Patent Publication Number: US-10769759-B2

Title: Image processing apparatus, imaging apparatus, image processing method, storage medium, and lens apparatus

Description:
BACKGROUND OF THE INVENTION 
     Field of the Invention 
     The present invention relates to an image processing apparatus configured to provide sharpening processing to an image. 
     Description of the Related Art 
     One conventionally known unsharp mask processing sharpens an image by adding to or subtracting from an original image a difference between the original image and a blurred image through an unsharp mask. The larger the difference between the blurred image and the input image is, the sharper the image becomes. Japanese Patent Laid-Open No. (“JP”) 2010-81263 discloses a method for applying an asymmetric one-dimensional filter to a pixel signal sequence arranged in an image height direction, and for reducing the influence of the Point Spread Function (“PSF”) of the optical system. 
     However, the conventional unsharp mask processing uses a rotationally symmetric filter for the unsharp mask, and has difficulties in sharpening an image deteriorated by the influence of the PSF having a complicated shape, such as an asymmetric aberration and a sagittal halo. An attempt to correct the aberration in the azimuth direction having a large aberration causes the undershoot in the azimuth direction having a small aberration. Conversely, an attempt to suppress the undershoot cannot fully correct the aberration. 
     The method in JP 2010-81263 considers only the asymmetry in the image height direction, uses a one-dimensional correction filter, and thus cannot improve the asymmetry in a non-image height direction. The image height direction is the meridional azimuth direction. In addition, this method adjusts the asymmetry of the filter by the number of minus tap coefficients for the filter, and the correction in the image height direction is different from the blurring state of the PSF of the optical system. Thus, the conventional method cannot provide sufficient sharpening. 
     SUMMARY OF THE INVENTION 
     The present invention provides an image processing method, an image processing apparatus, an imaging apparatus, a storage medium, and a lens apparatus, which can execute highly accurate sharpening processing while reducing an information amount required for sharpening processing. 
     An image processing apparatus according to one aspect of the present invention includes an acquirer configured to acquire a captured image generated through imaging by an optical system, a reconstruction processor configured to reconstruct a discretized point spread function of the optical system using coefficient data used to approximate the point spread function, and a sharpening processor configured to perform unsharp mask processing for the captured image based on information on the reconstructed point spread function. A discretization interval of the reconstructed point spread function is different from a pixel pitch in an image sensor used for the imaging. 
     An image processing method according to another aspect of the present invention includes the steps of acquiring a captured image generated through imaging by an optical system, reconstructing a discretized point spread function of the optical system using coefficient data used to approximate the point spread function, and performing unsharp mask processing for the captured image based on information on the reconstructed point spread function. A discretization interval of the reconstructed point spread function is different from a pixel pitch in an image sensor used for the imaging. A non-transitory computer readable storage medium for storing a computer program that enables a computer to execute the image processing method also constitutes another aspect of the present invention. 
     A lens apparatus according to another aspect of the present invention is attachable to an imaging apparatus having an image sensor and configured to communicate with the imaging apparatus. The lens apparatus includes an optical system, and a memory configured to store coefficient data used to approximate a point spread function of the optical system and a discretization coefficient used to adjust a discretization interval in reconstructing the point spread function so that the discretization interval is different from a pixel pitch in the image sensor. The imaging apparatus reconstructs the discretized point spread function using the coefficient data and the discretization coefficient transmitted from the lens apparatus, and performs unsharp mask processing for a captured image generated through imaging by the optical system based on information on the reconstructed point spread function. 
     Further features of the present invention will become apparent from the following description of exemplary embodiments with reference to the attached drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a flowchart illustrating an image processing method according to first and second embodiments. 
         FIG. 2  is a block diagram of an imaging apparatus according to each embodiment. 
         FIGS. 3A and 3B  schematically illustrate sharpening by unsharp mask processing according to each embodiment. 
         FIGS. 4A and 4B  schematically illustrate a PSF of an imaging optical system on an xy plane according to each embodiment. 
         FIGS. 5A to 5C  schematically illustrate sharpening processing with a rotationally symmetric unsharp mask according to each embodiment. 
         FIGS. 6A to 6C  schematically illustrate the sharpening processing with a rotationally asymmetric unsharp mask according to each embodiment. 
         FIGS. 7A and 7B  are a schematic view and a schematic sectional view of an unsharp mask according to each embodiment. 
         FIG. 8  schematically illustrates a Bayer array according to each embodiment. 
         FIG. 9  is a block diagram of the image processor according to each embodiment. 
         FIGS. 10A to 10F  are contour maps of an approximated PSF. 
         FIG. 11  is a flowchart illustrating a method of calculating coefficients according to each embodiment. 
         FIG. 12  illustrates the approximated PSF and designed values according to each embodiment. 
         FIGS. 13A and 13B  explain the rotation processing of a point spread function. 
         FIG. 14  is a sectional view of a reconstructed PSF. 
         FIGS. 15A and 15B  explain discretization processing according to each embodiment. 
         FIG. 16  illustrates a relationship between a pixel pitch and a discretization interval according to each embodiment. 
         FIG. 17  is a flowchart illustrating an image processing method according to a third embodiment. 
     
    
    
     DESCRIPTION OF THE EMBODIMENTS 
     Referring now to the accompanying drawings, a description will be given of embodiments of the present invention. 
     Input Image 
     An input image is a digital image (captured image) generated with an output from an image sensor that photoelectrically converts an object image (optical image) formed via an imaging optical system (optical system) in an imaging apparatus. The digital image is an image deteriorated by an optical transfer function (OTF: Optical Transfer Function) that contains the aberration of the optical system that includes optical elements, such as a lens and an optical filter. The image sensor includes a photoelectric conversion element, such as a CMOS sensor and a CCD sensor. The imaging optical system may include a mirror (reflective surface) having a curvature. The optical system may be attached to and detached from (or may be replaced from) the imaging apparatus. In the imaging apparatus, an imaging system includes the optical system, the image sensor, and an image processing circuit configured to generate a digital image (input image) using an output from the image sensor. 
     A color component of the input image contains information, for example, on RGB color components. The color component can be used by selecting a generally used color space, such as a brightness expressed by LCH, a hue, a chroma, a luminance expressed by YCbCr, and a color difference signal. Another color space may use, for example, XYZ, Lab, Yuv, and JCh, and the color temperature may also be used. 
     Information on an imaging condition (imaging condition information), such as a focal length, a F-number, an imaging distance, and an image height of the optical system in the imaging apparatus in generating (capturing) an input image, may be attached to an input image and an output image. Various correction information used to correct the input image may be attached to the input image and the output image. The imaging apparatus may output the input image to an image processing apparatus separately provided to the imaging apparatus, and the imaging condition information and the correction information may be attached to the input image for the image processing in the image processing apparatus. The imaging apparatus may directly or indirectly deliver the imaging condition information and the correction information to the image processing apparatus through communications. 
     Image Sharpening Processing 
       FIGS. 3A and 3B  are schematic diagrams of sharpening in the unsharp mask processing (image sharpening processing) according to this embodiment. In  FIG. 3A , a solid line denotes an input image, a broken line denotes an image made by blurring the input image with an unsharp mask, and a dotted line denotes a sharpened image. A solid line in  FIG. 3B  is a correction component. In each of  FIGS. 3A and 3B , an abscissa axis denotes a coordinate, and an ordinate axis denotes a pixel value or a luminance value.  FIGS. 3A and 3B  correspond to a section in a predetermined direction, such as an X direction, in  FIGS. 4A and 4B . 
     Where f(x, y) is an original image and h(x, y) is a correction component, a sharpened image g(x, y) can be represented by Expression (1):
 
 g ( x,y )= f ( x,y )± m×h ( x,y )  (1)
 
     In Expression (1), m is an adjustment coefficient to change a correction intensity, and the adjustment coefficient m can be varied to adjust a correction amount. The adjustment coefficient m may be constant irrespective of a position in the input image or may be variable according to the position (image height) in the input image. Thereby, the correction amount can be adjusted according to the position in the input image. The adjustment coefficient m(x, y) may vary depending on the imaging condition such as a focal length, an aperture value (F-number), or an object distance in the optical system. 
     Where USM(x, y) is the unsharp mask, the correction component h(x, y) is expressed as follows:
 
 h ( x,y )= f ( x,y )− f ( x,y )*USM( x,y )  (2)
 
where USM(x, y) is, for example, a tap value at a coordinate (x, y).
 
     The right side of Expression (2) is rewritten as below.
 
 h ( x,y )= f ( x,y )*(δ( x,y )−USM( x,y ))  (3)
 
     In Expressions (2) and (3), * represents a convolution (convolution integral, product sum), and symbol δ represents a delta function (ideal point image). The “delta function” is data whose number of taps is equal to that of USM(x, y) and whose value is zero except for a central value of one. Expression (3) can be expressed by modifying Expression (2), and thus Expressions (2) and (3) are equivalent with each other. For this reason, Expression (2) is used below to describe generation of the correction component. 
     Expression (2) calculates a difference between the captured image f(x, y) and an image obtained by unsharpening the captured image f(x, y) with the unsharp mask, and the correction component h(x, y) is generated based on this difference information. The typical unsharp mask processing uses for the unsharp mask a smoothing filter such as a Gaussian filter, a median filter, and a moving average filter. 
     For example, when the Gaussian filter is applied as the unsharp mask to the captured image f(x, y) illustrated with the solid line in  FIG. 3A , an image obtained by unsharpening the captured image f(x, y) is illustrated with the dashed line in  FIG. 3A . The correction component h(x, y) is thus the difference between the captured image f(x, y) and the unsharpened image as in Expression (2). Thus, subtracting a dashed line in  FIG. 3A  from a solid line in  FIG. 3A  yields a solid line in  FIG. 3B , which represents the correction component. The correction component thus calculated is used to calculate Expression (1) so as to sharpen the input image f(x, y) illustrated with the solid line in  FIG. 3A  and obtain the image illustrated with the short-dotted line in  FIG. 3A . 
     Next follows a description of image sharpening through the unsharp mask processing on an image degraded through an imaging optical system configured to form an optical image of an object. Where I(x, y) is a pre-captured image (object image) and psf(x, y) is a point spread function PSF as a function representing a response of the optical system to a point light source, the captured image f(x, y) formed through the optical system can be expressed as follows:
 
 f ( x,y )= I ( x,y )*psf( x,y )  (4)
 
     If the imaging optical system is a rotationally symmetrical coaxial optical system, a PSF corresponding to the central part in the image is rotationally symmetric. This enables the sharpening processing to make closer the captured image f(x, y) to the original image I(x, y) by applying a rotationally symmetric USM to the central part in the image. Since the correction amount is a difference value between the captured image and an unsharpened image obtained through the unsharp mask, a more accurate correction requires the use of an unsharp mask that is shaped more similarly to psf(x, y), not a simple smoothing filter. For example, when a captured image is degraded due to the spherical aberration, which has rotationally symmetric influence, a smoothing filter such as the Gaussian filter has a different distribution shape from that of the PSF affected by the spherical aberration. Thus, the use of the PSF of the optical system improves the correction in reducing the rotationally symmetric unsharpening. 
     This embodiment uses the PSF as the unsharp mask USM(x, y). Although the captured image f(x, y) in  FIG. 3A  has a symmetrical shape for convenience, the shape of the input image may not be symmetric. Even when the original image I(x, y) has an asymmetric shape, as long as a degradation function of the original image I(x, y) corresponding to psf(x, y) is rotationally symmetric, the captured image can still be sharpened with a rotationally symmetric unsharp mask. 
     On the other hand, the PSF is usually asymmetric at positions in non-central part in the image, even when the optical system is a rotationally symmetric coaxial optical system.  FIGS. 4A and 4B  schematically illustrate the PSF of the optical system on the xy plane:  FIG. 4A  illustrates the on-axis PSF, and  FIG. 4B  illustrates the off-axis PSF. 
     For example, if the original image (object) is an ideal point image, Expression (4) shows that the captured image f(x, y) is the PSF of the optical system. Assume that the ideal point image exists in an angle of view corresponding to  FIG. 4B  and the original image (object) is degraded due to the PSF of the optical system. Then, an image obtained as the input image is a blurred image having the shape illustrated in  FIG. 4B . Next follows a description of sharpening through the unsharp mask processing on the image thus asymmetrically blurred. 
       FIGS. 5A to 5C  and  FIGS. 6A to 6C  schematically illustrate the unsharp processing to an asymmetrically degraded image.  FIGS. 5A to 5C  illustrate that the unsharp processing uses a rotationally symmetric unsharp mask.  FIGS. 6A to 6C  illustrate that the unsharp processing uses a rotationally asymmetric unsharp mask. The ordinate axis and the abscissa axis are the same as those in  FIGS. 3A and 3B . 
     Solid lines in  FIGS. 5A and 6A  represent a section along the y direction in  FIG. 4B , and dotted lines represent images obtained by blurring captured images with the respective unsharp masks. The rotationally symmetric unsharp mask in  FIGS. 5A to 5C  uses the Gaussian filter. The PSF of the imaging apparatus applies the rotationally asymmetric unsharp mask in  FIGS. 6A to 6C . 
       FIGS. 5B and 6B  respectively illustrate correction components as differences between the images obtained by blurring the input image with the respective unsharp masks and the original input image. For illustration convenience, in  FIGS. 5A and 6A , an input image blurred by the PSF has an extended skirt on the plus side of the Y axis. 
     In  FIG. 5A , the difference between the unsharpened image and the original input image is smaller on the plus side with respect to the peak position of the solid line and larger on the minus side. Therefore, the correction component illustrated in  FIG. 5B  has a higher extreme value on the right side (plus side) of the central peak position than that on the left side (minus side). It is understood from a comparison between curves in  FIGS. 5A and 5B  that the correction component has a correction amount smaller on the plus side in the captured image and a correction amount larger on the minus side on which the skirt does not extend. Thus, the sharpening with Expression (1) cannot correct an asymmetric blur. 
       FIG. 5C  illustrates a sharpened result where m=1. It is understood that sharpening is improved for the solid line in  FIG. 5A  but the minus side is significantly concave relative to the plus side and the asymmetrical blur is not successfully corrected. For example, assume that the correction amount is adjusted by changing the adjustment coefficient m in Expression (1) without changing the unsharp mask. When a large adjustment coefficient m is used to sufficiently correct the plus side of the input image, the minus side of the input image is overcorrected (undershot). In contrast, when the adjustment coefficient m is set such that the minus side of the input image is appropriately corrected, the plus side of the input image is not sufficiently corrected. 
     This unsharp mask processing with the rotationally symmetric unsharp mask to an asymmetrically blurred input image has difficulties in improving the asymmetry and sharpening the image. The same difficulties occur when rotationally symmetric filters other than the Gaussian filter are used as the rotationally symmetric unsharp mask. 
     On the other hand, the difference between the unsharpened image and the original input image is lamer on the plus side with respect to the peak position illustrated by the solid line in  FIG. 6A  and is smaller on the minus side, and this tendency is opposite to the relationship in  FIG. 5A . Therefore, the correction component illustrated in  FIG. 6B  has a higher extreme value on the left side (minus side) with respect to the central peak position than that on the right side (plus side). When this correction component is applied to the captured image illustrated by the solid line in  FIG. 6A , a large correction amount on the plus side with respect to the peak position where a large blur exists and a small correction amount on the minus side where a small blur exists. The use of the asymmetric unsharp mask thus enables the blur of the input image and the correction amount of the correction component to have similar distributions, and reduces the excess-and-insufficient correction that occurs with the use of the rotationally symmetric unsharp mask.  FIG. 6C  illustrates a sharpened result where m=1. Sharpening is improved for the solid line in  FIG. 6A , and a difference of a concave balance is improved between the minus side and the plus side that stand out in  FIG. 5C . Moreover, this case is less likely to cause an overcorrection than the use of the rotationally symmetric unsharp mask, and thus a value of the adjustment coefficient m in Expression (1) can be set relatively large for more improved asymmetry and further sharpening. Since the correction amount of the correction component corresponds to the difference between the blurred image and the original image for more accurate corrections, a portion more blurred by the PSF of the optical system needs to be more blurred by the unsharp mask than other portions. Thus, it is ideal to use the PSF of the optical system as the unsharp mask for the more accurate corrections. 
     This embodiment illustrates an unsharp mask using the PSF as the image sharpening processing, but the PSF may be used for the image restoration processing represented by the Wiener filter and the image restoration processing of iterative processing represented by the RL method. The PSF can be used to generate a learned image for deep learning (DL) which has recently been developed. 
     Coefficient Data 
     Next follows a description of the coefficients used to generate the unsharp mask USM. As described above, each embodiment uses the PSF of the imaging optical system as an unsharp mask. Then, since the PSF of the imaging optical system varies depending on an imaging condition, such as an image height, a focal length, an F-number, and an imaging distance, it is necessary for the sharpening to generate an unsharp mask that matches imaging condition. In order to change the unsharp mask in accordance with the PSF which varies depending on the imaging condition, one conceivable method is to calculate all combinations and to select a PSF corresponding to the imaging condition. However, this method is not desirable in view of the processing speed in applying the unsharp mask and in storage capacity for saving the data. 
     Therefore, each embodiment saves coefficient data (coefficient data) for approximating the PSF of the imaging system, and uses the coefficient data to reconstruct (approximate) the (discretized) PSF in creating the unsharp mask. This configuration can maximize the sharpening effect while reducing a data amount to be stored. Each embodiment uses a continuous function and its coefficient data, as a method for approximating the PSF and for creating the unsharp mask, as described below. 
     Next follows a description of a continuous function used to approximate the PSF of the imaging optical system. As described above, in fitting a celestial body photometered in the field of astrophysics, the function P (x, y) represented by the following Expression (5) called a Moffat function is frequently used. 
     
       
         
           
             
               
                 
                   
                     P 
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               y 
                               2 
                             
                           
                           
                             α 
                             2 
                           
                         
                       
                       ) 
                     
                     
                       - 
                       β 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Herein, α and β in Expression (5) are coefficients and, in particularly, P(x, y) with β=1 is called a Lorentz function. For example, in modeling the PSF using Expression (5), the coefficients α and β are obtained by fitting the distribution of PSF measured or calculated by Expression (5). Then, the PSF can be modeled based on the calculated coefficients α and β and Expression (5). While the approximated PSF can be created based on Expression (5), Expression (5) is a function that can express only a rotationally symmetric distribution for the coordinates x and y. Thus, a rotationally asymmetric distribution or an N-fold rotationally symmetric distribution (where N is an integer) cannot be created with Expression (5). 
     Expression (6) (and (6a)) is called an Elliptical Moffat function that can express an elliptical shape (or a distribution of the two-fold rotationally symmetry) as a function obtained by modifying Expression (5). 
     
       
         
           
             
               
                 
                   
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                             ( 
                             
                               
                                 X 
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                               + 
                               
                                 
                                   Y 
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                                   γ 
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                       β 
                     
                   
                 
               
               
                 
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     In Expression (6), α, β, and γ are coefficients. Expression (6a) is a rotation matrix with respect to an angle θ. Expressions (6) and (6a) can be summarized and expressed as in the following Expression (7).
 
 P ( x,y )=+(σ+ ax   2 +2 bxy+cy   2 ) −β   (7)
 
     In Expression (7), a, b, c, σ, and β are coefficients. In order to maintain the elliptical shape in using Expression (7), it is necessary to satisfy a relationship of b 2 −ac&lt;0 for the coefficients a, b, and c. 
     Use of Expression (7) (or Expression (6)) can reproduce the distribution of the elliptical shape which cannot be represented by the function of Expression (5) and the correction becomes more accurate than that with the function of Expression (5). However, even when the function of Expression (7) is used for fitting to the PSF of the imaging optical system, a complicated shape, such as an asymmetric aberration and a sagittal halo, cannot be reproduced. In other words, use of Expression (7) can express a distribution having a rotational symmetry or the N-fold rotation symmetry but a rotationally asymmetric distribution cannot be expressed. 
     Accordingly, each of the embodiments uses a function represented by the following Expressions (8) ((8a)-(8c)) as reproducible functions of the PSF having the complicated shape, such as the asymmetric aberration and the sagittal halo of the imaging optical system. 
     When x≥0 and y≥0,
 
 P ( x,y )=+(σ+ ax   2 +2 bxy+cy   2 ) −β   −e   (8a)
 
     When x≥0 and y&lt;0,
 
 P ( x,y )=+(σ+ ax   2 +2 bdxy+cd   2   y   2 ) −β   −e   (8b)
 
     When x&lt;0,
 
 P ( x,y )= P (− x,y )  (8c)
 
     In Expressions (8), a, b, c, d, e, σ, and β are coefficients. Similar to Expression (7), the coefficients a, b, and c in Expressions (8) need to satisfy the relationship of b 2 −ac&lt;0. 
       FIGS. 10A to 10F  are illustrative distribution shapes that can be expressed by a function based on Expressions (8).  FIG. 10A  illustrates a rotationally symmetrical distribution in the XY coordinates, which can be expressed by any of functions of Expressions (5) to (8). When the imaging optical system is a coaxial system and the image point is located on the optical axis, the PSF also becomes rotationally symmetric. Therefore, the distribution shape of  FIG. 10A  can be expressed by the function of any one of Expressions (5) to (8). 
       FIGS. 10B and 10C  illustrate distributions of an elliptical shape (referred to as “elliptical distributions” hereinafter) in which a long axis and a short axis of the ellipse overlap the X axis and the Y axis. These elliptical distributions cannot be expressed by the function of Expression (5), and the approximation accuracy improves with any one of the functions of Expressions (6) to (8).  FIG. 10D  illustrates the elliptical distribution where none of the major axis and the minor axis of the ellipse overlap the X axis or the Y axis, and this elliptical distribution cannot be represented by the functions of Expressions (5) and (6). This elliptical distribution can be accurately approximated with either of the function of Expression (7) or (8). 
       FIGS. 10E and 10F  illustrate distributions that are symmetrical with respect to the Y axis (or in the X direction) and asymmetrical with respect to the X axis (or in the Y direction). The distribution shape in  FIG. 10E  corresponds to a distribution in which a portion on the +X side from the Y axis in the elliptical distribution illustrated in  FIG. 10D  is folded on the −X side with respect to the Y axis. The distribution shape in  FIG. 10F  is a portion above the X axis is made line-symmetrical with respect to the Y axis and a portion below the X axis is made concentrically semicircular in an elliptical distribution in which the long axis overlaps the Y axis and the short axis does not overlap the X axis. The functions of Expressions (5) to (7) cannot represent a line-symmetric distribution shape like  FIGS. 10E and 10F . On the other hand, the function of Expression (8) used for this embodiment can express a rotationally asymmetric distribution, and is used to accurately approximate the distribution shape illustrated in  FIGS. 10E and 10F . 
     As described above, the PSF at the image point on the optical axis in the imaging optical system has a rotationally symmetric distribution shape, but that is not the case at the off-axis image point within the plane (image plane) orthogonal to the optical axis. When the imaging optical system is a coaxial optical system, the PSF even at the off-axis image point has a symmetrical distribution shape in the direction (sagittal direction) orthogonal to a direction (meridional direction) of a line that extends from the image point to the optical axis in the image plane. Thus, the PSF of the imaging optical system does not always have a rotationally symmetric distribution shape, but is symmetrical in the sagittal direction. A complicated asymmetric aberration can be corrected by setting the x direction (the X direction in  FIGS. 10E and 10F ) in Expression (8) to the sagittal direction and the y direction (the Y direction in  FIGS. 10E and 10F ) to the meridional direction. 
     Next follows a detailed description of each coefficient in Expression (8). The coefficients a, b, and c among the coefficients in Expression (8) are used to generate an elliptical distribution in which none of the major axis and the minor axis overlap the X axis or the Y axis as illustrated in  FIG. 10D . The asymmetry in the X and Y directions of the elliptical distribution can be controlled by controlling these coefficients a, b, and c. In addition, this method can express an elliptical distribution, as illustrated in  FIGS. 10E and 10F , in which at least one of the major axis and the minor axis is set to an axis other than the X and Y axes and the distribution is made symmetrical only with respect to the Y axis, or an aberration such as the sagittal halo, which is hard for another function to express. 
     The coefficient d is used to make asymmetrical the elliptical distribution in the Y direction (specific direction), and controlling the coefficient d can correct the asymmetric aberration in the meridional direction. For example, the approximation accuracy for the coma (aberration) can be improved by controlling the coefficient d. 
     The coefficients e, σ, and β are used to control the spread of the elliptical distribution. The approximation accuracy can be improved by increasing the coefficient σ when the elliptical distribution to be approximated has a wide spread and by increasing the coefficient β when the shape of the elliptical distribution to be approximated abruptly changes near the peak. The coefficient e is used to limit the spread of the elliptical distribution. When the coefficient e is 0, the elliptical distribution gradually approaches to P(x, y)=0 on the peripheral side based on Expression (8). Therefore, when the elliptical distribution has a narrow spread, the approximation accuracy can be improved by setting the coefficient e to be larger than 0. 
     In order to approximate the PSF of the imaging optical system, the elliptical distribution needs to be P(x, y)≥0. Therefore, when e&gt;0, P(x, y)&lt;0 in the peripheral part, but then it is only necessary to clip and set P(x, y)=0. 
     Referring now to  FIG. 11 , a description of a method of calculating the coefficient data.  FIG. 11  is a flowchart illustrating a method of calculating the coefficient data. Each embodiment calculates a coefficient (coefficient data) for generating an unsharp mask using the designed value of the imaging optical system  101 . 
     First, in the step S 1 , information necessary to calculate the coefficients is acquired. More specifically, the imaging condition in imaging for generating a captured image as a target for calculating the coefficient, and a target value (a threshold of an evaluation value E) corresponding to a shift between the PSF (designed PSF) of the imaging optical system  101  and the approximated PSF. 
     Subsequently, in the step S 2 , the designed PSF is calculated based on the designed value data of the imaging optical system  101  corresponding to the imaging condition acquired in the step S 1 .  FIG. 12  illustrates a section of the designed PSF calculated in the step S 2  and the approximated PSF generated in the step S 4  described later. As illustrated in  FIG. 12 , the designed PSF calculated in the step S 2  is discretized with a division number (tap number) N and an interval (discretization interval or sampling pitch) Do. The size (kernel size) of the designed PSF can be expressed by the product of the interval Do and the tap number N, as is apparent from  FIG. 12 . In other words, if any two of the interval Do, the tap number N, and the kernel size are known in the discretized designed PSF, the remaining one is uniquely determined. For example, if the interval Do=2.0 μm and the tap number N=11, the kernel size Do×(N−1)=20 μm. Do×N may be called the kernel size, then it is 22 μm. The interval Do is the pitch in fitting and may be smaller than the pixel pitch in the existing image sensor. The previous fitting with a small pixel pitch is applicable to the image sensors of various pixel pitches. While the designed value data of the imaging optical system  101  is used for fitting, data may be used which is obtained by imaging a chart or the like and by estimating the PSF of the imaging optical system  101  based on the captured image. 
     Next, in the step S 3 , initial values are set to the coefficients a, b, c, d, e, σ, β in approximating the PSF. Since each coefficient is updated (converted) in the subsequent processing, the step S 3  sets a temporary value as an initial value. 
     Next, the step S 4  creates the approximated PSF by substituting coefficients for Expression (8) and by approximating the PSF. This processing discretizes the approximated PSF in order to derive the optimum coefficient by performing fitting for the designed value. The number of divisions and intervals in the discretization are adjusted to the designed PSF calculated in the step S 2 . 
     Next, the step S 5  evaluates the shift between the designed PSF calculated in the step S 2  and the approximated PSF created in the step S 4 . As an index in evaluating the shift between the designed PSF and the approximated PSF, for example, a root mean square of a difference between the designed PSF and the approximated PSF is calculated and used as the evaluation value E. The smaller the evaluation value E is, the closer to the designed PSF the approximated PSF is. 
     Next, the step S 6  makes a determination using the evaluation value E calculated in the step S 5  and the target value acquired in the step S 1 . The evaluation value E may include not only information on the shift between the designed PSF and the approximated PSF but also information on the coefficients of the approximated PSF. This embodiment uses Expression (8) as a function (model) for approximating the PSF, but as described above, the coefficients a, b, and c must satisfy the relationship b 2 −ac&lt;0. Hence, when the coefficients a, b, and c do not satisfy this relationship or satisfy b 2 −ac≥0, a desired result cannot be obtained and setting a weight so as to increase the evaluation value E can more efficiently provide fitting. Where there is a restriction on a range that each coefficient can take, the evaluation value E is similarly changed, the fitting can become efficient, and the accuracy improves. 
     In the step S 6 , the thus-calculated evaluation value E is compared with a preset target value. When the evaluation value E is equal to or less than the target value, the fitting or the generation of the approximated PSF is completed, and the coefficient data that provides the approximated PSF is output. On the other hand, when the evaluation value E exceeds the target value, the fitting has not yet been fully performed and the flow proceeds to the step S 7 . 
     In the step S 7 , the coefficient (coefficient data) is changed (updated). Then, only one coefficient may be updated or a plurality of coefficients may be updated. After the coefficient is updated, the flow returns to the step S 4 , the approximated PSF is calculated again, and the evaluation value E is calculated in the step S 5 . Then, in the step S 6 , the processing from the step S 4  to the step S 7  is repeated until the evaluation value E converges below the target value. Unless the updated evaluation value E is smaller than the evaluation value E before the coefficient is updated, the flow may return to the pre-update coefficient and restart the processing from the step S 4 , or return to the step S 3  so as to escape from the local solution and reset the initial value. 
     Through the above calculation processing of the coefficients (coefficient data), the PSFs for various imaging conditions and for the various imaging optical systems (if exchangeable)  101  are previously converted into coefficients, and the data (coefficient data) can be stored in the memory  112 . When the coefficients for the approximated PSF are previously calculated in this manner, the approximated PSF according to the imaging optical system  101  and imaging conditions can be reconstructed (reproduced) only by acquiring the calculated coefficient data in the stage of the sharpening processing. 
     Discretization Processing 
     Referring now to  FIGS. 15A and 15B , a description will be given of discretization processing in creating discrete data from a continuous function used to approximate the PSF of the imaging optical system.  FIG. 15A  is a sectional view of an approximated PSF illustrating the discretization processing for the continuous function, where a solid line represents the continuous function, a white dot represents discrete data, an abscissa axis denotes a coordinate (pixel position), and an ordinate axis denotes the luminance. One rectangular section from P 1  to P 9  on the abscissa axis corresponds to the pixel size (pixel pitch) of the image sensor. A white dot is obtained from a median value in each section from a continuous function for each pixel position, and the discrete data can be made from the continuous function by simply taking a median value as illustrated by the white circle in  FIG. 15A . 
     Since the pixel has a finite size, accurate discrete data cannot be obtained only by taking the value of the continuous function located at the center of the pixel at pixel pitch intervals. In order to acquire accurate discrete data based on the pixel size, it is necessary to perform an integration in each pixel section. In other words, the discretization can be properly performed by repeating the integral calculation in the x-axis and the y-axis directions for Expressions (8). However, this integration processing has a lame processing load and the imaging apparatus has difficulties in executing it. 
     Accordingly, the discretization processing of the approximated PSF according to this embodiment more accurately discretizes while suppressing the processing load. The discretization caused error is small, if a difference between an integral value (average value) in the pixel section and the median value is small. Therefore, a more linear change in the continuous function in the section can reduce the error, and if the change is linear, the median value and the average value accords with each other. Conversely, the error increases if the change is nonlinear in the section and the slope in the continuous function monotonically increases or decreases. In general, the PSF of the imaging optical system changes significantly near the peak value, and the kurtosis is often large. In  FIG. 15A , in P 5  having the peak value of the continuous function in the section, the shift between the median value and the average value is large, and the median value is larger than the median value. 
     In this way, if there is a shift and the median value is taken as illustrated by the white dot in  FIG. 15A , the discretized approximated PSF data has a smaller distribution than original one due to the normalization processing. Herein, the normalization processing is processing of maintaining the sum of discrete data to 1, and a detailed description will be given later. 
     Referring now to  FIG. 15B , a description will be given of an adjustment method (discretization processing) according to this embodiment. A cross mark in  FIG. 15B  indicates a representative point acquired in the discretization. In  FIGS. 15A and 15B , sp represents the discretization interval (sampling pitch), the discretization is made with the median value in each interval in  FIG. 15A , and the discretization interval sp corresponds to the pixel pitch in the image sensor. On the other hand, the discretization interval sp in  FIG. 15B  is discretized at intervals different from the pixel pitch of the image sensor. More specifically, an interval smaller than the pixel pitch is set to the cross mark in  FIG. 15B . For example, the interval (sp=0.9 p) is 0.9 times as long as the pixel pitch p for the discretization. 
     As described above, since the PSF has a rapidly increasing distribution value towards the peak value, the discretization at intervals smaller than the pixel pitch makes the representative point indicated by the cross mark has a value larger than the median value of the white dot in  FIG. 15A . In other words, when the cross mark and the white dot are compared with each other, the value in the section P 5  including the peak value does not change and the value of the cross mark becomes larger in each of the other sections, so that the discretized distribution more widely spreads relatively. Thus, this embodiment reduces the discretization error by widening the distribution through the discretization at intervals different from the pixel pitch, in order to solve the problem of the distribution having the spread narrower than the original one in the method of obtaining the median value. 
     The image sensor having the Bayer array has the same colors alternately. Based on the Bayer array, the discretization interval sp in  FIG. 15A  may be twice as long as the pixel pitch p (sp=2p). Then, the discretization interval sp in  FIG. 15B  also needs to be doubled, and in adjusting with the interval that is 0.9 times as long as the pixel pitch as in the above example, the discretization interval sp is doubled or 1.8 times (Sp=1.8p). In  FIG. 15B , each section has a corresponding representative value. However, it is not always necessary to set a representative value in each section and, for example, a representative value corresponding to the section P 9  may be located in a section of the section P 8 . 
     Referring now to  FIG. 16 , a description will be given of processing when the pixel pitch changes and the imaging apparatus  100  is a lens interchangeable camera.  FIG. 16  illustrates the relationship between the pixel pitch and the discretization interval. In  FIG. 16 , the abscissa axis represents the pixel pitch p, the ordinate axis represents the adjusted discretization interval sp, and the pixel pitches p 1  and p 2  represent the minimum pixel pitch and the maximum pixel pitch in the reconstruction. Where p is the pixel pitch and f(p) is the adjusted discretization function (discretization interval), f(p) can be expressed by a linear function as in the following Expression (9).
 
 F ( p )= Ap+B   (9)
 
     In Expression (9), A and B are discretized coefficients (discretization adjustment coefficients) in the discretization. While the discretization caused error has been described with reference to  FIGS. 15A and 15B , the discretization also depends on the size of the pixel pitch. When each section in  FIGS. 15A and 15B  is sufficiently small, the change of the continuous function in the section can be regarded as an approximately linear change. Thus, the average value and the median value in each section almost accord with each other and, as illustrated by the white dot in  FIG. 15A , the discretization caused error is small even if the discretization is made only with the median value. In other words, the smaller the pixel pitch is, the smaller the difference is between when the inside of the pixel is integrated for the accurate calculation and where only the median value in the section is taken. Thus, in the correction (adjustment) illustrated in  FIG. 15B , the correction (adjustment) level increases for a larger pixel pitch and the correction (adjustment) level decreases for a smaller pixel pitch. 
     For a correction (adjustment) according to the pixel pitch, this embodiment uses the function of Expression (9). With Expression (9), the discretization coefficient A is a slope for the pixel pitch p and can be adjusted by a ratio to the pixel pitch p. As in the above example, when the discretization interval sp is 0.9 times as long as the pixel pitch p, the discretization coefficients A and B of the discretization function f (p) are A=0.9 and B=0. In adjusting the pixel pitch with Expression (9), the discretization interval needs to be smaller in a region with a large pixel pitch to be adjusted and the discretization coefficient A may fall within a range between 0 and 1. 
     However, this embodiment is not limited to the discretization interval set to be smaller than the pixel pitch, and the discretization interval may be larger than the pixel pitch if necessary. For example, the frequency characteristic can be controlled in the unsharp mask processing by setting the discretization interval to be different from the pixel pitch. When the discretization interval is made larger than the pixel pitch, the gain of the frequency characteristic decreases, and when the discretization interval is made smaller than the pixel pitch, the gain of the frequency characteristic increases. Hence, in order to reduce the gain of the frequency characteristic in the unsharp mask processing, for example, the discretization interval may be set twice as long as the pixel pitch so that the discretization interval becomes larger than the pixel pitch. Then, the discretization coefficients A and B of the discretization function f(p) are A=2.0 and B=0, respectively. 
     The discretization coefficient B is independent of the pixel pitch, and all pixel pitches can be uniformly adjusted by setting the value of B while setting A=0. The discretization function (discretization interval) f(p) in Expression (9) is expressed as a linear function using the discretization coefficients A and B, but the linearity is not always necessary and a nonlinear power series function, an exponential function, a logarithmic function, etc. may also be used. 
     Specific embodiments will be described below. 
     First Embodiment 
     Referring now to  FIG. 2 , a description will be given of an imaging apparatus according to a first embodiment of the present invention.  FIG. 2  is a block diagram of the imaging apparatus  100  according to this embodiment. The imaging apparatus  100  includes a camera body (imaging apparatus body)  116  and an interchangeable lens (lens apparatus)  102  attachable to the camera body  116 . In the camera body  116 , a program for the sharpening processing (image processing method) to an input image (captured image) is installed in the memory (or storage unit)  109 , such as a ROM (memory) or a hard disk drive, and the sharpening processing is performed by an image processor (image processing apparatus)  105 . Instead of the memory  109 , the program of the image processing method according to this embodiment may be storage in a storage unit inside the image processor  105 . A circuit corresponding to the program may be designed, and the sharpening processing may be executed by operating the circuit. 
     The interchangeable lens  102  includes an imaging optical system  101 , an optical system controller  107 , a memory (or storage unit)  112 , a lens CPU  113 , and a contact unit  114 . The imaging optical system  101  includes an aperture stop (or diaphragm)  101   a  and a focus lens  101   b . In this embodiment, the interchangeable lens  102  is interchangeable to the camera body  116 . However, this embodiment is not limited to this example and is applicable to an imaging apparatus integrated with the camera body  116 . The memory  112  is a rewritable nonvolatile memory. The data stored in the memory  112  mainly includes information on an optical characteristic unique to the interchangeable lens  102 . The camera body  116  acquires this information from the interchangeable lens  102  and corrects the captured image based on the acquired information. 
     The memory  112  stores the coefficient data used to reconstruct (approximate) the PSF of the imaging optical system  101  and information on the PSF such as the adjustment coefficient m. The information is transmitted via the contact units  114  and  115  from the interchangeable lens  102  to the camera body  116  by communications. The camera body  116  generates a filter based on the information on the PSF transmitted from the interchangeable lens  102 , executes the correction processing, and generates a sharpened image. The information on the PSF may be stored in the memory  109  in the camera body  116 . When the interchangeable lens  102  is attached to the camera body  116 , the coefficient data and the like stored in the memory  112  in the initial communication can be transferred to and stored in the memory  109  via the lens CPU  113  and the camera CPU  111 . In this embodiment, the information on the PSF such as the coefficient data and the adjustment coefficient m stored in the memory  112  or the memory  109 , is information corresponding to the imaging optical system  101  in the interchangeable lens  102 . 
     The lens CPU  113  serves as a communication circuit (communicator) for communications between the interchangeable lens  102  and the camera body  116 , reset exception processing, an A/D converter, a timer, an input/output port, a built-in ROM, and a built-in RAM. The communication circuit communicates between the interchangeable lens  102  and the camera body  116  by a communication method including control information corresponding to an imaging mode (motion imaging mode, still imaging mode). The optical system controller  107  is a lens controller that controls each component in the interchangeable lens  102 , and driving of an optical element, such as a lens and a diaphragm, based on the instruction from the lens CPU  113  and control information obtained via the communication circuit. The contact unit  114  is a connector that includes a plurality of metal contacts for communications between the interchangeable lens  102  and the camera body  116 , and electrically connects the lens CPU  113  and the camera CPU  111  to each other. 
     The camera body  116  includes an optical low-pass filter  103 , an image sensor  104 , an image processor  105 , a display unit  106 , a state detector  108 , a memory  109 , an image recording medium  110 , a camera CPU  111 , and a contact unit  115 . The image sensor  104  is a two-dimensional image sensor, such as a CCD (Charge Coupled Device) sensor or a CMOS (Complementary Metal-Oxide Semiconductor) sensor. The image sensor  104  photoelectrically converts an object image (optical image, imaged light) obtained via the imaging optical system  101  and the optical low-pass filter  103  to generate a captured image. The object image is photoelectrically converted by the image sensor  104  and is converted into an analog signal (electric signal). The analog signal is converted into a digital signal by an unillustrated A/D, and the digital signal is input to the image processor  105 . 
     The image processor  105  is an image processor that performs predetermined processing and predetermined unsharp mask processing for a digital signal. In this embodiment, the image processor  105  in the camera body  116  performs the sharpening processing, but a personal computer (PC) or a dedicated apparatus may perform the sharpening processing as an image processing apparatus. The image processor  105  acquires the imaging condition (imaging condition information) of the imaging apparatus  100  from the state detector  108 . The imaging condition information is information on an image height, an aperture value (F-number), an imaging distance, and a focal length of the zoom lens. The state detector  108  can acquire the imaging condition information directly from the camera CPU  111 , but the present invention is not limited to this example. For example, the imaging condition information on the imaging optical system  101  can be acquired from the optical system controller  107  via the lens CPU. 
     As illustrated in  FIG. 9 , the image processor  105  includes an acquirer  1051 , a reconstruction processor  1052 , and a sharpening processor  1053 , and performs the image sharpening processing for an input image (captured image). The output image processed by the image processor  105  is stored in the memory  109  in a predetermined format. The memory  109  serves to store the relationship between the imaging condition of the imaging optical system  101  and the PSF of the imaging optical system. Where an image processing apparatus that executes unsharp mask processing is separate from the image processor  105 , the camera CPU  111  may store aberration information in association with the captured image. 
     The display unit  106  can display an image obtained by performing predetermined processing for display use after the sharpening process. The display unit  106  may display an image obtained by performing simple processing for high-speed displaying. The series of above processing is controlled by the camera CPU  111 . 
     The imaging optical system  101  may include an optical element, such as a low-pass filter and an infrared cut filter. Where an optical element is used that affects the PSF, such as a low-pass filter, the accurate sharpening processing is available when the influence of this optical element is considered in creating the unsharp mask. Even when an infrared cut filter is used, since each PSF of the RGB channel (RGB color component) which is the integral value of the PSF of the spectral wavelength, in particular, the PSF of the R channel is influenced, the influence may be considered in creating the unsharp mask. 
     Referring now to  FIG. 1 , a description will be given of an image processing method according to this embodiment.  FIG. 1  is a flowchart illustrating the image processing method according to this embodiment. The flowchart illustrated in  FIG. 1  can be implemented as a program (image processing program) that enables a computer to execute the function of each step. This is true of other flowcharts in other embodiments. Each step in  FIG. 1  is executed by the image processor  105  based on a command from the camera CPU  111 . 
     First, in the step S 11 , the image processor  105  (acquirer  1051 ) acquires a captured image as an input image generated via the optical system (imaging optical system  101 ). The color component data as a correction target used as the input image is, for example, G-channel image data after demosaicing. Alternatively, the R-channel image data, the B-channel image data, and all RGB-channel image data, or image data before demosaicing may be used. 
       FIG. 8  is a schematic diagram of a Bayer array as a discrete regular array. For example, the processing may be applied to the input image that is data of each channel of RGB or that is data of a particular channel. Alternatively, as illustrated in  FIG. 8 , the G channel may be divided into two or G 1  and G 2  so as to provide four channels in total. The configuration where the G channel is divided into two enables the image data for each of R, G 1 , G 2 , and B to have the same resolution, and facilitates image processing and data processing. 
     Next, in the step S 12  in  FIG. 1 , the image processor  105  (acquirer  1051 ) acquires data (coefficient data) of the coefficients a, b, c, d, e, σ, and β of the function (predetermined function) in Expression (8) used to reconstruct the PSF of the imaging optical system  101  according to the imaging condition. In order to generate an approximated PSF corresponding to a certain image point, it is not always necessary to acquire data of all of these coefficients. For example, the above rotationally symmetrical shape is obtained for the on-axis PSF, and a=c, b=0 and d=1 are established. 
     The coefficient β is an exponential term and causes a heavy processing load when the coefficient β is made variable according to the PSF. Thus, β may be fixed to 1. When the coefficient β is fixed, a shape range that can be expressed becomes narrower than that when the coefficient β is not fixed but a coefficient data amount stored in the memory  112  or  109  and the processing load can be reduced. The number of coefficients may be increased in order to improve the approximation accuracy. For example, it is difficult to provide highly accurate fitting with a continuous function for the PSF having a narrowly spread distribution and a high peak, and thus the peak value or a value near the peak of the PSF may be directly set as the coefficient. The approximation accuracy can be improved by thus directly setting to the coefficient an area in which the distribution abruptly changes, and by setting the area to be reproduced with the function to another area. 
     In the step S 12 , the image processor  105  (acquirer  1051 ) may acquire the adjustment coefficient m for the sharpening processing in addition to the coefficient data. In addition, the step S 12  acquire the discretization coefficient (discretion adjustment coefficient) to be used to reconstruct the approximated PSF. The discretization adjustment coefficient is the coefficient data contributing to the reduction of the discretization caused error in discretizing the continuous function, and when Expression (9) is adjusted, the discretization coefficients A and B are acquired. 
     Next follows a description of the unsharp mask USM with reference to  FIGS. 7A and 7B .  FIG. 7A  is a schematic view of the unsharp mask, and  FIG. 7B  is a schematic sectional view of the unsharp mask. The number of taps for the unsharp mask is determined depending on the aberration characteristics of the imaging optical system and the required sharpening accuracy. The unsharp mask USM illustrated in  FIG. 7A  is, for example, a two-dimensional mask having 11×11 taps. The unsharp mask USM illustrated in  FIG. 7A  is a two-dimensional filter in which filter coefficients are rotationally asymmetrically distributed. 
       FIG. 7A  omits a value (corresponding to the filter coefficient) in each tap, and  FIG. 7B  illustrates one section of the unsharp mask. In  FIG. 7B , the solid line represents the section of the unsharp mask USM, the abscissa axis represents the tap, and the ordinate axis represents the tap value. An ideal distribution of a signal value (PSF of the imaging optical system) that spreads due to the aberration is a distribution of each tap value (coefficient value) of the unsharp mask USM. 
     According to this embodiment, the image processor  105  generates the approximated PSF using the coefficient data, and an unsharp mask USM corresponding to the approximated PSF. Hence, a data amount to be stored can be much smaller while the correction accuracy is maintained than that where the data corresponding to the PSF of the imaging optical system  101  is directly stored. For example, as illustrated in  FIGS. 7A and 7B , the unsharp mask USM with 11×11 taps needs to have data of 121 tap values. In order to separately store the RGB data, a data amount becomes a triple and it is necessary to store data of 363 tap values. On the other hand, when the coefficients are stored, Expression (8) has seven coefficients or twenty-one coefficients for the RGB colors. Thus, storing the coefficient data results in saving a storage data amount. 
     Next, in the step S 13  in  FIG. 1 , the image processor  105  (reconstruction processor  1052 ) reconstructs the PSF (discretized PSF) with the coefficient data acquired in the step S 12  (reconstruction processing). The PSF is reconstructed based on the coefficient data and Expression (8) as a function utilized to calculate the coefficient data, and this embodiment utilizes the reconstructed PSF for the unsharp mask. 
       FIG. 14  is a sectional view of the reconstructed PSF. In reproducing an area A as the unsharp mask in  FIG. 14 , the coefficients may be generated by fitting a little wider area B. Thereby, when a tap number and pitch are later changed through an interchangeable lens  102  etc., a change may be made so as to widen the area. 
     This embodiment uses the discretization adjustment coefficient acquired in the step S 12 , in discretizing the continuous function such as Expression (8). The discretization interval is determined based on Expression (9) and the pixel pitch in the image sensor  104 . For example, when A=0.2 and B=1.6, the discretization interval is 2.0 [μm] if the pixel pitch is 2.0 [μm], and the pixel pitch is 6.0 [μm] if the pixel pitch is 2.8 [μm]. The lamer the pixel pitch is, the greater the influence of the discretization caused error is. Therefore, in the step S 13 , the adjustment processing is performed to reduce the influence. The above example does not execute the adjustment processing, when the pixel pitch is 2.0 [μm] and accords with the discretization interval. On the other hand, when the pixel pitch is 6.0 [μm], the discretization interval is about half of the pixel pitch and the original pixel pitch is discretized at fine intervals. 
     Next, the image processor  105  (reconstruction processor  1052 ) performs normalization processing for the discretized approximated PSF. The total value of data discretized from the continuous function at certain intervals differs according to the discretization interval and the tap number. However, when the discretized data is treated as the PSF, the sum total of the PSF data must be 1. The normalization processing according to this embodiment initially calculates the sum total of the tap values in the approximated PSF data. Next, the sum total of the calculated tap values is divided by the value of each tap in the approximated PSF data. Due to this division, the sum total of the tap values is always 1 for the approximated PSF data. This is used for the normalized approximated PSF data. Processing instead of the division may integrate a reciprocal of the sum total. Due to the influence of the decimal accuracy of the value stored in each tap, the sum total of the tap values may not be equal to 1 in the normalization, but a slight shift is permissible and this embodiment may provide an adjustment so that the sum total of the tap values is about 1. Although this example utilizes the normalization through the division, a non-division method may be used. One non-division method is to calculate the sum total of the tap values and then add the difference between 1 and the sum total to a specific tap (such as the tap having the maximum value and the center tap). 
     In generating an approximated PSF from a continuous function, such normalization processing is indispensable, so that the discrete data of the approximated PSF varies according to the normalization processing. In other words, as illustrated in  FIG. 15A , after the discretization, the tap value having the peak value becomes larger than the original value, and an entire lifting amount by the normalization processing is reduced by that amount, so that the approximated PSF after the normalization processing has a smaller distribution shape than the original one. This embodiment adjusts the discretization interval as illustrated in  FIG. 15B , in order to alleviate the influence, and generates an approximated PSF close to the original result obtained by integrating the inside of the pixel. Since this error is also influenced by the shape of the continuous function, the discretization coefficients A and B may be properly determined according to the performance of the lens and the like. Thus, the memory  112  in the interchangeable lens  102  may store the information on the discretization of the approximated PSF such as the discretization coefficients A and B. 
       FIG. 13A  illustrates a relationship between the position of the generated unsharp mask and the input image. A white dot illustrates a position of the generated unsharp mask. The input image is divided as illustrated in  FIG. 13A , and an unsharp mask is generated at eighty-one points. Then, the interpolation processing is performed for the unsharp mask, and thereby the unsharp mask can be generated at an arbitrary position in the input image so as to handle with the image height change of the PSF. Herein, the number of divisions is 9×9 in  FIG. 13A , but may be smaller for the lightweight purposes or larger for the accuracy purposes. Each point of the white dot in  FIG. 13A  may be generated through the interpolation rather than acquiring the direct PSF. 
       FIG. 13B  illustrates one example, and the unsharp mask is generated by the interpolation at each point. A black dot in  FIG. 13B  illustrates the unsharp mask to be generated in the step S 13 . In general, the PSF of the imaging optical system can be rotationally symmetrical and thus the unsharp mask is also rotationally symmetrical. Based on this characteristic, the example in  FIG. 13B  generates unsharp masks at ten points from the image center in the down direction, and the unsharp mask through the interpolation at a position corresponding to each white dot by rotating them around the image center. Thereby, it is unnecessary to produce the unsharp mask one by one at each point in the input image, and thus the processing load can reduce. The image height is treated as described above. 
     Next, in the step S 14  in  FIG. 1 , the image processor  105  (sharpening processor  1053 ) executes the sharpening processing for the captured image with the unsharp mask USM generated in the step S 13 . This embodiment uses the PSF of the imaging optical system for the unsharp mask USM, and thus can accurately correct and sharpen the input image even when the image is deteriorated by the asymmetrical PSF of the imaging optical system as seen in the periphery of the input image. 
     The image g(x, y) after the sharpening processing can be expressed as in the following Expressions (10), (11), and (12) based on Expression (1) and (3).
 
 g ( x,y )= f ( x,y )+ m×{f ( x,y )− f ( x,y )*USM( x,y )}  (10)
 
 g ( x,y )= f ( x,y )+ m×f ( x,y )*{δ( x,y )−USM( x,y )}  (11)
 
 g ( x,y )= f ( x,y )*{δ( x,y )+ m ×(δ( x,y )−USM( x,y ))}  (12)
 
     A brace in Expression (12) will be referred to as a sharpening filter for convenience. The sharpening filter can be generated with the unsharp mask USM and the adjustment coefficient m. The adjustment coefficient m is determined based on the overcorrection and the insufficient correction of the sharpening and the noises in the image. In the step S 14 , the sharpening processor  1053  executes the sharpening processing for the input image based on Expression (12) and the unsharp mask shaped in the step S 13 . 
     This embodiment discretely maintains the unsharp mask USM for the input image as illustrated in  FIG. 13A . The corresponding unsharp mask USM or sharpening filter is necessary for sharpening processing at the position other than the white dot in  FIG. 13A . This embodiment can perform the sharpening processing at an arbitrary position through a linear interpolation to the discretely generated unsharp mask USM. More specifically, the unsharp mask USM corresponding to a certain position is generated through a linear interpolation of the unsharp masks at four white dots near the certain point, and the sharpening processing is executed based on Expression (12). This configuration can provide the sharpening processing at an arbitrary position in the image, the continuously changing sharpening effect in the image, and a naturally sharpened image. The linear interpolation may be performed by the sharpening filter instead of the unsharp mask USM. 
     This embodiment explains the sharpening processing based on Expression (12), but may perform the sharpening processing using Expression (10) or Expression (11) for similar effects. Expression (10) or (11) adds a correction component to the input image but this expression is established where the adjustment coefficient m is positive. Where the adjustment coefficient m is negative, the correction component is subtracted from the input image. Thus, although the code of the adjustment coefficient m is different, this operation expresses the same meaning and thus any one of the operations may be used as lone as the code of the adjustment coefficient is properly adjusted. 
     This embodiment uses Expression (8) corresponding to the PSF of the imaging optical system and the approximated PSF generated from the coefficient for the unsharp mask USM. Therefore, it is possible to accurately sharpen the deterioration caused by the asymmetric PSF of the imaging optical system as seen in the periphery of the input image. In addition, this embodiment can generate an approximated PSF with a reduced discretization error rather than that with the simple discretization at the pixel pitch interval using the discretization adjustment coefficient, in reconstructing the approximated PSF. As described above, sharpening is performed using an approximated PSF in which the influence of the discretization error is reduced, whereby the sharpening processing accuracy can also be improved. 
     As described above, the image processing method according to this embodiment can execute accurate sharpening processing while reducing the information amount required for sharpening processing. 
     Second Embodiment 
     Next follows a description of an imaging apparatus according to a second embodiment of the present invention. This embodiment is different from the first embodiment in discretization method in the reconstruction processing, but the imaging apparatus according to this embodiment has the same configuration as that according to the first embodiment and a detailed description thereof will be omitted. 
     The image processing method according to this embodiment is different from the flowchart illustrated in  FIG. 1  in step S 13 , and thus a description will be given of the processing. In the step S 13 , the image processor  105  (reconstruction processor  1052 ) reconstructs the discretized PSF using the coefficient data acquired in the step S 12  (reconstruction processing). The PSF is reconstructed based on the coefficient data and Expression (8) as a function used in calculating the coefficient data. Assume that (i, j) is a discretized coordinate to calculate the discretized PSF. A relationship between (x, y) in Expression (8) and the discretized coordinate (i, j) can be expressed by the following Expressions (13a) and (13b). 
     
       
         
           
             
               
                 
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     Assume in Expressions (13a) and (13b) that A is a discretization interval, Δ f  is an interval in fitting, (2M+1) is the tap number in fitting, and (2n+1) is the tap number of the discretized PSF where N f  and n are integers. The discretized PSF can be expressed using Expressions (8), (13a), and (13b), and each tap value can be calculated by sequentially substituting integer values from −n to n for i and j. While the first embodiment performs the discretization processing in this way, this embodiment changes the discretization interval according to the position (discretizes at irregular intervals) in order to further improve the correction accuracy. More specifically, instead of substituting integer values for i and j, values are substituted at irregular intervals using decimal numbers without changing the range from −n to n. 
     As understood from the section of the PSF in  FIG. 15A , the vicinity of the peak value of the PSF is basically nonlinear and has a rapid change, and as the distance from the peak value decreases, the change amount decreases. The discretization caused error increases as the change in one pixel is large and nonlinear, and the discretization caused error also decreases when the change amount is small or linear. In the PSF in  FIG. 15A , the discretization caused error is likely to be large near P 5 , and as the distance is larger from the peak position as in P 1  or P 9 , the influence of the error reduces. This embodiment more finely discretized the vicinity of the peak position (or makes small the discretization interval), and makes longer the interval as the position is closer to the periphery (or increases the discretization interval). When the PSF is approximated by Expression (8), the peak is located at x=0 and y=0, the approximated PSF is generated by densely discretizing the vicinity of i=0 and j=0. 
     This embodiment performs the discretization processing at irregular intervals according to the shape of the PSF (or the position of the PSF), and can generate an approximated PSF with a reduced discretization error. 
     Third Embodiment 
     Next follows a description of an imaging apparatus according to a third embodiment of the present invention. The configuration of the imaging apparatus according to this embodiment is the same as that of the first and second embodiments, and a description thereof will be omitted. 
     Referring now to  FIG. 17 , a description will be given of an image processing method according to this embodiment.  FIG. 17  is a flowchart illustrating an image processing method according to this embodiment. The image processing method according to this embodiment differs from that of the first embodiment only in step S 33 . The steps S 31 , S 32 , S 34 , and S 35  are similar to the steps S 11  to S 14  in  FIG. 1 , respectively, and a description thereof will be omitted. 
     In the step S 33 , the image processor  105  (reconstruction processor  1052 ) updates (converts) the coefficient data acquired in the step S 32 . Based on Expressions (8), (13a), and (13b), the discretized PSF data can be converted as in the following Expression (14) (Expressions (14a) to (14c)). 
     When i≥0 and j≥0,
 
 P ( i,j )=(σ+ aγ   2   i   2 +2 bγ   2   ij+cγ   2   j   2 ) −β   −e   (14a)
 
     When i≥0 and j&lt;0,
 
 P ( i,j )=+(σ+ ay   2   x   2 +2 bdyγ   2   ij+cd   2 γ 2   j   2 ) −β   −e   (14b)
 
     When i&lt;0,
 
 P ( i,j )= P (− i,j )  (14c)
 
     In Expressions (14), γ=Δ/Δ f N f  and the discretization interval Δ corresponds to f(p) in Expression (9). Since Δ f  and N f  are coefficient data determined when the PSF is converted into a function, γ is determined by the pixel pitch in the image sensor  104 . The coefficients a, b, and c are multiplied by a square of γ in Expressions (14), and it is considered that these coefficients change due to the change in the pixel pitch. The step S 33  updates (converts) the coefficient data (generates updated coefficient data) by multiplying the coefficients a, b, and c among the coefficient data acquired in the step S 32  by the square of γ determined according to the pixel pitch and the discretization adjustment coefficient. For the coefficients other than the coefficients a, b, and c, there is no influence even when the pixel pitch changes. Hence, this embodiment uses the pre-update coefficient for the post-update coefficient data. 
     When the image sensor  104  generates the PSF data for each color in the Bayer array, it is necessary to double the discretization interval Δ since the same color is arranged every other pixel. In that case, doubling γ can handle this similarly. While the first embodiment performs the discretization processing at intervals corresponding to the pixel pitch in the reconstruction, this embodiment previously updates the coefficient data according to the pixel pitch and the reconstruction does not depend on the pixel pitch. The method according to the first embodiment and the method according to the third embodiment have the same reconstructed PSF data and sharpening result, but this embodiment includes the step of updating the coefficient data and further simplifies the subsequent reconstruction processing. 
     The processing following the step S 34  performs the reconstruction processing in the step S 13  according to the first embodiment and the sharpening processing in the step S 14  using the updated coefficient data generated in the step S 33 , and thereby executes accurate sharpening processing. 
     Other Embodiments 
     Embodiment(s) of the present invention can also be realized by a computer of a system or apparatus that reads out and executes computer executable instructions (e.g., one or more programs) recorded on a storage medium (which may also be referred to more fully as a ‘non-transitory computer-readable storage medium’) to perform the functions of one or more of the above-described embodiment(s) and/or that includes one or more circuits (e.g.; application specific integrated circuit (ASIC)) for performing the functions of one or more of the above-described embodiment(s), and by a method performed by the computer of the system or apparatus by, for example, reading out and executing the computer executable instructions from the storage medium to perform the functions of one or more of the above-described embodiment(s) and/or controlling the one or more circuits to perform the functions of one or more of the above-described embodiment(s). The computer may comprise one or more processors (e.g., central processing unit (CPU), micro processing unit (MPU)) and may include a network of separate computers or separate processors to read out and execute the computer executable instructions. The computer executable instructions may be provided to the computer, for example, from a network or the storage medium. The storage medium may include, for example, one or more of a hard disk, a random-access memory (RAM), a read only memory (ROM), a storage of distributed computing systems, an optical disk (such as a compact disc (CD), digital versatile disc (DVD), or Blu-ray Disc (BD)™), a flash memory device, a memory card, and the like. 
     The embodiments can provide an image processing apparatus, an imaging apparatus, an image processing method, a storage medium, and a lens apparatus which can perform accurate sharpening processing while reducing the amount of information required for sharpening processing. 
     While the present invention has been described with reference to exemplary embodiments, it is to be understood that the invention is not limited to the disclosed exemplary embodiments. The scope of the following claims is to be accorded the broadest interpretation so as to encompass all such modifications and equivalent structures and functions. 
     This application claims the benefit of Japanese Patent Application No. 2018-000742, filed on Jan. 5, 2018 which is hereby incorporated by reference herein in its entirety.