the equation generator  803  generates the normal equation based on the selected model. 
   In step S 804 , the adder  804  sets the pixel values of the foreground component image in the generated normal equation. In step S 805 , the adder  804  determines whether the pixel values of all the pixels corresponding to the unit of processing are set. If it is determined that the pixel values of all the pixels corresponding to the unit of processing are not yet set, the process returns to step S 804 , and the processing for setting the pixel values in the normal equation is repeated. 
   If it is determined in step S 805  that the pixel values of all the pixels corresponding to the unit of processing are set, the process proceeds to step S 806 . In step S 806 , the calculator  805  calculates the pixel values of the foreground in which the amount of motion blur is adjusted based on the normal equation in which the pixel values are set supplied from the adder  804 . The processing is then completed. 
   As discussed above, the motion-blur adjusting unit  107  is able to adjust the amount of motion blur of the foreground image containing motion blur based on the shutter time, the motion vector within the shutter time, and the area information. 
   That is, it is possible to adjust the amount of motion blur contained in the pixel values, that is, contained in sampled data. 
     FIG. 77  is a block diagram illustrating another example of the configuration of the motion-blur adjusting unit  107 . The motion vector within the shutter time and the positional information thereof supplied from the mixture-ratio calculator  104  are supplied to a unit-of-processing determining portion  901  and an adjusting portion  905 . The area information supplied from the area specifying unit  103  is supplied to the unit-of-processing determining portion  901 . The foreground component image supplied from the foreground/background separator  105  is supplied to a calculator  904 . 
   The unit-of-processing determining portion  901  generates a unit of processing based on the motion vector within the shutter time and the positional information thereof and the area information, and supplies the unit of processing, together with the motion vector within the shutter time, to a model-forming portion  902 . 
   The model-forming portion  902  forms a model based on the motion vector within the shutter time and the input unit of processing. More specifically, for example, the model-forming portion  902  stores in advance a plurality of models in accordance with the number of pixels contained in the unit of processing, the number of virtual divided portions of the pixel value in the time direction, and the number of foreground components for each pixel. The model-forming portion  902  then selects, the model in which the correlation between the pixel values and the foreground components is designated, such as that in  FIG. 78 , based on the unit of processing and the number of virtual divided portions of the pixel value in the time direction. 
   It is now assumed, for example, that the number of pixels corresponding to the unit of processing is 12, and that the amount of movement v within the shutter time is 5. Then, the model-forming portion  902  sets the number of virtual divided portions to 5, and selects a model formed of eight types of foreground components so that the leftmost pixel contains one foreground component, the second pixel from the left contains two foreground components, the third pixel from the left contains three foreground components, the fourth pixel from the left contains four pixel components, the fifth pixel from the left contains five foreground components, the sixth pixel from the left contains five foreground components, the seventh pixel from the left contains five foreground components, the eighth pixel from the left contains five foreground components, the ninth pixel from the left contains four foreground components, the tenth pixel from the left contains three foreground components, the eleventh pixel from the left contains two foreground components, and the twelfth pixel from the left contains one foreground component. 
   Instead of selecting a model from the prestored models, the model-forming portion  902  may generate a model based on the motion vector within the shutter time and the unit of processing when the motion vector within the shutter time and the unit of processing are supplied. 
   An equation generator  903  generates an equation based on the model supplied from the model-forming portion  902 . 
   A description is now given, with reference to the models of foreground component images shown in  FIGS. 78 through 80 , of an example of the equation generated by the equation generator  903  when the number of foreground components is 8, the number of pixels corresponding to the unit of processing is 12, and the amount of movement v within the shutter time is 5. 
   When the foreground components contained in the foreground component image corresponding to the shutter time/v are F 01 /v through F 08 /v, the relationships between F 01 /v through F 08 /v and pixel values C 01  through C 12  can be expressed by equations (74) through (85), as stated above. 
   By considering the pixel values C 12  and C 11 , the pixel value C 12  contains only the foreground component F 08 /v, as expressed by equation (107), and the pixel value C 11  consists of the product sum of the foreground component F 08 /v and the foreground component F 07 /v. Accordingly, the foreground component F 07 /v can be found by equation (108).
 
 F 08/ v=C 12  (107)
 
 F 07/ v=C 11− C 12  (108)
 
   Similarly, by considering the foreground components contained in the pixel values C 10  through C 11 , the foreground components F 06 /v through F 01 /v can be found by equations (109) through (114), respectively.
 
 F 06/ v=C 10− C 11  (109)
 
 F 05/ v=C 09− C 10  (110)
 
 F 04/ v=C 08− C 09  (111)
 
 F 03/ v=C 07− C 08+ C 12  (112)
 
 F 02/ v=C 06− C 07+ C 11− C 12  (113)
 
 F 01/ v=C 05− C 06+ C 10− C 11  (114)
 
   The equation generator  903  generates the equations for calculating the foreground components by the difference between the pixel values, as indicated by the examples of equations (107) through (114). The equation generator  903  supplies the generated equations to the calculator  904 . 
   The calculator  904  sets the pixel values of the foreground component image in the equations supplied from the equation generator  903  so as to obtain the foreground components based on the equations in which the pixel values are set. For example, when equations (107) through (114) are supplied from the equation generator  903 , the calculator  904  sets the pixel values C 05  through C 12  in equations (107) through (114). 
   The calculator  904  calculates the foreground components based on the equations in which the pixel values are set. For example, the calculator  904  calculates the foreground components F 01 /v through F 08 /v, as shown in  FIG. 79 , based on the calculations of equations (107) through (114) in which the pixel values C 05  through C 12  are set. The calculator  904  supplies the foreground components F 01 /v through F 08 /v to the adjusting portion  905 . 
   The adjusting portion  905  multiplies the foreground components supplied from the mixture-ratio calculator  104  by the amount of movement v within the shutter time contained in the motion vector within the shutter time supplied from the unit-of-processing determining portion  901  so as to obtain the foreground pixel values from which motion blur is eliminated. For example, when the foreground components F 01 /v through F 08 /v are supplied from the calculator  904 , the adjusting portion  905  multiples each of the foreground components F 01 /v through F 08 /v by the amount of movement v within the shutter time, i.e., 5, so as to obtain the foreground pixel values F 01  through F 08  from which motion blur is eliminated, as shown in  FIG. 80 . 
   The adjusting portion  905  supplies the foreground component image consisting of the foreground pixel values without motion blur calculated as described above to a motion-blur adder  906  and a selector  907 . 
   The motion-blur adder  906  is able to adjust the amount of motion blur by using the amount v′ by which motion blur is adjusted, which is different from the amount of movement v within the shutter time generated based on the shutter time, for example, the amount v′ by which motion blur is adjusted, which is one half the value of the amount of movement v within the shutter time, or the amount v′ by which motion blur is adjusted, which is irrelevant to the amount of movement v within the shutter time. For example, as shown in  FIG. 73 , the motion-blur adder  906  divides the foreground pixel value Fi without motion blur by the amount v′ by which motion blur is adjusted so as to obtain the foreground component Fi/v′. The motion-blur adder  906  then calculates the sum of the foreground components Fi/v′, thereby generating the pixel value in which the amount of motion blur is adjusted. For example, when the amount v′ by which motion blur is adjusted is 3, the pixel value C 02  is set to (F 01 )/v′, the pixel value C 3  is set to (F 01 +F 02 )/v′, the pixel value C 04  is set to (F 01 +F 02 +F 03 )/v′, and the pixel value C 05  is set to (F 02 +F 03 +F 04 )/v′. 
   Based on the shutter time, the motion-blur adder  906  supplies the foreground component image in which the amount of motion blur is adjusted to the selector  907 . 
   The selector  907  selects either the foreground component image without motion blur supplied from the adjusting portion  905  or the foreground component image in which the amount of motion blur is adjusted supplied from the motion-blur adder  906  based on a selection signal reflecting a user&#39;s selection, and outputs the selected foreground component image. 
   As discussed above, the motion-blur adjusting unit  107  is able to adjust the amount of motion blur based on the selection signal and the amount v′ by which motion blur is adjusted. 
   The processing for adjusting the amount of motion blur of the foreground executed by the motion-blur adjusting unit  107  configured as shown in  FIG. 77  is described below with reference to the flowchart of  FIG. 81 . 
   In step S 901 , the unit-of-processing determining portion  901  of the motion-blur adjusting unit  107  generates the unit of processing based on the motion vector within the shutter time and the area information, and supplies the generated unit of processing to the model-forming portion  902  and the adjusting portion  905 . 
   In step S 902 , the model-forming portion  902  of the motion-blur adjusting unit  107  selects or generates the model according to the amount of movement v within the shutter time and the unit of processing. In step S 903 , the equation generator  903  generates, based on the selected or generated model, the equations for calculating the foreground components by the difference between the pixel values of the foreground component image. 
   In step S 904 , the calculator  904  sets the pixel values of the foreground component image in the generated equations, and extracts the foreground components by using the difference between the pixel values based on the equations in which the pixel values are set. In step S 905 , the calculator  904  determines whether all the foreground components corresponding to the unit of processing have been extracted. If it is determined that all the foreground components corresponding to the unit of processing have not been extracted, the process returns to step S 904 , and the processing for extracting the foreground components is repeated. 
   If it is determined in step S 905  that all the foreground components corresponding to the unit of processing have been extracted, the process proceeds to step S 906 . In step S 906 , the adjusting portion  905  adjusts each of the foreground components F 01 /v through F 08 /v supplied from the calculator  904  based on the amount of movement v within the shutter time so as to obtain the foreground pixel values F 01 /v through F 08 /v from which motion blur is eliminated. 
   In step S 907 , the motion-blur adder  906  calculates the foreground pixel values in which the amount of motion blur is adjusted, and the selector  907  selects the image without motion blur or the image in which the amount of motion blur is adjusted, and outputs the selected image. The processing is then completed. 
   As described above, the motion-blur adjusting unit  107  configured as shown in  FIG. 77  is able to more speedily adjust motion blur of the foreground image containing motion blur according to simpler computations. 
   A known technique for partially eliminating motion blur, such as a Wiener filter, is effective when being used in the ideal state, but is not sufficient for an actual image quantized and containing noise. In contrast, it is proved that the motion-blur adjusting unit  107  configured as shown in  FIG. 77  is sufficiently effective for an actual image quantized and containing noise. It is thus possible to eliminate motion blur with high precision. 
     FIG. 82  illustrates the configuration of the synthesizer  108 . A background component generator  1021  generates a background component image based on the mixture ratio α and a certain background image, and supplies the background component image to a mixed-area-image synthesizing portion  1022 . 
   The mixed-area-image synthesizing portion  1022  combines the background component image supplied from the background component generator  1021  with the foreground component image so as to generate a mixed-area synthesized image, and supplies the generated mixture-area synthesized image to an image synthesizing portion  1023 . 
   The image synthesizer  1023  combines the foreground component image, the mixed-area synthesized image supplied from the mixed-area-image synthesizing portion  1022 , and the certain background image based on the area information so as to generate a synthesized image, and outputs it. 
   As discussed above, the synthesizer  108  is able to combine the foreground component image with a certain background image. 
   The image obtained by combining a foreground component image with a certain background image based on the mixture ratio α, which is the feature quantity, appears more natural compared to an image obtained by simply combining pixels. 
   As described above, the image processing apparatus configured as shown in  FIG. 2  is capable of separating a foreground component image from an input image, adjusting the amount of motion blur included in the separated foreground component image, and combining it with a desired background image. 
     FIG. 83  is a block diagram illustrating still another configuration of the function of the image processing apparatus for adjusting the amount of motion blur. The image processing apparatus shown in  FIG. 2  sequentially performs the area-specifying operation and the calculation for the mixture ratio α. In contrast, the image processing apparatus shown in  FIG. 84  simultaneously performs the area-specifying operation and the calculation for the mixture ratio α. 
   The functional elements similar to those in the block diagram of  FIG. 2  are designated with like reference numerals, and an explanation thereof is thus omitted. 
   An input image is supplied to the object extracting unit  101 , the area specifying unit  103 , a mixture-ratio calculator  1101 , and a foreground/background separator  1102 . 
   The area specifying unit  103  generates area information based on the input image, and supplies the area information to the foreground/background separator  1102 , the motion-blur adjusting unit  107 , and a synthesizer  1103 . 
   The mixture-ratio calculator  1101  calculates, based on the input image, the estimated mixture ratio when it is assumed that each pixel contained in the input image belongs to the covered background area, and the estimated mixture ratio when it is assumed that each pixel contained in the input image belongs to the uncovered background area, and supplies the estimated mixture ratio calculated with the assumption that each pixel belongs to the covered background area and the estimated mixture ratio calculated with the assumption that each pixel belongs to the uncovered background area to the foreground/background separator  1102 . 
   The mixture-ratio calculator  1101  generates a motion vector within the shutter time based on the estimated mixture ratio calculated with the assumption that each pixel belongs to the covered background area and the estimated mixture ratio calculated with the assumption that each pixel belongs to the uncovered background area, and supplies the motion vector within the shutter time to the shutter-time calculator  106  and the motion-blur adjusting unit  107 . 
     FIG. 85  is a block diagram illustrating an example of the configuration of the mixture-ratio calculator  1101 . 
   An estimated-mixture-ratio processor  401  shown in  FIG. 85  is the same as the estimated-mixture-ratio processor  401  shown in  FIG. 49 . An estimated-mixture-ratio processor  402  shown in  FIG. 85  is the same as the estimated-mixture-ratio processor  402  shown in  FIG. 49 . 
   The estimated-mixture-ratio processor  401  calculates the estimated mixture ratio for each pixel by the computation corresponding to a model of the covered background area based on the input image, and outputs the calculated estimated mixture ratio. Based on the estimated mixture ratio, the estimated-mixture-ratio processor  401  calculates an estimated motion vector, and supplies the estimated mixture ratio and the estimated motion vector to a selector  1111 . 
   The estimated-mixture-ratio processor  402  calculates the estimated mixture ratio for each pixel by the computation corresponding to a model of the uncovered background area based on the input image, and outputs the calculated estimated mixture ratio. Based on the estimated mixture ratio, the estimated-mixture-ratio processor  402  calculates an estimated motion vector, and supplies the estimated mixture ratio and the estimated motion vector to the selector  1111 . 
   The selector  1111  selects an estimated motion vector corresponding to an estimated mixture ratio exceeding 0 and below 1, and sets the estimated motion vector as the motion vector within the shutter time and outputs it. 
   The foreground/background separator  1102  generates the foreground component image from the input image based on the estimated mixture ratio calculated when it is assumed that the pixel belongs to the covered background area supplied from the mixture-ratio calculator  1101 , the estimated mixture ratio calculated when it is assumed that the pixel belongs to the uncovered background area supplied from the mixture-ratio calculator  1101 , and the area information supplied from the area specifying unit  103 , and supplies the generated foreground component image to the motion-blur adjusting unit  1103 . 
     FIG. 85  is a block diagram illustrating an example of the configuration of the foreground/background separator  1102 . 
   The elements similar to those of the foreground/background separator  105  shown in  FIG. 61  are indicated by like reference numerals, and an explanation thereof is thus omitted. 
   A selector  1121  selects, based on the area information supplied from the area specifying unit  103 , either the estimated mixture ratio calculated when it is assumed that the pixel belongs to the covered background area supplied from the mixture-ratio calculator  1101  or the estimated mixture ratio calculated when it is assumed that the pixel belongs to the uncovered background area supplied from the mixture-ratio calculator  1101 , and supplies the selected estimated mixture ratio to the separating portion  601  as the mixture ratio α. 
   The separating portion  601  extracts the foreground components and the background components from the pixel values of the pixels belonging to the mixed area based on the mixture ratio α supplied from the selector  1121  and the area information, and supplies the extracted foreground components to the synthesizer  603  and also supplies the foreground components to the synthesizer  605 . 
   The separating portion  601  can be configured similarly to the counterpart shown in  FIG. 66 . 
   The synthesizer  603  synthesizes the foreground component image and outputs it. The synthesizer  605  synthesizes the background component image and outputs it. 
   The motion-blur adjusting unit  107  adjusts the amount of motion blur included in the foreground component image supplied from the foreground/background separator  1102  based on the area information supplied from the area specifying unit  103 , the motion vector within the shutter time supplied from the mixture-ratio calculator  1101 , and the shutter time supplied from the shutter-time calculator  106 , and outputs the foreground component image in which the amount of motion blur is adjusted to the synthesizer  1103 . 
   The synthesizer  1103  combines a certain background image with the foreground component image supplied from the motion-blur adjusting unit  107 , in which motion blur is adjusted, based on the estimated mixture ratio calculated when it is assumed that the pixel belongs to the covered background area and the estimated mixture ratio calculated when it is assumed that the pixel belongs to the uncovered background area supplied from the mixture-ratio calculator  1101 , and the area information supplied from the area specifying unit  103 , and outputs the synthesized image. 
     FIG. 86  is a block diagram illustrating the configuration of the synthesizer  1103 . 
   The elements similar to those shown in  FIG. 82  are designated with like reference numerals, and explanation thereof is thus omitted. 
   A selector  1131  selects, based on the area information supplied from the area specifying unit  103 , either the estimated mixture ratio calculated when it is assumed that the pixel belongs to the covered background area supplied from the mixture-ratio calculator  1101  or the estimated mixture ratio calculated when it is assumed that the pixel belongs to the uncovered background area supplied from the mixture-ratio calculator  1101 , and supplies the selected estimated mixture ratio to the background component generator  1021  as the mixture ratio α. 
   As described above, the image processing apparatus configured as shown in  FIG. 83  is able to adjust the amount of motion blur included an image corresponding to a foreground object in an input image before output. 
   The embodiment has been discussed above by setting the mixture ratio α to the ratio of the background components contained in the pixel values. However, the mixture ratio α may be set to the ratio of the foreground components contained in the pixel values. 
   The embodiment has been discussed above by setting the moving direction of the foreground object to the direction from the left to the right. However, the moving direction is not restricted to the above-described direction. 
   In the above description, a real-space image having a three-dimensional space and time axis information is projected onto a time space having a two-dimensional space and time axis information by using a video camera. However, the present invention is not restricted to this example, and can be applied to the following case. When a greater amount of first information in one-dimensional space is projected onto a smaller amount of second information in a two-dimensional space, distortion generated by the projection can be corrected, significant information can be extracted, or a more natural image can be synthesized. 
   The sensor is not restricted to a CCD, and may be another type of sensor, such as a solid-state image-capturing device, for example, a CMOS, (Complementary Metal Oxide Semiconductor), a BBD (Bucket Brigade Device), a CID (Charge Injection Device), or a CPD (Charge Priming Device). Also, the sensor does not have to be a sensor in which detection devices are arranged in a matrix, and may be a sensor in which detection devices are arranged in one line. 
   A recording medium in which a program for performing the signal processing of the present invention is recorded may be formed of a package medium in which the program is recorded, which is distributed for providing the program to a user separately from the computer, as shown in  FIG. 1 , such as the magnetic disk  51  (including a floppy (registered trade name) disk), the optical disc  52  (CD-ROM (Compact Disc-Read Only Memory) and a DVD (Digital Versatile Disc)), the magneto-optical disk  53  (including MD (Mini-Disc) (registered trade name)), or the semiconductor memory  54 . The recording medium may also be formed of the ROM  22  or a hard disk contained in the storage unit  28  in which the program is recorded, such recording medium being provided to the user while being prestored in the computer. 
   The steps forming the program recorded in a recording medium may be executed chronologically according to the orders described in the specification. However, they do not have to be executed in a time-series manner, and they may be executed concurrently or individually. 
   INDUSTRIAL APPLICABILITY 
   According to the present invention, detection of an exposure time of an image that has already been captured is allowed.