Patent Publication Number: US-2011063435-A1

Title: Position measuring target, position measurement system, calculation device for position measurement and computer-readable medium

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is based on and claims priority under 35 USC 119 from Japanese Patent Application No. 2009-210730 filed on Sep. 11, 2009. 
     BACKGROUND 
     1. Technical Field 
     The present invention relates to a position measuring target, a position measurement system, a calculation device for position measurement, and a computer-readable medium. 
     2. Related Art 
     Various technologies have been proposed as means for measuring the three-dimensional position of an object. 
     SUMMARY 
     According to an aspect of the invention, a position measuring target includes four or more reference points and a plurality of shading pattern portions. The four or more reference points are defined on a plane. The references points have positional relationships among them. The plurality of shading pattern portions corresponds a plurality of geometric curved surfaces in relation to the degree of shading used for defining the reference points. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Exemplary embodiments of the invention will be described in detail based on the following figures, wherein: 
         FIG. 1  is a diagram representing a position measurement system according to an embodiment of the present invention; 
         FIG. 2A  is a diagram representing a position measuring target according to an embodiment of the present invention; 
         FIG. 2B  is a diagram representing the degree of shading of a path denoted by X-Y shown in  FIG. 2A ; 
         FIG. 2C  is a diagram representing image positions and the degree of shading of a picked-up image; 
         FIG. 2D  is a diagram representing a geometric curved surface in a three-dimensional space which is denoted by the image positions and the degree of shading of a picked-up image; 
         FIGS. 3A and 3B  are diagrams representing an example of a method of defining a reference point for position measurement; 
         FIG. 4  is a block diagram representing a configuration example of a calculation device shown in  FIG. 1 ; 
         FIG. 5  is a flowchart representing an example of a process that is performed by a computer; 
         FIG. 6  is a diagram illustrating an example of a method of calculating a three-dimensional position of an object that has four reference points; 
         FIG. 7A  is a diagram of a position measuring target according to another embodiment of the present invention; 
         FIG. 7B  is a diagram representing the degree of shading of a path denoted by X-Y shown in  FIG. 7A ; 
         FIG. 8A  is a diagram of a position measuring target according to another embodiment of the present invention; 
         FIG. 8B  is a diagram representing the degree of shading of a path denoted by X-Y shown in  FIG. 8A ; 
         FIG. 9A  is a diagram of a position measuring target according to another embodiment of the present invention; 
         FIG. 9B  is a diagram representing the degree of shading of a path denoted by X-Y shown in  FIG. 9A ; 
         FIG. 10  is a diagram of a position measuring target according to another embodiment of the present invention; 
         FIG. 11  is a diagram of a position measuring target according to another embodiment of the present invention; and 
         FIGS. 12A and 12B  are diagrams of position measuring targets according to other embodiments of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a diagram representing a position measurement system according to an embodiment of the present invention. This system, as shown in the figure, includes: a position measuring target  1  that has a plurality of shading pattern portions forming a geometric curved surface in relation to the degree of shading used for defining four reference points a 1 , b 1 , c 1 , and dl; an image pickup device  3  that has a two-dimensional image pickup element  2  that picks up an image of the position measuring target  1 ; and a calculation device  4  that calculates four reference points a 1 , b 1 , c 1 , and d 1  based on the image of the image measuring target  1  that is picked up by the image pickup device  3  and calculates at least one of a three-dimensional position or an angle of the position measuring target  1  based on the calculated reference points. The reference points a 1 , b 1 , c 1 , and d 1  have the function of measurement markers that are used for measuring the three-dimensional position of the position measuring target  1 . 
     Here, as the image pickup device  3 , a digital camera in which a two-dimensional image pickup element  2  such as a CCD or a CMOS sensor is built is used. However, the image pickup device  3  is not limited thereto. The calculation device  4  is connected to communication means, not shown in the figure, of the image pickup device  3  in a wireless or wired manner for communicating with the image pickup device  3 . As the calculation device  4 , a computer such as a personal computer (PC) is used. However, the calculation device  4  is not limited thereto. In the example represented in  FIG. 1 , the number of the reference points is four. However, the number of the reference points may be five or more. This applies the same to examples described below. The configuration of the position measuring target  1  will be described in detail as below. A method of calculating the reference points by using a picked up image and a method of calculating the three-dimensional position or the angle of the target will be described later. 
       FIG. 2A  is a diagram representing a position measuring target according to an embodiment of the present invention.  FIG. 2B  is a diagram representing the degree of shading of a path denoted by X-Y shown in  FIG. 2A .  FIG. 2C  is a diagram representing the image position and the degree of shading of a picked-up image. As represented in  FIG. 2A , the position measuring target  1  includes an object  5  having four or more reference points a 1 , b 1 , c 1 , and d 1  of which the positional relationship defined on a plane is known and a plurality of shading pattern portions  21  to  24  that form a geometric curved surface in relation to the degree of shading used for defining the reference points. As the object  5 , a plate-shaped member such as a card or a substrate can be used. However, the object  5  is not limited thereto. Here, the geometric curved surface includes a planar surface as a special case of the curved surface. In this example, as represented in  FIG. 2B , the degrees of shading of the shading pattern portions  21  to  24  form a planar surface  25 . Accordingly, a plurality of intersection lines  27  are formed between the planar surface  25  of the shading pattern portions and the planar surface  26  of the object  5 . Here, the intersection line includes an extended line thereof. This applies the same to examples described below. At points in which the intersection lines  27  cross, the reference points a 1 , b 1 , c 1 , and d 1  for position measurement are defined. The reason for this will be described as below. 
     For example, the position measuring target  1  such as a pattern card having plural shading pattern portion is picked up by a camera as an example of the image pickup device  3 . The calculation device  4  obtains image information of the pattern card based on a imaging signal picked up by the camera. Then, the calculation device  4  extracts characteristic points based on the image information of the pattern card, and defines the characteristic points as reference points for position measurement. For extracting the characteristic points, for example, the angle, the center of a circle, an intersection of straight lines or curves, or the like of a target is corresponded.  FIG. 3A  is a diagram representing an example of this case.  FIG. 3B  is a diagram representing the degree of shading of a path denoted by X-Y shown in  FIG. 3A . In the example represented in  FIG. 3 , the reference points a 1 , b 1 , c 1 , and d 1  are extracted at the intersections of straight lines  31  to  34 . In this method, edge information of the image information is used. Edge information includes a position in which the intensity of the image level rapidly changes, and the position can be easily influenced by image noise. 
     Consequently, the image information including the degree of shading is used. The image information including the degree of shading is obtained from a position measuring target portion which has the plural shading pattern portions. The characteristic points are extracted by using the image information including the degree of shading, and the extracted characteristic points are set as the reference points for position measurement. Since the amount of information is large on the entire image planar surface or the entire image curved surface, the information is used. In this case, the image information including the degree of shading is composed of plural pixel. As shown in  FIG. 2D , each pixel  28  is represented by three-dimensional space. That is, each pixel  28  has three pieces of information, which are the pixel positions x and y, and the degree of shading z. Further, in the three-dimensional space, geometric curved surface is represented by the plural pixel  28 . The geometric curved surface includes a curved surface represented on the three-dimensional space by a whole series of the plural pixel  28  which are obtained based on one shading pattern portion of the position measuring target  1 , and the three-dimensional space is denoted by the pixel positions x and y, and the degree of shading z. The geometric curved surface includes at least one reference point. 
     The description will be followed with reference back to  FIGS. 2A to 2D . In  FIG. 2C , a plurality of black circles represent pixels  28  of the picked up image, and each pixel  28  is represented by three-dimensional coordinates of the pixel positions x and y and the degree of shading z. The planar surface  25  acquired from the picked up image is acquired based on three-dimensional positional information on each pixel  28 . For example, the calculation of the reference points a 1 , b 1 , c 1 , and d 1  can be performed as described below. However, the method of calculation of the reference points is not limited thereto, and a different method may be used. Now, the planar surface  25  will be focused. The equation of the planar surface is ax+by+cz=1, wherein a, b, and c are coefficients. The coefficients of the equation of the planar surface are determined by using the pixel position (xi, yi) and the degree of shading of each pixel of an image that belongs to the planar surface. 
         axi+byi+czi= 1 
     Here, in a case where image data of i=1 to N is used, [a b c]·Mi=1. 
     
       
         
           
             
               
                 
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     [a b c]·Mi·Mi −1 =1·Mi −1  (Mi −1  is an inverse matrix of Mi) 
     [a b c]=Mi −1    
     Accordingly, the coefficients a, b, and c can be calculated by using a least squares method. Then, an intersection line  27  of the planar surface  25  acquired here and the planar surface  26  of the object  5  is acquired. The intersection line is calculated by setting points at which the intersection lines  27  cross as the reference points a 1 , b 1 , c 1 , and d 1  for position measurement. This is an example of calculation of a planar surface. However, a curved surface such as a sphere surface can be calculated similarly. Then, the three-dimensional position or the angle of the target is calculated based on the calculated reference points. This calculation method will be described later. 
       FIG. 4  is a block diagram representing a configuration example of the calculation device shown in  FIG. 1 . The calculation device  4  includes: an input unit  41  to which an image of the position measuring target  1  picked up by the image pickup device  3  having the two-dimensional image pickup element  2  is input; a calculation unit (CPU)  42  that calculates four or more reference points for position measurement based on the input image and acquires at least one of the three-dimensional position or the angle of the position measuring target  1  based on the calculated reference points; and an output unit  43  that outputs to a display device, such as a monitor, at least one of the three-dimensional position and the angle of the position measuring target  1 , which has been calculated. 
     To the calculation unit  42 , a memory unit  44  is connected. Thus, information is transmitted and received between the calculation unit  42  and the memory unit  44 . In the memory unit  44 , a table, in which the positional information of four reference points a 1 , b 1 , c 1 , and d 1  is stored, is prepared. The calculation unit  42  acquires the stored positional information of the reference points from the memory unit  44  and calculates at least one of the three-dimensional position and the angle of the target based on the reference points that are calculated based on the picked up image. In addition, the memory unit  44  stores a program that is executed in the calculation unit  42  or various types of information used therein and can be configured as an internal memory. However, the memory unit  44  is not limited thereto and may be an externally connected memory device. 
     The above-described process can be performed by allowing a computer to execute the following program.  FIG. 5  is a flowchart representing an example of the processes that are performed by the computer. In other words, this program performs processes of the Step  51 , Step  52  and Step  53  to the computer. In the Step  51 , the process is performed to input an image of the position measuring target  1  which is picked up by the imaging device  3  having the two-dimensional image pickup element  2 . Specifically, the process is performed to obtain coordinates of the plural pixel including the plural shading pattern portion of the position measuring target  1 . Each pixel has coordinate information which are the pixel position x and y, and the degree of shading z. In the Step  52 , the process is performed to calculate four or more reference points for position measurement based on the input image. In the Step  53 , the process is performed to acquire at least one of the three-dimensional position and the angle of the position measuring target  1  based on the calculated reference points. Each shading pattern portion corresponds to the geometric curved surface. In this example, an embodiment in which the program is stored in the memory unit of the calculation device has been described. However, the program can be provided by being stored on a storage medium such as a CDROM or can be provided by the communication means. Hereinafter, an example of the method of calculating the three-dimensional position of the target will be described. 
       FIG. 6  is a diagram illustrating an example of a method of calculating a three-dimensional position of an object that has four reference points. In this example, the four reference points are defined, for example, as square angles, and two combinations of three reference points out of the four reference points will be considered. Then, two solutions are derived based on the following calculation by using three points, respectively. In one of the two solutions, the reference points represent the same values, and thus the solution is regarded as a correct solution. Accordingly, the position and the angle of the target can be determined. 
     First, as represented in  FIG. 6 , direction vectors Di (i=1, 2, or 3) of the positions of the reference points in a camera coordinate system are calculated based on the relationship between the positions k 1 , k 2 , and k 3  of the image on the image surface  10  (the two-dimensional image pickup surface of the camera) of the reference points a 1 , b 1 , and c 1  and the optical center  20  of the camera. Here, Di is a normalized unit vector. 
     When the spatial position vectors of the reference points a 1 , b 1 , and c 1  are p 1 , p 2 , and p 3 , the position vectors exist on extended lines of Di. Thus, the position vectors can be represented as in Numeric Expression 2-1 by using coefficients of t 1 , t 2 , and t 3 . 
     
       
         
           
             
               
                 
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     The shapes of the triangles are known in advance. Thus, when the lengths are assumed to be as represented in Numeric Expression 2-2, the following equations are acquired. 
     
       
         
           
             
               
                 
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     When being organized, Numeric Expression 2-4 is acquired. 
     
       
         
           
             
               
                 
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                                
                               2 
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                      
                                     x 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                   + 
                                   
                                     y 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                      
                                     y 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                   + 
                                   
                                     z 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                      
                                     z 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               t 
                                
                               
                                   
                               
                                
                               
                                 2 
                                 ⋀ 
                               
                                
                               2 
                             
                             - 
                             
                               L 
                                
                               
                                   
                               
                                
                               
                                 1 
                                 ⋀ 
                               
                                
                               2 
                             
                           
                           = 
                           0 
                         
                       
                     
                     
                       
                         
                           
                             
                               t 
                                
                               
                                   
                               
                                
                               
                                 2 
                                 ⋀ 
                               
                                
                               2 
                             
                             - 
                             
                               2 
                                
                               t 
                                
                               
                                   
                               
                                
                               2 
                                
                               t 
                                
                               
                                   
                               
                                
                               3 
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     x 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                   
                                   + 
                                   
                                     y 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     y 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                   
                                   + 
                                   
                                     z 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     z 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               t 
                                
                               
                                   
                               
                                
                               
                                 3 
                                 ⋀ 
                               
                                
                               2 
                             
                             - 
                             
                               L 
                                
                               
                                   
                               
                                
                               
                                 2 
                                 ⋀ 
                               
                                
                               2 
                             
                           
                           = 
                           0 
                         
                       
                     
                     
                       
                         
                           
                             
                               t 
                                
                               
                                   
                               
                                
                               
                                 3 
                                 ⋀ 
                               
                                
                               2 
                             
                             - 
                             
                               2 
                                
                               t 
                                
                               
                                   
                               
                                
                               3 
                                
                               t 
                                
                               
                                   
                               
                                
                               1 
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                      
                                     x 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                   + 
                                   
                                     y 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                      
                                     y 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                   + 
                                   
                                     z 
                                      
                                     
                                         
                                     
                                      
                                     3 
                                      
                                     z 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               t 
                                
                               
                                   
                               
                                
                               
                                 1 
                                 ⋀ 
                               
                                
                               2 
                             
                             - 
                             
                               L 
                                
                               
                                   
                               
                                
                               
                                 3 
                                 ⋀ 
                               
                                
                               2 
                             
                           
                           = 
                           0 
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   
                     2 
                      
                     
                       - 
                     
                      
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
     Thus, the following equations are formed, wherein sign “sqrt” denotes a square root. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Numeric 
                      
                     
                         
                     
                      
                     Expression 
                      
                     
                         
                     
                      
                     2 
                      
                     
                       - 
                     
                      
                     5 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             t 
                              
                             
                                 
                             
                              
                             1 
                           
                           = 
                           
                             
                               A 
                                
                               
                                   
                               
                                
                               
                                 1 
                                 · 
                                 t 
                               
                                
                               
                                   
                               
                                
                               2 
                             
                             ± 
                             
                               sqrt 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             A 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               1 
                                               ⋀ 
                                             
                                              
                                             2 
                                           
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                       · 
                                       t 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       2 
                                       ⋀ 
                                     
                                      
                                     2 
                                   
                                   + 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     
                                       1 
                                       ⋀ 
                                     
                                      
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             t 
                              
                             
                                 
                             
                              
                             2 
                           
                           = 
                           
                             
                               A 
                                
                               
                                   
                               
                                
                               
                                 2 
                                 · 
                                 t 
                               
                                
                               
                                   
                               
                                
                               3 
                             
                             ± 
                             
                               sqrt 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             A 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               2 
                                               ⋀ 
                                             
                                              
                                             2 
                                           
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                       · 
                                       t 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       3 
                                       ⋀ 
                                     
                                      
                                     2 
                                   
                                   + 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     
                                       2 
                                       ⋀ 
                                     
                                      
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             t 
                              
                             
                                 
                             
                              
                             3 
                           
                           = 
                           
                             
                               A 
                                
                               
                                   
                               
                                
                               
                                 3 
                                 · 
                                 t 
                               
                                
                               
                                   
                               
                                
                               1 
                             
                             ± 
                             
                               sqrt 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             A 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               3 
                                               ⋀ 
                                             
                                              
                                             2 
                                           
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                       · 
                                       t 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       1 
                                       ⋀ 
                                     
                                      
                                     2 
                                   
                                   + 
                                   
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     
                                       3 
                                       ⋀ 
                                     
                                      
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   
                     2 
                      
                     
                       - 
                     
                      
                     5 
                   
                   ) 
                 
               
             
           
         
       
     
     Accordingly, A 1 , A 2 , and A 3  are as the following equations. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Numeric 
                      
                     
                         
                     
                      
                     Expression 
                      
                     
                         
                     
                      
                     2 
                      
                     
                       - 
                     
                      
                     6 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             A 
                              
                             
                                 
                             
                              
                             1 
                           
                           = 
                           
                             
                               x 
                                
                               
                                   
                               
                                
                               1 
                                
                               x 
                                
                               
                                   
                               
                                
                               2 
                             
                             + 
                             
                               y 
                                
                               
                                   
                               
                                
                               1 
                                
                               y 
                                
                               
                                   
                               
                                
                               2 
                             
                             + 
                             
                               z 
                                
                               
                                   
                               
                                
                               1 
                                
                               z 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             A 
                              
                             
                                 
                             
                              
                             2 
                           
                           = 
                           
                             
                               x 
                                
                               
                                   
                               
                                
                               2 
                                
                               x 
                                
                               
                                   
                               
                                
                               3 
                             
                             + 
                             
                               y 
                                
                               
                                   
                               
                                
                               2 
                                
                               y 
                                
                               
                                   
                               
                                
                               3 
                             
                             + 
                             
                               z 
                                
                               
                                   
                               
                                
                               2 
                                
                               z 
                                
                               
                                   
                               
                                
                               3 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             A 
                              
                             
                                 
                             
                              
                             3 
                           
                           = 
                           
                             
                               x 
                                
                               
                                   
                               
                                
                               3 
                                
                               x 
                                
                               
                                   
                               
                                
                               1 
                             
                             + 
                             
                               y 
                                
                               
                                   
                               
                                
                               3 
                                
                               y 
                                
                               
                                   
                               
                                
                               1 
                             
                             + 
                             
                               z 
                                
                               
                                   
                               
                                
                               3 
                                
                               z 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   
                     2 
                      
                     
                       - 
                     
                      
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     There are real roots, and thus the inside of the square root represented in Numeric Expression 2-5 becomes positive. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Numeric 
                      
                     
                         
                     
                      
                     Expression 
                      
                     
                         
                     
                      
                     2 
                      
                     
                       - 
                     
                      
                     7 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             t 
                              
                             
                                 
                             
                              
                             1 
                           
                           ≤ 
                           
                             sqrt 
                              
                             
                               ( 
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 
                                   3 
                                   ⋀ 
                                 
                                  
                                 
                                   2 
                                   / 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         A 
                                          
                                         
                                             
                                         
                                          
                                         
                                           3 
                                           ⋀ 
                                         
                                          
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             t 
                              
                             
                                 
                             
                              
                             2 
                           
                           ≤ 
                           
                             sqrt 
                              
                             
                               ( 
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 
                                   1 
                                   ⋀ 
                                 
                                  
                                 
                                   2 
                                   / 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         A 
                                          
                                         
                                             
                                         
                                          
                                         
                                           1 
                                           ⋀ 
                                         
                                          
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             t 
                              
                             
                                 
                             
                              
                             3 
                           
                           ≤ 
                           
                             sqrt 
                              
                             
                               ( 
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 
                                   2 
                                   ⋀ 
                                 
                                  
                                 
                                   2 
                                   / 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         A 
                                          
                                         
                                             
                                         
                                          
                                         
                                           2 
                                           ⋀ 
                                         
                                          
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   
                     2 
                      
                     
                       - 
                     
                      
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
     By sequentially substituting real numbers t 1 , t 2 , and t 3  that satisfy these conditions in Numeric Expression 2-5, all t 1 , t 2 , and t 3  that satisfy Numeric Expression 2-5 are calculated. Next, p 1 , p 2 , and p 3 , that is, the three-dimensional positions of the reference points are calculated from the above-described Numeric Expression 2-1. In a case where there are three reference points, one position has two solutions. However, there are four reference points in this example. Thus, the above-described calculation is performed for the other three reference points, for example, a 1 , b 1 , and d 1 , so that other two solutions are derived. In one solution of the two solutions, the positions of the reference points represent a same value, and thus the solution is regarded as a correct solution. Even in a case where there are four or more reference points, the above-described process is performed similarly. As described above, the three-dimensional position of the target can be determined. The angle of the target can be acquired as a direction in which the target faces from the three-dimensional position. The method of calculating the three-dimensional position of the target is not limited to the above-described method, and a different method may be used. 
       FIG. 7A  is a diagram of a position measuring target according to another embodiment of the present invention.  FIG. 7B  is a diagram representing the degree of shading of a path denoted by X-Y shown in  FIG. 7A . As represented in  FIG. 7A , the position measuring target  1  of this example includes an object  5  having four or more reference points a 1 , b 1 , c 1 , and d 1  of which the positional relationship defined on a plane is known and a plurality of shading pattern portions  21  to  24  that form a geometric curved surface in relation to the degree of shading used for defining the reference points. In this example, other shading pattern portions  71  to  74  that are adjacent to the shading pattern portions  21  to  24  and form a geometric curved surface in relation to the degree of shading are included. Accordingly, a plurality of intersection lines  77  are formed by the geometric curved surfaces  21  to  24  and the geometric curved surfaces  71  to  74  adjacent thereto, and the reference points a 1 , b 1 , c 1 , and d 1  for position measurement are defined at points at which the intersection lines  77  cross each other. The three-dimensional position or the angle of the position measuring target  1  is calculated based on the reference points. 
       FIG. 8A  is a diagram of a position measuring target according to another embodiment of the present invention.  FIG. 8B  is a diagram representing the degree of shading of a path denoted by X-Y shown in  FIG. 8A . As represented in  FIG. 8A , the position measuring target  1  of this example includes an object  5  having four or more reference points a 1 , b 1 , c 1 , and d 1  of which the positional relationship defined on a plane is known and a plurality of shading pattern portions  81  to  84  that form a geometric curved surface in relation to the degree of shading used for defining the reference points. In this example, the geometric curved surface that is formed by the shading pattern portions  81  and  84  is, as represented in  FIG. 8B , a conic surface  85 . The reference points a 1 , b 1 , c 1 , and d 1  are defined as vertexes of the conic surface  85 . The three-dimensional position or the angle of the position measuring target  1  is calculated based on the reference points. 
       FIG. 9A  is a diagram of a position measuring target according to another embodiment of the present invention.  FIG. 9B  is a diagram representing the degree of shading of a path denoted by X-Y shown in  FIG. 9A . As represented in  FIG. 9A , the position measuring target  1  of this example includes an object  5  having four or more reference points a 1 , b 1 , c 1 , d 1 , and e 1  of which the positional relationship defined on a plane is known and a plurality of shading pattern portions  91  to  95  that form a geometric curved surface in relation to the degree of shading used for defining the reference points. In this example, the geometric curved surface that is formed by the shading pattern portions  91  and  95  is, as represented in  FIG. 9B , an ellipsoid of revolution  96 . The reference points a 1 , b 1 , c 1 , d 1 , and e 1  are defined as vertexes of the ellipsoid of revolution  96 . The three-dimensional position or the angle of the position measuring target  1  is calculated based on the reference points. The planar surface  97  of the object  5  of this example has, as represented in  FIG. 9B , the degree of shading that is the same as the vertex of the ellipsoid of revolution  96 . In this example, the geometric curved surface of the shading pattern portions  91  to  95  is the ellipsoid of revolution  96 . However, the geometric curved surface is not limited thereto and, for example, may be a spherical surface or the like. 
       FIG. 10  is a diagram of a position measuring target according to another embodiment of the present invention. The position measuring target  1  of this example has a basic configuration that is the same as that represented in  FIG. 2A . However, the shading pattern portions  101  to  104  are configured by using retroreflective members, which is different from that represented in  FIG. 2A . The retroreflective member has a structure in which incident light is reflected back to the incident side. The retroreflective member, for example, may be a concave corner cube or the like. However, the retroreflective member is not limited thereto. In this example, the shading pattern portions  101  to  104  are formed based on the sizes and the density of disposition of a plurality of reflective elements (in the figure, rectangles in which only characters are represented to be white) configuring the retroreflective member. However, the shading pattern portions  101  to  104  are not limited thereto. Thus, the shading pattern portions  101  to  104  may be configured based on one of the size and the density of the disposition of the reflective elements. In this example, same as represented in  FIG. 2B , the shading pattern portions  101  to  104  (the reflective elements are actually disposed to be more delicate with higher density than illustrated in the figure) form a planar surface in relation to the degree of shading. Accordingly, a plurality of intersection lines are formed by the planar surface of each shading pattern portion and the planar surface of the object  5 , and reference points a 1 , b 1 , c 1 , and d 1  for position measurement are defined at points at which the intersection lines cross each other. The three-dimensional position or the angle of the position measuring target  1  is calculated based on the reference points. 
       FIG. 11  is a diagram of a position measuring target according to another embodiment of the present invention. The position measuring target  1  of this example has a basic configuration that is the same as that represented in  FIG. 9A . However, the shading pattern portions  111  to  115  are configured by using retroreflective members, which is different from that represented in  FIG. 9A . In this example, the shading pattern portions  111  to  115  are formed based on the sizes and the density of disposition of a plurality of reflective elements (in the figure, rectangles in which only characters are represented to be white) configuring the retroreflective member. However, the shading pattern portions  111  to  115  are not limited thereto. Thus, the shading pattern portions  111  to  115  may be configured based on one of the size and the density of the disposition of the reflective elements. In this example, same as represented in  FIG. 9B , the shading pattern portions  111  to  115  (the reflective elements are actually disposed to be more delicate with higher density than illustrated in the figure) forms an ellipsoid of revolution in relation to the degree of shading. The reference points a 1 , b 1 , c 1 , d 1 , and e 1  are defined as vertexes of the ellipsoid of revolution. The three-dimensional position or the angle of the position measuring target  1  is calculated based on the reference points. 
       FIGS. 12A and 12B  are diagrams of position measuring targets according to other embodiments of the present invention. The position measuring target  1  of this example is used as a checker board that is used for correcting distortion of a camera lens. In the example represented in  FIG. 12A , a plurality of shading pattern portions  121  are arranged on an object  5 . The shading pattern portion  121  includes a plurality of shading pattern portions that form a planar surface in relation to the degree of shading as represented in the above-described  FIG. 2A . In this example, similarly to the case represented in the above-described  FIG. 2B , a plurality of intersection lines are formed by the planar surface of each shading pattern portion  121  and the planar surface of the object  5 , and a plurality of reference points  122  are defined at points at which the intersection lines cross each other. By setting the positions of actual reference points  122  to be in correspondence with the positions of the reference points  122  of a picked up image, the distortion of the lens can be corrected. In the example represented in  FIG. 12B , a plurality of the shading pattern portions  123  are arranged on the object  5 . The geometric curved surface that is formed by the shading pattern portion  123  is an ellipsoid of revolution as represented in the above-described  FIG. 9A . The reference points  124  are defined as the vertexes of the ellipsoid of revolution. By setting the positions of actual reference points  124  to be in correspondence with the positions of the reference points  124  of the picked up image, the distortion of the lens can be corrected. By disposing the shading pattern portions  121  and  123  with a precision higher than that illustrated in the figure, the advantage of correction for the distortion of the lens is improved. 
     The foregoing description of the exemplary embodiments of the present invention has been provided for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise forms disclosed. Obviously, many modifications and variations will be apparent to practitioners skilled in the art. The embodiments were chosen and described in order to best explain the principles of the invention and its practical applications, thereby enabling others skilled in the art to understand the invention for various embodiments and with the various modifications as are suited to the particular use contemplated. It is intended that the scope of the invention be defined by the following claims and their equivalents.