ECE 549CS 543: COMPUTER VISON LECTURE 20

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ECE 549/CS 543 COMPUTER VISON LECTURE
20 PROJECTIVE STRUCTURE FROM MOTION I
From Affine to Euclidean SFM The Projective SFM
Problem The Projective Ambiguity of Projective
SFM Projective SFM from two Pictures Projective
SFM from Multiple Pictures

• 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

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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

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What if we could factorize D? (Tomasi and
Kanade, 1992)
Affine SFM is solved!
Singular Value Decomposition
We can take
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Relative reconstruction error 2.8
Mean reprojection error 2.4pixel
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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.
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Relative reconstruction error 3.0
Mean reprojection error 2.4pixel
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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.
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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!
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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.
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Motion estimation from fundamental matrices
Q
Once M and M are known, P can be computed with
LLS.
Facts
b can be found using LLS.
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Relative reconstruction error 1.2
Mean reprojection error 0.8pixel
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Projective Structure from Motion and Factorization
Factorization??

• Algorithm (Sturm and Triggs, 1996)
• Guess the depths
• Factorize D
• Iterate.

Does it converge? (Mahamud and Hebert, 2000)
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Relative reconstruction error 0.2
Mean reprojection error 0.9pixel
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Bundle adjustment (Photogrammetry)
Minimize
with respect to the matrices Mi and the point
positions Pj .
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Relative reconstruction error 0.2
Mean reprojection error 0.8pixel
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From Projective to Euclidean Images
If z , P , R and t are solutions, so are l z
, l P , R and l t .
Absolute scale cannot be recovered! The Euclidean
shape (defined up to an arbitrary similitude) is
the best that can be recovered.
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From uncalibrated to calibrated cameras
Perspective camera
Calibrated camera
Problem what is Q ?
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From uncalibrated to calibrated cameras II
Perspective camera
Calibrated camera
Problem what is Q ?
Example known image center
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Relative reconstruction error 1.2
Mean reprojection error 0.9pixel
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From uncalibrated to calibrated cameras II
Perspective camera
Calibrated camera
Problem what is Q ?
Example known image center
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Relative reconstruction error 1.5
Mean reprojection error 0.9pixel
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(Pollefeys, Koch and Van Gool, 1999)
Reprinted from Self-Calibration and Metric 3D
Reconstruction from Uncalibrated
Image Sequences, by M. Pollefeys, PhD Thesis,
Katholieke Universiteit, Leuven (1999).