ECE 549CS 543: COMPUTER VISON LECTURE 20

1
ECE 549/CS 543 COMPUTER VISON LECTURE
20 AFFINE STRUCTURE FROM MOTION II
Affine SFM from Multiple Images From Affine to
Euclidean SFM The Projective SFM Problem The
Projective Ambiguity of Projective SFM

• Reading Chapters 12 and 13
• A list of potential projects is at
• http//www-cvr.ai.uiuc.edu/ponce/fall04/project
  s.pdf
• Homework Affine SFM due Th. Nov. 11.
• http//www-cvr.ai.uiuc.edu/ponce/fall04/hw4/hw4
  .pdf

2
The Affine Ambiguity of Affine SFM
When the intrinsic and extrinsic parameters are
unknown
So are M and P where
i
j
and
Q is an affine transformation.
3
Affine Epipolar Geometry
4
An Affine Trick..
Algebraic Scene Reconstruction Method
5
First reconsruction. Mean reprojection error
1.6pixel
Second reconsruction. Mean reprojection error
7.8pixel
6
Suppose we observe a scene with m fixed cameras..
u11 u12 u1n v11 v12 v1n um1 um2
umn vm1 vm2 vmn
A1 A2 Am
P1 P2 Pn

7
What if we could factorize D? (Tomasi and
Kanade, 1992)
Affine SFM is solved!
Singular Value Decomposition
We can take
8
Relative reconstruction error 2.8
Mean reprojection error 2.4pixel
9
From uncalibrated to calibrated cameras
Weak-perspective camera
Calibrated camera
Problem what is Q ?
Note Absolute scale cannot be recovered. The
Euclidean shape (defined up to an arbitrary
similitude) is recovered.
10
Relative reconstruction error 3.0
Mean reprojection error 2.4pixel
11
Reconstruction Results (Tomasi and Kanade, 1992)
Reprinted from Factoring Image Sequences into
Shape and Motion, by C. Tomasi and T. Kanade,
Proc. IEEE Workshop on Visual Motion (1991). ?
1991 IEEE.
12
The Projective Structure-from-Motion Problem
Given m perspective images of n fixed points P
we can write
j
2mn equations in 11m3n unknowns
Overconstrained problem, that can be solved using
(non-linear) least squares!
13
The Projective Ambiguity of Projective SFM
When the intrinsic and extrinsic parameters are
unknown
and Q is an arbitrary non-singular 4x4 matrix.
Q is a projective transformation.