TODAY: Affine Structure from Motion III

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TODAY Affine Structure from Motion III

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

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

• Reading Affine Structure from Motion
• This file on feather.ai.uiuc.edu in
  pub/ponce/lect18.ppt.gz

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Affine Coordinates

• Coordinate system for U

• Coordinate system for YOU

• Affine coordinates

• Coordinate system for Y

• Barycentric
• coordinates

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When do m1 points define a p-dimensional
subspace Y of an n-dimensional affine space X
equipped with some coordinate frame basis?
Rank ( D ) p1, where
Writing that all minors of size (p2)x(p2) of D
are equal to zero gives the equations of Y.
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Affine Transformations

• Bijections from X to Y that
• map m-dimensional subspaces of X onto
  m-dimensional
• subspaces of Y
• map parallel subspaces onto parallel subspaces
  and
• preserve affine (or barycentric) coordinates.

• Bijections from X to Y that
• map lines of X onto lines of Y and
• preserve the ratios of signed lengths of
• line segments.

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In E they are combinations of rigid
transformations, non-uniform scalings and shears.
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Affine Transformations II

• Given two affine spaces X and Y of dimension m,
  and two
• coordinate frames (A) and (B) for these spaces,
  there exists
• a unique affine transformation mapping (A) onto
  (B).

• Given an affine transformation from X to Y, one
  can always
• write

• When coordinate frames have been chosen for X
  and Y,
• this translates into

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Affine projections induce affine transformations
from planes onto their images.
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Affine Shape
Two point sets S and S in some affine space X
are affinely equivalent when there exists an
affine transformation y X X such that X
y ( X ).
Affine structure from motion affine shape
recovery.
recovery of the corresponding motion
equivalence classes.
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Geometric affine scene reconstruction from two
images (Koenderink and Van Doorn, 1991).
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Affine Structure from Motion
(Koenderink and Van Doorn, 1991)
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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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