Abstract:
An apparatus for measuring three-dimension information of an object has an image reader, a corresponding point detector and a three-dimension information calculator. The image reader reads a pair of images recorded in a recording medium in the apparatus. Note that, the object is cylindrical and a pair of occluding contours is reflected in each of the pair of images. The corresponding point detector detects at least one pair of corresponding points, which is an imaginary pair of images of at least one measuring point positioned on a central axis, on a pair of bisecting lines. The three-dimension information calculator calculates the three-dimension position of the at least one measuring point on the basis of the at least one pair of corresponding points by applying a triangulation method.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a method and apparatus for measuring 3-D (Three-Dimension) information of an object from a pair of object images, which is obtained by capturing the object from two different directions. 
     2. Description of the Related Art 
     A “stereo”, or so called “binocular vision” is known as a method for measuring 3-D information of an object, such as position and form, from images. According to the stereo method, the object is captured from two different directions so that a pair of images is obtained. Then, a pair of corresponding points, each of which is an image of one point of the 3-D object, is obtained from the pair of images. Based on the pair of corresponding points and the distance between two capturing points, the position of the object is calculated using triangulation. The stereo method is, for example, utilized for photogrammetry, such as aerial photogrammetry and photogrammetry for a traffic accident spot. 
     When calculating the position of the object, it is important to detect the pair of corresponding points correctly. This corresponding points determining problem has been an important technical problem in the stereo method and various ways to overcome it have been proposed. 
     Among some of the ways is a way that detects the pair of corresponding points on the basis of a specific portion of the object, such as an edge or a ridge-line. However, when the shape of the object is a curved surface, such as a cylinder, as the specific portion on the object cannot be easily identified, incorrect corresponding points are detected, or the corresponding points cannot be detected. Consequently, the actual 3-D information of the object cannot be obtained. 
     SUMMARY OF THE INVENTION 
     Therefore, an object of the present invention is to provide a method and apparatus for correctly detecting a pair of corresponding points and measuring 3-D information of the object when an object has a curved surface. 
     An apparatus for measuring three-dimension information of an object according to the present invention has an image reader, a corresponding point detector, and a three-dimension information calculator. This apparatus is applied for photogrammetry, or computer vision in the AI (Artificial Intelligence) field. For example, in the case of photogrammetry for a traffic accident spot, an exclusive still camera with an image sensor is used. The camera is arranged at two capturing points in order to capture the object from two directions. The object to be captured is cylindrical, in other words, the object is a body of revolution. The object has substantially rotational symmetry with respect to a central axis of the object, and a given cross-section perpendicular to the central axis is a circle. For example, the object is a cylinder-shaped object or a frustum-shaped object and so on. When the object is captured, the image reader reads a pair of images recorded in a recording medium. For example, in the case of photogrammetry, the pair of images is recorded in a memory card detachably installed in the camera and apparatus. When the memory card is installed in the apparatus, the pair of images is read from the recording medium and is then temporarily stored in a memory, such as a RAM (Random Access Memory), by the image reader. The pair of images is obtained by capturing the object from two capturing points such that apair of contour lines, called “occluding contours” is reflected in each pair of images. 
     The object image in each of the obtained pair of images has line symmetry with respect to an imaginary projected image of the central axis, which bisects the object image. According to the apparatus, the corresponding point detector detects at least one pair of corresponding points on a pair of bisecting lines defined in the pair of images. The pair of corresponding points is an imaginary pair of images of a measuring point positioned on the central axis, and the pair of corresponding points is uniquely determined in the pair of images. Each of the pair of bisecting lines is an imaginary projected image of the central axis. The three-dimension information calculator calculates a three-dimension position of the at least one measuring point on the basis of the at least one pair of corresponding points. To calculate the three-dimension information, a triangulation method is applied. 
     According to the present invention, the imaginary pair of bisecting lines is defined, and then at least one pair of corresponding points is detected on the pair of bisecting lines. Consequently, the pair of corresponding points is detected correctly and 3-D information of the object is obtained correctly. 
     To detect the at least one pair of corresponding points correctly, preferably, an epipolar line, which is used in the photogrammetry or in the computer vision, is defined. The corresponding point detector defines at least one first image point on one of the pair of bisecting lines in one of the pair of images, and sets at least one epipolar line, corresponding to the at least one first image point, in the other image of the pair of images. Then, the corresponding point detector defines at least one second image point, which is a crossing point of the other of the pair of bisecting lines and the at least one epipolar line. The at least one first image point and the at least one second image point is defined as the at least one pair of corresponding points. 
     In the case of the photogrammetry, usually, an operator performs a given process for calculating the 3-D information using peripheral equipment, such as a keyboard or mouse. To calculating the 3-D information with the support of an operator, preferably, the apparatus includes a display and an indicating point inputting device. When the pair of images is read from the recording medium, the pair of images is displayed on the display. The indicating point inputting device is operated for inputting two pairs of indicating points on the pair of occluding contours in each of the pair of images. The corresponding point detector detects the two pairs of indicating points input by the operator, and calculates the pair of bisecting lines and the at least one pair of corresponding points in accordance with the two pairs of indicating points in each of the pair of images. In this case, the at least one pair of corresponding points is not automatically detected by a given image-processing method, such as an edge detecting process, but detected on the basis of the input points. 
     To measure a radius of the object with the 3-D position, preferably, the apparatus includes a radius calculator. In this case, the corresponding point detector detects two pairs of corresponding points, and the three-dimension information calculator calculates the positions of two measuring points on the basis of the two pair of corresponding points. The radius calculator firstly calculates a plane, in which a vector, perpendicular to the central axis and passing one of the two capturing points, is a normal vector and the central axis is included, from the positions of the two measuring points. Next, the radius calculator calculates an edge point, which is on the plane and a curved surface of the object, on the basis of an image point. The position along the central axis with respect to the edge point is the same as the position of one of the two measuring points. The image point is on one of the pair of occluding contours and corresponds to the edge point. Then, the radius calculator calculates a radius from the edge point and one of the two measuring points. 
     According to another aspect of the present invention, a method for measuring three-dimension information of an object includes steps of: 1) reading a pair of images recorded in a recording medium, the pair of images being obtained by capturing the object from two capturing points, the object being cylindrical, a pair of occluding contours being reflected in each of the pair of images; 2) detecting at least one pair of corresponding points, which is an imaginary pair of images of at least one measuring point positioned on the central axis and is uniquely determined in the pair of images, on a pair of bisecting lines defined in the pair of images, each pair of bisecting lines being imaginary projected image of the central axis; 3) calculating a three-dimension position of the at least one measuring point on the basis of the at least one pair of corresponding points by applying a triangulation method. 
     According to another aspect of the present invention, a memory medium that stores a program for measuring three-dimension information of an object. The program includes steps of: 1) reading a pair of images recorded in a recording medium in the apparatus, the pair of images being obtained by capturing the object from two capturing points, the object being cylindrical, a pair of occluding contours being reflected in each of the pair of images; 2) detecting at least one pair of corresponding points, which is an imaginary pair of images of at least one measuring point positioned on the central axis and is uniquely determined in the pair of images, on a pair of bisecting lines defined in the pair of images, each of the pair of bisecting lines being imaginary projected image of the central axis; 3) calculating a three-dimension position of the at least one measuring point on the basis of the at least one pair of corresponding points by applying a triangulation method. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present invention will be better understood from the description of the preferred embodiment of the invention set fourth below together with the accompanying drawings, in which: 
     FIG. 1 is a view showing a camera, an object, image information processor, and peripheral equipments. 
     FIG. 2 is a block diagram of the image information processor and peripheral equipment. 
     FIG. 3 is a view showing a pair of projected images. 
     FIG. 4 is a view showing a flowchart of a 3-D information calculating process. 
     FIG. 5 is a view showing a parallel stereo compensation. 
     FIG. 6 is view showing a pair of images displayed on a monitor. 
     FIG. 7 is a view showing a subroutine of a 3-D position calculating process. 
     FIG. 8 is a view showing one of the pair of images associated with a bisecting line. 
     FIG. 9 is a view showing the other of the pair of images associated with a bisecting line. 
     FIG. 10 is a view showing the pair of images associated with a pair of corresponding points. 
     FIG. 11 is a view showing a subroutine of a radius calculating process. 
     FIG. 12 is a view showing the object and a plane having normal vector. 
     FIG. 13 is a view showing a projected image formed by a weak perspective projection. 
     FIG. 14 is a view showing one of the pair of images associated with the radius calculation. 
     FIG. 15 is a view showing another of the projected images different from the images shown in FIG.  6 . 
     FIG. 16 is a view showing projected images of a frustum. 
     FIG. 17 is a view showing projected images of a body of revolution. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Hereinafter, the preferred embodiment of the present invention is described with reference to the attached drawings. 
     FIG. 1 is a schematic view showing a camera for photogrammetry, an image information processor for calculating 3-D (three-dimension) information of an object, and peripheral equipment. 
     In this embodiment, an object S is a cylinder and a camera is used for photogrammetry. Firstly, the camera  42  is arranged at a first capturing point PA and the object S is captured by the camera  42  so that an image including an object image is obtained. At this time, the object is captured such that a pair of occluding contours, which is a pair of contour line images, is reflected or projected in the image. Secondly, the camera  42  is arranged at a second capturing point PB and the object S is captured, similarly to the first capturing point PA. Consequently, a pair of images is obtained. Note that, a center of the camera  42 , namely, a lens-center of the camera  42  is positioned at the capturing points PA and PB respectively. Further, capturing data including a distance between the capturing points PA and PB, is recorded in the memory card  36  when capturing the object S. 
     The camera  42  is a digital still camera with a CCD  41  (Charge-Coupled Device). The pair of object images, which is formed on the CCD  41  via a lens (not shown) at the first and second capturing points PA and PB, are recorded in a memory card  36  in the camera  42 , respectively. The memory card  36  is an auxiliary memory device, which is detachably installed in the camera  42 . When the memory card  36  is installed in the card slot  11  of the image information processor  10 , the pair of images and the capturing data are read from the memory card  36 . Herein, the memory card  36  is a compact flash memory. After the capturing is finished, the image information processor  10  is used for calculating 3-D information of the object S. 
     The image information processor  10  calculates 3-D information of the object S, and a monitor  30 , a keyboard  32 , and a mouse  34  are connected to the image information processor  10 . A card slot  11  is provided on the image information processor  10 , the memory card  36  is detached from the camera  42  and then installed in the card slot  11 . 
     When a keyboard  32  is operated by an operator, the pair of images is displayed on the monitor  30 . Further, the mouse  34  is operated such that the 3-D information of the object S is obtained. 
     FIG. 2 is a block diagram of the image information processor  10  and peripheral equipment. 
     A system control circuit  12  including a CPU (Central Processing Unit)  12 A controls the image processor  10  and performs a calculation of the 3-D information associated with the object S. A program for calculating the 3-D information is stored in a ROM  14  in advance. A signal transmission between the system control circuit  12  and peripheral equipment, namely, the monitor  30 , the keyboard  32 , the mouse  34 , and the memory card  36 , are performed via an interface circuit  16 . 
     As described above, when the memory card  36  is installed into the card slot  11 , the pair of images and the capturing data are read from the memory card  36  and are temporarily stored in a RAM  18  via the interface circuit  16  and the system control circuit  12 . The keyboard  32  is operated to display the pair of object images, thus the pair of images is read from the RAM  18  and is subjected to various processes so that the image signals (video signals) for displaying the object image are generated at an image processing circuit (not shown) in the system control circuit  12 . The image signals are fed to the monitor  30  via the interface circuit  16 , thus the pair of images are displayed on the monitor  30 . The mouse  34  is operated to calculate the 3-D information of the object S, and the process for calculating the 3-D information is performed by the CPU  12 A in accordance with input information. The calculated 3-D information is displayed on the monitor  30  and is temporarily stored in the RAM  18 . 
     FIG. 3 is a view showing an object S and the pair of object images. 
     Herein, an image, in which the object image corresponding to the first capturing point PA is included, is represented as “π1” and an image, in which the object image corresponding to the second capturing point PB is included, is represented by “π2”. In the image π 1 , a pair of contour line images of the object image S 1  is represented by “MA1” and “MA2”. On the other hand, a pair of contour line images of the object image S 2  in the image π 2  is represented by “MB1” and “MB2”. The contour line images are called occluding contours in the stereo field. Further, the object images of the first and second capturing points PA and PB are represented by “S1” and “S2” respectively. 
     As well known, to calculate the 3-D position of the object S, a pair of corresponding points should be detected. Each of the corresponding points corresponds to an image of one point of the object S. When an outer surface of an object is a curved surface, it is difficult to correctly determine the pair of corresponding points from the pair of captured images. For example, as for a point JA, which is on a circumference of an upper surface UA, an image point JA′ of the point JA is on the contour line MA 2  in the image π 1 . On the other hand, in the image π 2 , an image point JB′ of a point JB, which is on the circumference, is on the contour line MB 2 . Though the point JA is different from the point JB, the image point JA′ is on the contour lines MA 2  and the image point JB′ is on the contour line MB 2 . Therefore, when determining the image points JA′ and JB′ as the pair of corresponding points, incorrect 3-D information is calculated. 
     The object S is cylindrical and has rotational symmetry with respect to a central axis SU of the object S. Therefore, considering imaginary projected images of the central axis SU in the images π 1 , π 2  (actually these image is not represented), the object images S 1  and S 2  become line symmetry images with respect to imaginary central axis images respectively. Herein, the central axis image in the image π 1  is represented by “SU1” and the central axis image in the image π 2  is represented by “SU2”. As the object S is a cylinder, the pair of contour lines, or the pair of occluding contours MA 1  and MA 2  is parallel to the central axis image SU 1  and the object image S 1  is bisected by the central axis image SU 1 . The object image S 2  is also bisected by the central axis image SU 2 . Therefore, as shown in FIG. 3, image points “C1” and “C2” in the images π 1  and π 2 , which are imaginary images of a point “C” on the central axis SU, are on the central axis images SU 1  and SU 2 , respectively. 
     Further, also as for any other position except for the first and second capturing points PA and PB, an obtained object image has line symmetry with respect to a projected image of the central axis SU, which bisects the obtained object image. Accordingly, in this embodiment, a given point on the central axis SU is defined as a measuring point having 3-D information of the object S. In this embodiment, a 3-D position of the object S is expressed by the position of the measuring point. For the measuring point on the central axis SU, the pair of corresponding points on the central axis images SU 1  and SU 2  are uniquely detected. Therefore, the measuring point can be obtained from the pair corresponding points by applying a triangulation. 
     FIG. 4 is a view showing a flowchart of a 3-D information calculating process. FIG. 5 is a view showing a compensated pair of images and FIG. 6 is a pair of images displayed on the monitor  30 . When the operator performs a given operation on the keyboard  32 , the 3-D information calculating process is started. 
     In Step  101 , the pair of captured images is read from the memory card  36  and is temporarily stored in the RAM  18 . Then, in Step  102 , the capturing data including camera positions and a camera posture is read from the memory card  36 . 
     In Step  103 , a distortion compensation and a parallel stereo compensation are performed at the system control circuit  12 . The distortion compensation compensates the distortion of the images π 1  and π 2 , which is caused by the characteristics of the lens in the camera  42 . In the parallel stereo compensation, an “affine transformation” is performed so that affine-transformed images π 1 ′ and π 2 ′ are obtained. The geometrical relationship between images π 1  and π 2  shown in FIG. 3 is changed to a relationship between the images π 1 ′ and π 2 ′, as shown in FIG.  5 . When defining 3-D coordinates (x, y, z) are defined at the capturing points PA and PB, a shift vector “SV” connecting the first capturing point PA and the second capturing point PB coincides with a direction of the x-coordinate in the transformed image π 1 ′ and the transformed image π 2 ′. This is different from coordinates-relationship between the images π 1 , π 2  (See FIG.  3 ). Note that, a revolutionary affine transformation is herein performed, and the distortion compensation and the parallel stereo compensation are well known process. 
     In Step  104 , a pair of images, obtained by performing the distortion compensation and the parallel stereo compensation, is displayed on the monitor  30 , as shown in FIG.  6 . Hereinafter, an image corresponding to the image π 1 ′ (See FIG. 5) is referred to a “first image IA” and an image corresponding to the image π 2 ′ is referred to a “second image IB”. After the pair of images IA and IB is displayed, the mouse  34  is operated by the operator to input a series of indicating points “Pa1 to Pd1” and “Pa2 to Pd2”. For the first image IA, the indicating points Pa 1  and Pb 1  are set on the occluding contour MA 1  and the indicating points Pc 1  and Pd 1  are set on the occluding contour MA 2 . Similarly, for the second image IB, the indicating points Pa 2  and Pb 2  are set on the occluding contour MB 1  and indicating points Pc 2  and Pd 2  are set on the occluding contour MB 2 . Note that, the indicating points Pa 1  and Pc 1  and the indicating points Pb 1  and Pd 1  are a pair respectively and the indicating points Pa 2  and Pc 2  and the indicating points Pb 2  and Pd 2  are also a pair respectively. When the indicating points are input by the mouse  34 , the positions of the indicating points on the images IA and IB are detected. After Step  104  is performed, the process goes to Step  105 . 
     In Step  105 , a position of the object S, namely, the position of the measuring point expressed by 3-D coordinates, is calculated. Then, in Step  106 , a radius of the object S is calculated on the basis of the 3-D position. 
     FIG. 7 is a view showing a subroutine of Step  105  in FIG.  4 . FIGS. 8 and 9 are views showing the first image IA and the second image IB, and FIG. 10 is a view showing the pair of images, in which an epipolar line is defined. For each of the first and second images IA and IB, the screen coordinates (X, Y) are defined and the origin is set to a left and upper corner of the first and second images IA and IB respectively. A pixel number of the first and second images IA and IB is “W×H”. Note that, the pixel number along the X-coordinate is “W” and the pixel number along the Y-coordinate is “H”. 
     In Step  201 , a straight line La 1  passing through the indicating points Pa 1  and Pb 1  is calculated in the first image IA, as shown in FIG.  8 . This straight line La 1  is on the occluding contour MA 1 . In Step  202 , a straight line Lb 1  passing through the indicating points Pc 1  and Pd 1  is calculated in the first image IA. This straight line Lb 1  is on the occluding contour MA 2 . Note that, the straight lines La 1  and Lb 1  are expressed by the screen coordinates (X, Y). 
     In Step  203 , a straight line Q 1  passing through the indicating points Pa 1  and Pc 1  is calculated in the first image IA (See FIG.  8 ). Similarly, In Step  204 , a straight line Q 2  passing through the indicating points Pb 1  and Pd 1  is calculated in the first image IA. In Step  205 , based on the straight lines La 1  and Lb 1 , a bisecting line Le 1  is calculated. As described above, the bisecting line Le 1  bisects the object image S 1  and corresponds to an imaginary projected image of the central axis SU (See FIG.  3 ). As the object S is a cylinder, the straight lines La 1  and Lb 1  are both parallel to the bisecting line Le 1 , further, a distance from the straight line La 1  to the bisecting line Le 1  is the same as the distance from the straight line Lb 1  to the bisecting line Le 1 . After Step  205  is performed, the process goes to Step  206 . 
     In Step  206 , as shown in FIG. 9, a straight line La 2 , passing through the indicating points Pa 2  and Pb 2 , is calculated. In Step  207 , a straight line Lb 2 , passing through the indicating points Pc 2  and Pd 2 , is calculated. The straight lines La 2  and Lb 2  correspond to the occluding contours MB 1  and MB 2  respectively. Then, In Step  208 , a bisecting line Le 2 , bisecting the object image S 2 , is obtained on the basis of the straight lines La 2  and Lb 2 . The bisecting line Le 2  also bisects the object image S 2  and corresponds to an imaginary projected image of the central axis SU, similarly to the bisecting line Le 1 . After step  208  is performed, the process goes to Step  209 . 
     In Step  209 , as shown in FIG. 10, a crossing point Pe 1  expressed by coordinates (Xa 1 , Ya 1 ), on which the straight line Q 1  intersects the bisecting line Le 1 , is calculated. In Step  210 , a crossing point Pf 1  expressed by coordinates (Xb 1 , Yb 1 ), on which the straight line Q 2  intersects the bisecting line Le 1 , is calculated. In Step  211 , an epipolar line EP 1  corresponding to the crossing point Pe 1  in the image IA is set in the second image IB, and a crossing point Pe 2  expressed by coordinates (Xa 2 , Ya 2 ), on which the epipolar line EP 1  intersects the bisecting line Le 2 , is calculated. The crossing point Pe 1  and the crossing point Pe 2  are defined as the pair of corresponding points, which is an imaginary image of a specific point on the central axis SU. Note that, as the parallel stereo compensation is performed at Step  103  (See FIGS.  4  and  5 ), the epipolar line EP 1  is parallel to the coordinate in the image IB and the passes the Y-coordinate “Ya1”, which is the Y-coordinate of the crossing point Pe 1  in the image IA. After Step  211  is performed, the process goes to Step  212 . 
     In Step  212 , an epipolar line EP 2 , which is parallel to the X-coordinate and passes through the Y-coordinate “Yb1”, is set. The Y-coordinate “Yb1” corresponds to the Y-coordinate of the crossing point Pf 1  in the image IA. Then, a crossing point Pf 2  expressed by coordinates (Xb 2 , Yb 2  (=Yb 1 )), on which the epipolar line EP 2  intersects the bisecting line Le 2 , is calculated. The pair of the corresponding points composed of the crossing points Pf 1  and Pf 2  is different from the pair of corresponding points Pe 1  and Pe 2 . When the pair of corresponding points Pe 1  and pe 2  and the pair of corresponding points Pf 1  and Pf 2  are calculated, the process goes to Step  213 . 
     In Step  213 , a coordinate transform is performed to calculate the 3-D position of the object S by applying the triangulation. Firstly, the coordinates (X, Y) defined on the first and second image IA and IB are transformed to CCD-coordinates (u, v) defined on an image-forming area of the CCD  41 , which corresponds to images π 1 ′ and π 2 ′ shown in FIG.  5 . In this transform, a translation for matching the origin position and scale transform are performed so that the pair of corresponding points Pe 1  and Pe 2  and the pair of corresponding points Pf 1  and Pf 2  are expressed by the CCD-coordinates (u, v) in place of the screen coordinates (X,Y). The coordinate transform is performed using the following formula (1). The coordinates (Xa 1 , Ya 1 ), (Xa 2 , Ya 2 ), (Xb 1 , Yb 1 ), (Xb 2 , Yb 2 ) are transformed to the CCD-coordinates (ua 1 , va 1 ), (ua 2 , va 2 ), (ub 1 , vb 1 ), (ub 2 , vb 2 ), respectively. 
     
       
           P′=R·P   (1) 
       
     
     Note that,        R   =     [             -   PitchX     ,           0   ,           PitchX   ·     W   /   2                 0   ,           PitchY   ,             -   PitchY     ·     H   /   2                 0   ,           0   ,         F         ]             P   =         [         X           Y           1         ]                     P   ′       =     [         u           v           F         ]                              
     In the formula (1), the “PitchX” indicates a ratio of a width of the image-forming area to a width of the image IA (or the image IB). The width of the image IA corresponds to the X-direction length. The “PitchY” indicates a ratio of a length of the image-forming area to a length of the image IA (or the image IB) corresponding to the Y-direction length. The “F” indicates a focal length of the camera  42 . Note that, the origin point of the CCD-coordinate (u, v) is defined at a central point of the image-forming area. Further, the unit of the CCD-coordinates (u, v) is a millimeter (mm) and the unit of the screen-coordinates (X, Y) is one pixel. As can be seen from the formula (1), a depth direction is treated in the CCD-coordinates, namely, the CCD-coordinates are expressed by 3-D coordinates (u, v and F). When the coordinates of the corresponding points Pe 1  and Pe 2  and the corresponding points Pf 1  and Pf 2  are transformed, the process goes to Step  214 . 
     In Step  214 , based on the CCD-coordinates (ua 1 , va 1 ) and (ua 2 , va 2 ), which correspond to the pair of corresponding points Pe 1  and Pe 2 , and the distance between the first and second capturing points PA and PB, a position of a measuring point “P1” is calculated by following formula. The measuring point P 1  is on the central axis SU and the crossing point Pe 1  and the crossing point Pe 2  are both an imaginary projected image of the measuring point P 1 . The distance between the first and second capturing points PA and PB is herein represented by “C”. The 3-D coordinates (x, y, z) are defined at the first capturing point PA, namely, the original position of the 3-D coordinates (x, y, z) are set to the first capturing point PA.                [         x           y           z         ]     =     [           u   ·     C   /   d                 v   ·     C   /   d                 F   ·     C   /   d             ]             (   2   )                                
     Note that,        d   =     {           ua1   -   ua2             or             ub1   -   ub2                                    
     The formula (2) is based on the triangulation, and the 3-D coordinates (x 1 , y 1 , z 1 ) of the measuring point P 1  are obtained by the formula (2). The z-coordinate “z1” of the measuring point P 1  indicates a depth from the first capturing point PA. In Step  215 , the coordinates (x 2 , y 2 , z 2 ) of a measuring point P 2  on the central axis SU, corresponding to the pair of corresponding points Pf 1  and Pf 2 , is obtained by the formula (2). The z-coordinate “z2” is the same as the z-coordinate “z1”. The two coordinates (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) are displayed on the monitor  30  and temporarily stored in the RAM  18 . After Step  215  is performed, this subroutine is terminated and the process returns to Step  105  in FIG.  4 . 
     Note that, while Steps  201  to  205  are performed, Data including the calculated straight line, the crossing point and so on, is temporarily stored and the stored data is read as required. 
     FIG. 11 is a view showing a subroutine of Step  106  in FIG.  4 . FIG. 12 is a view showing the object and a plane having a normal vector. FIG. 13 is a view showing a projected image formed by a weak perspective projection. FIG. 14 is a view showing one of the pair of images associated with the radius calculation. 
     In Step  301  shown in FIG. 11, firstly, a straight line “ml”, expressed by the 3-D coordinates (x, y, z) , is obtained from the coordinates (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) of the two measuring points P 1  and P 2 . The straight line “ml” is calculated by the following formula.                x   a     =       y   b     =     z   c               (   3   )                                
     Note that, 
     
       
         ( a,b,c )=( x   2 − x   1 ,  y   2 − y   1 ,  z   2 − z   1 ) 
       
     
     Then, as shown in FIG. 12, a vector “V=(e, g, f)” is calculated on the basis of the straight line “ml”. The vector “V” is a vector of a line extending from the origin of the 3-D coordinates (x, y, z), namely, the first capturing point PA to the straight line “ml”. The vector is perpendicular to the straight line “ml”. The vector “V” is calculated by the following formula, which indicates a vertical relationship between the straight line “ml” and the vector “V”. 
     
       
         ( e,f,g )·( a,b,c )=0  (4) 
       
     
     When the vector “V” is obtained, the process goes to Step  302 . 
     In Step  302 , a plane “R” is calculated, where the straight line “ml”, namely, the central axis SU is included and the vector “V=(e, f, g)” is normal-vector (See FIG.  12 ). The plane “R” is obtained by the following formula. 
     
       
           ex+fy+gz+d= 0  (5) 
       
     
     Note that,              d   =     -     (     ex1   +   fy1   +   gz1     )                   =     -     (     ex2   +   fy2   +   gz2     )                                    
     When the plane “R” is calculated, the process goes to Step  303 . 
     In FIG. 13, a relationship between a given point on the plane “R” and on the curved surface of the object S and a given point on the pair of occluding contours MA 1  and MA 2  in the image IA corresponding to the first capturing point PA, is shown. Herein, a direction of the vector “V” coincides with the depth direction, namely, the z-direction of the 3-D coordinates (x, y, z) for ease of explanation. On the image IA, a point on the contour line MA 1  is defined as an “edge point T1”. On the image-plane of the CCD  41 , an image point T 1 ′ corresponding to the edge point T 1  is defined. A straight line “N”, passing the origin position “O (=PA)” and the image point T 1 ′, is further defined. 
     In this embodiment, a “weak perspective projection” is applied as a projection method. The weak perspective projection is a combination of the orthographic projection and the perspective projection. Firstly, the object S is subjected to the orthographic projection for a plane “τ”, which is defined adjacent to the object S. Then, the projected image is subjected to the perspective projection for the origin position “O”. Note that, in the case of the weak perspective projection, it is regarded that a distance from the origin position “O” to the object S, represented by “M” in FIG. 13, is much longer than a radius “r” of the object S, in other words, the distance “M” is much longer than a length of the object S along the depth direction (z-direction). As the orthographic projection is performed along an optical axis direction of the camera  42 , namely, the normal vector “V”, the plane “τ” is parallel to the plane “R”. 
     When the point which is on the surface of the object S and on the plane “R” is defined as an “edge point T” and an image point of the edge point T on the plane “τ” is represented as an image point “T′”, the straight line N passes the image point T 1 ′ on the image-forming area in the CCD  41  and the image point “T′” on the plane “τ”. Based on the characteristic of the weak perspective projection, the straight line N is regarded as a straight line “N′” (shown by broken line), which passes the edge point “T” of the object S. Therefore, when the y-coordinate in the 3-D coordinates (x, y, z) with respect to the edge point “T” is the same as the y-coordinate of the measuring point P 1  or P 2  of the object S, the radius “r” is obtained by calculating the straight line “N” and the 3-D coordinates of the edge point “T”. Accordingly, in this embodiment, the edge point “T” is firstly calculated. 
     In Step  303 , a straight line Lg 1 , which is perpendicular to the bisecting line Le 1  and passes the crossing point Pe 1 , is calculated in the image IA (See FIG.  14 ). In Step  304 , a straight line Lh 1 , which is perpendicular to the bisecting line Le 1  and passes the crossing point Pf 1 , is calculated. In Step  305 , a screen boundary point Pg 1 , which is a crossing point of the straight line Lg 1  and the straight line La 1 , is obtained. In Step  306 , a screen boundary point Ph 1 , which is a crossing point of the straight line Lh 1  and the straight line La 1 , is obtained. 
     In Step  307 , the coordinate transform is performed for the crossing points Pg 1  and Ph 1 , similarly to Step  213  in FIG.  7 . Namely, the screen coordinates (X, Y) is transformed to the CCD-coordinates (u, v, F). In Step  308 , a straight line LA, which passes an image point P′g 1  on the image-forming area, corresponding to the screen boundary point Pg 1 , and passes the origin “O”, is calculated. In Step  309 , a straight line LB, which passes an image point P′h 1  on the image-forming area, corresponding to the screen boundary point Ph 1 , and passes the origin “O”, is calculated. The straight line LA or LB corresponds to the straight line N (N′) shown in FIG.  13 . When the straight lines LA and LB are obtained, the process goes to Step  310 . 
     In Step  310 , an edge point “PC”, which is a crossing point of the straight line LA and the object S, and is on the plane “R”, is calculated on the basis of the formula (5) and the straight line LA expressed by the 3-D coordinates (x, y, z). In Step  311 , an edge point “PD”, which is a crossing point of the straight line LB and the object S, and is on the plane “R”, is calculated on the basis of the formula (5) and the straight line LB expressed by the 3-D coordinates (x, y, z). The edge points “PC” and “PD” correspond to the edge point “T” shown in FIG.  13 . The y-coordinate of the edge points PC and PD coincides with the y-coordinate of the measuring points P 1  and P 2 , respectively. 
     In Step  312 , a distance “r1” from the edge point PC to the measuring point P 1  is calculated. In Step  313 , a distance “r2” from the edge point PD to the measuring point P 2  is calculated. As the object S is the cylinder, the distance “r1” is the same as the distance “r2”. The “r1” and “r2” indicate the radius of the object S. When Step  313  is performed, this subroutine is terminated. 
     With reference to FIGS. 15 to  17 , projected images and a shape of the object will be explained. 
     FIG. 15 is a view showing another captured images of the object S. When the pair of corresponding points is obtained as described above, the 3-D information can be calculated even if an upper surface or a bottom surface of the object S is not reflected in the captured images. 
     FIG. 16 is a view showing an object different from the cylinder. 
     According to the 3-D calculation process described above, 3-D information of a frustum E shown in FIG. 16, which has rotational symmetry with respect to a central axis ES, can be measured. In this case, straight lines Q 1  and Q 2  are firstly defined, then, a straight line passing a middle point between indicating points Pa 1  and Pc 1 , and passing a middle point between indicating points Pb 1  and Pd 1 , is defined as the bisecting line Le 1 . In the image IB, straight lines Q 1 ′ and Q 2 ′ are defined, then, a straight line passing a middle point between indicating points Pa 2  and Pc 2 , and passing a middle point between indicating points Pb 2  and Pd 2 , is defined as the bisecting line Le 2 . Then, the pair of corresponding points Pe 1  and Pe 2  is obtained by defining an epipolar line EP 1 . When the object S is the frustum, the radius corresponding to the pair of corresponding points Pe 1  and Pe 2  is different from that of the pair of corresponding points Pf 1  and Pf 2 . 
     FIG. 17 is a view showing an object different from the cylinder and the frustum. 
     The object RO shown in FIG. 17 is cylindrical having rotational symmetry with respect to a central axis RS. Across section perpendicular to the central axis RS is a circle. According to the 3-D calculation process described above, 3-D information of the object RO can be also calculated. In this case, the bisecting lines Le 1  and Le 2  are defined and the pair of corresponding points Pe 1  and Pe 2  is obtained by defining an epipolar line EP 1 , similarly to the object E shown in FIG.  16 . 
     In this way, in this embodiment, the positions of the measuring points P 1  and P 2  are obtained in accordance with the pair of corresponding points Pe 1  and Pe 2 , and the pair of corresponding points Pf 1  and Pf 2 . Further, the radius of the object is calculated on the basis of the measuring points P 1  and P 2 . 
     Note that, when the radius is not calculated, only the pair of corresponding points Pe 1  and Pe 2  (or Pf 1  and Pf 2 ) may be detected for calculating the position of the measuring point P 1  (or P 2 ). Further, one point of the pair of corresponding points “Pe2 (or Pf2)” may be detected without defining the epipolar line EP 1 . In this case, the point on the bisecting line Le 2 , the Y-coordinate of which is the same as that of the corresponding point Pe 1  (or Pf 1 ) is defined as the corresponding point Pe 2  (or Pf 2 ). 
     The pair of corresponding points Pe 1  (Pe 2 ) may be detected without the input-operation using the monitor  30  and the mouse  34 . Namely, the pair of corresponding points is automatically calculated. In this case, the pair of occluding contours MA 1  and MA 2  and the pair of occluding contours MB 1 , MB 2  are detected by a line detecting process, such as an edge detecting process, without displaying the pair of images IA and IB and then the pair of corresponding points is detected. 
     In this embodiment, a measurement of the 3-D information using the stereo method is applied for the photogrammetry, however, the measurement may be applied to a “computer vision”. In this case, two still or movie cameras are prepared and an object is captured from two directions by two cameras. Then, the 3-D information of the object is calculated from the pair of images. 
     In this embodiment, the parallel stereo compensation is performed, however, the positions of the measuring points may be calculated without the parallel stereo compensation. In this case, an epipolar line different from the epipolar line EP 1  shown in FIG. 10 is defined. 
     Finally, it will be understood by those skilled in the art that the foregoing description is of preferred embodiments of the device, and that various changes and modifications may be made to the present invention without departing from the spirit and scope thereof. 
     The present disclosure relates to subject matters contained in Japanese Patent Application No. P2000-218608 (filed on Jul. 19, 2000) which is expressly incorporated herein, by reference, in its entirety.