TODAY: Affine Structure from Motion IV

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TODAY Affine Structure from Motion IV

• Affine Structure from Motion from two Views
• Affine Epipolar Geometry
• An Algebraic Approach

• Affine Structure from Motion from two Views
• An SVD-Based Approach Factorization

• Computing Euclidean Upgrades

• Reading Projective Structure from Motion
• This file on feather.ai.uiuc.edu in
  pub/ponce/lect19.ppt.gz
• Programming assignment on feather in
  pub/ponce/prog19.tex,
• due next Thursday.

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The Affine Epipolar Constraint
Note the epipolar lines are parallel.
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Affine Epipolar Geometry
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The Affine Fundamental Matrix
where
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An Affine Trick..
Algebraic Scene Reconstruction Method
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The Affine Structure of Affine Images
Suppose we observe a scene with m fixed cameras..
The set of all images of a fixed scene is a 3D
affine space!
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has rank 4!
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From Affine to Vectorial Structure
Idea pick one of the points (or their center of
mass) as the origin.
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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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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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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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Projective Spaces (Semi-Formal) Definition
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A Model of P( R )