# Single-view geometry - UNC Computer Science

Shared by:
Categories
Tags
-
Stats
views:
0
posted:
1/31/2013
language:
English
pages:
27
Document Sample

```							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)
Camera parameters
• Intrinsic parameters
•   Principal point coordinates
•   Focal length
•   Pixel magnification factors
•   Skew (non-rectangular pixels)
• 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

```
Other docs by dfhdhdhdhjr
PowerPoint Presentation - The Radclyffe School