3. Robert Collins
CSE486, Penn State
Review : Forward Projection
World Camera Film Pixel
Coords Coords Coords Coords
U X x u
V Mext Y Mproj Maff
y v
W Z
U X Mint u
V Y v
W Z
U u
M
V m11 m12 m13 m14 v
W m21 m22 m23 m24
m31 m31 m33 m34
4. Robert Collins
World to Camera Transformation
CSE486, Penn State
PC PW
W
Y
X
U
R Z V
C
Rotate to Translate by - C
align axes (align origins)
P C = R ( PW - C )
= R PW + T
5. Robert Collins
CSE486, Penn State
Perspective Matrix Equation
X (Camera Coordinates)
x= f
Z
y= f
Y X
Z x' f 0 0 0
y ' = 0 f Y
0 0
Z
z' 0
0 1 0
1
p = M int ⋅ PC
6. Robert Collins
CSE486, Penn State
Film to Pixel Coords
2D affine transformation from film
coords (x,y) to pixel coordinates (u,v):
X
u’ a11 a12 xa
' 13 f 0 0 0
v’ ya = 0
a21 a22 ' 23 f Y
0 0
w’
Z
0 0 z1
' 0
0 1 0
1
Maff Mproj
u = Mint PC = Maff Mproj PC
7. Robert Collins
Projection of Points on Planar Surface
CSE486, Penn State
Perspective
projection
y
Film coordinates
x
Point
on plane
Rotation +
Translation
9. Robert Collins
Projection of Planar Points (cont)
CSE486, Penn State
Homography H
(planar projective
transformation)
10. Robert Collins
Projection of Planar Points (cont)
CSE486, Penn State
Homography H
(planar projective
transformation)
Punchline: For planar surfaces, 3D to 2D perspective
projection reduces to a 2D to 2D transformation.
Punchline2: This transformation is INVERTIBLE!
11. Robert Collins
CSE486, Penn State
Special Case : Frontal Plane
What if the planar surface is perpendicular to
the optic axis (Z axis of camera coord system)?
Then world rotation matrix simplies:
12. Robert Collins
CSE486, Penn State
Frontal Plane
So the homography for a frontal plane simplies:
Similarity Transformation!
13. Robert Collins
CSE486, Penn State
Convert to Pixel Coords
Internal camera a11 a12 a13 v pixels
parameters a21 a22 a23
0 0 1
u
14. Robert Collins
Planar Projection Diagram
CSE486, Penn State
Here’s where
transformation
groups get useful!
15. Robert Collins
General Planar Projection
CSE486, Penn State
H1
H
A1
16. Robert Collins
General Planar Projection
CSE486, Penn State
H1
H-1
A1
17. Robert Collins
Frontal Plane Projection
CSE486, Penn State
S1
A
A1
18. Robert Collins
Frontal Plane Projection
CSE486, Penn State
S1
A-1
A1
19. Robert Collins
General Planar Projection
CSE486, Penn State
H1 H2
H
20. Robert Collins
CSE486, Penn State
Summary: Planar Projection
Pixel coords
Internal u
params
Perspective v
projection
y
Homography
x
Point on plane
Rotation + Translation
21. Robert Collins
Applying Homographies to Remove
CSE486, Penn State
Perspective Distortion
from Hartley & Zisserman
4 point correspondences suffice for
the planar building facade
24. Robert Collins
CSE486, Penn State
Two Practical Issues
How to estimate the homography given
four or more point correspondences
(will derive L.S. solution now)
How to (un)warp image pixel values to
produce a new picture (last class)
26. Robert Collins
CSE486, Penn State
Degrees of Freedom?
There are 9 numbers h11,…,h33 , so are there 9 DOF?
No. Note that we can multiply all hij by nonzero k
without changing the equations:
27. Robert Collins
CSE486, Penn State
Enforcing 8 DOF
One approach: Set h33 = 1.
Second approach: Impose unit vector constraint
Subject to the
constraint:
28. Robert Collins
CSE486, Penn State
L.S. using Algebraic Distance
Setting h33 = 1
Multiplying through by denominator
Rearrange
29. Robert Collins
CSE486, Penn State
Algebraic Distance, h33=1 (cont)
2N x 8 8x1 2N x 1
Point 1
Point 2
Point 3
Point 4
additional
points
30. Robert Collins
CSE486, Penn State
Algebraic Distance, h33=1 (cont)
Linear 2Nx8 8x1 2Nx1
equations A h = b
8x2N 2Nx8 8x1 8x2N 2Nx1
Solve:
AT A h = AT b
8x8 8x1 8x1
(AT A) h = (AT b)
-1
h = (AT A) (AT b)
Matlab: h = Ab
31. Robert Collins
CSE486, Penn State
Caution: Numeric Conditioning
R.Hartley: “In Defense of the Eight Point Algorithm”
Observation: Linear estimation of projective transformation
parameters from point correspondences often suffer from poor
“conditioning” of the matrices involves. This means the solution
is sensitive to noise in the points (even if there are no outliers).
To get better answers, precondition the matrices by performing
a normalization of each point set by:
• translating center of mass to the origin
• scaling so that average distance of points from origin is sqrt(2).
• do this normalization to each point set independently
32. Robert Collins
CSE486, Penn State
Hartley’s PreConditioning
PointSet1 H = T2-1 S2H?Hnorm S1 T1
-1
PointSet2
Translate
T1 center of mass T2
to origin
Scale so average
S1 point dist is sqrt(2) S2
Estimate Hnorm
33. Robert Collins
CSE486, Penn State
A More General Approach
What might be wrong with setting h33 = 1?
If h33 actually = 0, we can’t get the right answer.
34. Robert Collins
CSE486, Penn State
Algebraic Distance, ||h||=1
||h|| = 1
Multiplying through by denominator
Rearrange
=0
=0
35. Robert Collins
Algebraic Distance, ||h||=1 (cont)
CSE486, Penn State
2N x 9 9x1 2N x 1
4
P
O
I
N
T
S
additional
points
36. Robert Collins
Algebraic Distance, ||h||=1 (cont)
CSE486, Penn State
Homogeneous 2Nx9 9x1 2Nx1
equations A h = 0
9x2N 2Nx9 9x1 9x2N 2Nx1
Solve:
AT A h = AT 0
9x9 9x1 9x1
(AT A) h = 0
SVD of ATA = U D UT
Let h be the column of U (unit eigenvector)
associated with the smallest eigenvalue in D.
(if only 4 points, that eigenvalue will be 0)