Colorado School of Mines
Computer Vision
Professor William Hoff
Dept of Electrical Engineering &Computer Science
Colorado School of Mines
Computer Vision
http://inside.mines.edu/~whoff/
1
3D-2D Coordinate Transforms
Additional material
Colorado School of Mines
Computer Vision
2
Special Case – Viewing a Plane
• Consider a plane observed from two viewpoints
• It can be shown that the image points are transformed using a
homography (i.e., an arbritrary 3x3 transformation)
image 1
image 0
~
x1 ~ H ~
x0
2D points in
image 1
(homogeneous
coords)
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2D points in
image 0
(homogeneous
coords)
 xa   h11
  
 xb    h21
 x  h
 c   31
Computer Vision
h12
h22
h32
h13  x0 
 
h23  y0 
h33  1 
 x1   xa / xc 
  

 y1    xb / xc 
1  1 
  

3
Derivation of Homography
• The equation of a plane is
n
nT P  d
– where n is the surface normal, P is
any point on the plane, and d is the
perpendicular distance to the plane
– In other words, the dot product of n
and P (i.e., the projection of P onto
unit vector n) is the distance d
• Equivalently,
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P
d
nT P
1
d
Computer Vision
4
Derivation of Homography (continued)
• Consider a point P = (X,Y,Z). If we know the rigid body
transformation from camera 0 to camera 1, then
1
P  01 R 0 P  1t 0org
• Now, the projection of P onto each image is just
p~0  K 0 P, p~1  K 1P
• and
0
• So
P ~ K 1 p~0 ,
1
P ~ K 1 p~1
The notation A ~ B means that A equals B to
within an arbitrary scale factor
0 T 0
0 T






n
P
n
   K  01 R  1t 0org 
K 1P  K  01 R 0 P  1t 0org 


d


 d



0 T


n
1~
p ~ K  01 R  1t 0org 
 d

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 0
  P


  1 0 ~
 K p


Computer Vision
5
Derivation of Homography (continued)
• So
1
p~ ~ M 0 p~
– where M is the homography matrix given by
1
t 0org n  1
K
M  K 0 R 


d


1
0 T
•
•
•
0 nT
is the surface normal of the plane,
expressed in the coordinate system of
camera 0
1t
0org is the origin of camera 0, expressed
in the coordinate system of camera 1
d is the perpendicular distance to the
plane (from camera 0)
• Thus, if a plane is observed from two viewpoints, the
projected points are transformed from one image to the
other with a homography
Colorado School of Mines
Computer Vision
6
Special Case – Small Planar Patch
• Small planar patch
– Often we want to track a small patch on an object
– We want to know how the image of that patch transforms as the
object rotates
• Assume
– Size of patch small compared to distance -> weak perspective
– Rotation is small -> small angle approximation
– Patch is planar
• It can be shown that the patch undergoes affine
transformation
 xB   a11 a12
  
 yB    a21 a22
1  0
0
  
Colorado School of Mines
Computer Vision
t x  x A 
 
t y  y A 

1 
 1 
7
To show this
• Write the rotation matrix RzRyRx, but assume that
for small angles
– cos(theta) ≈ 1
– sin(theta) ≈ theta
– theta*theta ≈ 0
Colorado School of Mines
Computer Vision
Colorado School of Mines
Computer Vision
Colorado School of Mines
Computer Vision