Single-view geometry - UNC Computer Science

							Single-view geometry




  Odilon Redon, Cyclops, 1914
Our goal: Recovery of 3D structure
• Recovery of structure from one image is
  inherently ambiguous




                                          X?
                                     X?
                                X?




                      x
Our goal: Recovery of 3D structure
• Recovery of structure from one image is
  inherently ambiguous
Our goal: Recovery of 3D structure
• Recovery of structure from one image is
  inherently ambiguous
Our goal: Recovery of 3D structure
• We will need multi-view geometry
 Recall: Pinhole camera model




• Principal axis: line from the camera center
  perpendicular to the image plane
• Normalized (camera) coordinate system: camera
  center is at the origin and the principal axis is the z-axis
Recall: Pinhole camera model




      ( X ,Y , Z )  ( f X / Z , f Y / Z )

 X                         X
    f X f             0 
 Y                      Y            x  PX
 Z  fY        f     0 
                               Z
    Z               1 0 
 1                      1 
                            
    Principal point




• Principal point (p): point where principal axis intersects the
  image plane (origin of normalized coordinate system)
• Normalized coordinate system: origin is at the principal point
• Image coordinate system: origin is in the corner
• How to go from normalized coordinate system to image
  coordinate system?
Principal point offset


                           principal point:   ( px , p y )



   ( X , Y , Z )  ( f X / Z  px , f Y / Z  p y )

    X                                          X
       f X  Z px   f            px      0  
    Y                                      Y 
     Z    f Y  Z py       f   py      0  
                                                  Z
                                         0  
    1          Z                 1         1 
                                                
Principal point offset


                            principal point:   ( px , p y )


                                          X
    f X  Zpx   f        p x  1   0  
                              1     Y 
    f Y  Zp y      f   py       0  
                                        Z
        Z                1    1 0  
                                         1 
                                           
   f       px 
K 
       f   p y  calibration matrix
                                        P  KI | 0
   
           1  
Pixel coordinates


                                                1   1
                                    Pixel size:   
                                                mx m y


mx pixels per meter in horizontal direction,
 my pixels per meter in vertical direction

      mx             f       p x   x        x 
   K 
           my       
                          f   py   
                                         y     y 
                                                     
      
                   1 
                              1             1
                                                   
         pixels/m           m            pixels
Camera rotation and translation

                                            • In general, the camera
                                              coordinate frame will
                                              be related to the world
                                              coordinate frame by a
                                              rotation and a
                                              translation


                         ~
                                    
                                  ~ ~
                         Xcam  R X - C    
coords. of point                                 coords. of camera center
in camera frame                                       in world frame
                            coords. of a point
                   in world frame (nonhomogeneous)
   Camera rotation and translation

                                      In non-homogeneous
                                           coordinates:
                                          ~        ~ ~
                                                       
                                          Xcam  R X - C     

                          ~ ~           ~
                    R  RC X  R  RC
            X cam           
                            1           X
                    0   1    0    1 


                           ~
                             
 x  KI | 0Xcam  K R | RC X       P  KR | t ,
                                                                 ~
                                                           t  RC

Note: C is the null space of the camera projection matrix (PC=0)
 Camera parameters
• Intrinsic parameters
  •   Principal point coordinates
                                       mx      f       p x   x      x 
  •   Focal length                  K    my       f   py      y   y 
                                                                        
  •   Pixel magnification factors      
                                             1 
                                                        1           1
                                                                           
  •   Skew (non-rectangular pixels)
  •   Radial distortion
 Camera parameters
• Intrinsic parameters
  •   Principal point coordinates
  •   Focal length
  •   Pixel magnification factors
  •   Skew (non-rectangular pixels)
  •   Radial distortion
• Extrinsic parameters
  • Rotation and translation relative to world coordinate system
Camera calibration
• Given n points with known 3D coordinates Xi
  and known image projections xi, estimate the
  camera parameters

                                      Xi



                               xi




                        P?
Camera calibration
                                                 xi  P1 X i 
                                                          T

                                                 y   PT X   0
 x i  PX i            x i  PX i  0           i  2 i
                                                 1  P3T X i 
                                                            

                0           XT        yi X T  P1 
               
                                i            i
                                               T
                                                   
                Xi
                      T
                              0         xi X i  P2   0
                 yi X T
                       i   xi X   T
                                   i       0  P3 
                                                 
           Two linearly independent equations
Camera calibration
     0T   XT
            1     y1X 
                      T
                      1
     T                   
    X1    0T
                  x1X  P1 
                      T
                      1
                             
                  P2   0        Ap  0
     T                 T 
    0     XTn    yn X n  P3 
                                
    XT    0T     xn X T 
     n                 n


• P has 11 degrees of freedom (12 parameters, but
  scale is arbitrary)
• One 2D/3D correspondence gives us two linearly
  independent equations
• Homogeneous least squares
• 6 correspondences needed for a minimal solution
Camera calibration
     0T   XT
            1     y1X 
                      T
                      1
     T                   
    X1    0T
                  x1X  P1 
                      T
                      1
                             
                  P2   0     Ap  0
     T                 T 
    0     XTn    yn X n  P3 
                                
    XT    0T     xn X T 
     n                 n


• Note: for coplanar points that satisfy ΠTX=0,
  we will get degenerate solutions (Π,0,0),
  (0,Π,0), or (0,0,Π)
Camera calibration
• Once we’ve recovered the numerical form of
  the camera matrix, we still have to figure out
  the intrinsic and extrinsic parameters
• This is a matrix decomposition problem, not
  an estimation problem (see F&P sec. 3.2, 3.3)
Two-view geometry
•   Scene geometry (structure): Given
    projections of the same 3D point in two or
    more images, how do we compute the 3D
    coordinates of that point?

•   Correspondence (stereo matching):
    Given a point in just one image, how does it
    constrain the position of the corresponding
    point in a second image?

•   Camera geometry (motion): Given a set of
    corresponding points in two images, what
    are the cameras for the two views?
Triangulation
• Given projections of a 3D point in two or more
  images (with known camera matrices), find
  the coordinates of the point


                        X?




                                   x2
             x1



      O1                                 O2
Triangulation
• We want to intersect the two visual rays
  corresponding to x1 and x2, but because of
  noise and numerical errors, they don’t meet
  exactly
                  R2        R1
                       X?




                                  x2
             x1



     O1                                 O2
Triangulation: Geometric approach
• Find shortest segment connecting the two
  viewing rays and let X be the midpoint of that
  segment


                  X



                                   x2
             x1



      O1                                 O2
Triangulation: Linear approach

1 x1  P1X       x1  P1X  0       [x1 ]P1X  0
2 x 2  P2 X    x 2  P2 X  0     [x 2 ]P2 X  0

 Cross product as matrix multiplication:

              0       az    a y  bx 
                                   
     a  b   az      0      a x  by   [a ]b
              a y           0  bz 
                     ax            
Triangulation: Linear approach

1 x1  P1X      x1  P1X  0     [x1 ]P1X  0
2 x 2  P2 X   x 2  P2 X  0    [x 2 ]P2 X  0


      Two independent equations each in terms of
      three unknown entries of X
Triangulation: Nonlinear approach
Find X that minimizes

          d ( x1 , P X )  d ( x2 , P2 X )
            2
                    1
                             2




                     X?



          x’1
                                       x2
                x1
                                 x’2


     O1                                      O2