Omnidirectional epipolar geometry

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Omnidirectional epipolar geometry
Kostas Daniilidis and Chris Geyer University of
Pennsylvania
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Inside-out immersive
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And recently
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Omnidirectional input

Porta del Sol, Tiwanaku Bolivia
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Central Catadioptric Projection

• is a double projectionFirst on the mirror, then
  on the image plane.

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Unifying Theorem

• All central catadioptric projections are
  equivalent to double projection through the
  sphere.
• Corollary Conventional cameras are a just a
  singularity.

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Equivalence with the sphere

• Image of object obtained on image plane
  identical to catadioptric projection

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Two facts

• 1. Parabolic projection central projection to
  the sphere then stereo-graphic projection to a
  plane
• 2. Perspective projection central projection to
  the sphere followed by central projection to a
  plane from the same center ! Our model covers all
  conventional perspective cameras!!

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• Inside-Out-Inside Motion estimation

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• The projection of a line in space is a conic
  section and in parabolic mirrors it is a circle.

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A new representation of image features

• While the projective plane captures both points
  and lines, we do not have a space suitable for
  points and circles. We need a
• CIRCLE SPACE!

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Lift a circleline projection in parabolic
omnicameras
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Take inverse stereographic image
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Construct cone tangent to locusP is the circle
representation
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By varying the radius we model points, circles,
and imaginary circles!
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• Not every circle is a line projection (it has to
  be projection of a great circle). All these
  feasible lines lie on a plane circle space.

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Image of the absolute conic
calibrating conic
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Transformations of circle space

• Motivation In the perspective case the group of
  transformations is the set of collineations, i.e.
  non-singular matrices in PGL(3)
• Goal find the natural transformation group of
  circle space.

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A translation in the plane.

• If the sphere has projective quadratic form
• Then for A to preserve the sphere we must have
• (Note similarity with )

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The Lorentz group

• What is the general set of 44 matrices
  satisfying
• Actually since Q is projective we only need

Lorentz Group O(3,1) It is a six dimensional Lie
group
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The Lorentz group
rotationabout x-axis
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The Lorentz group
rotationabout y-axis
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The Lorentz group
rotationabout z-axis
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The Lorentz group
x translation
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The Lorentz group
y translation
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The Lorentz group
scale
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Linear transformation from uncalibrated pixels to
calibrated rays.
Such a linear transformation exists and its
kernel contains the parameters of this mapping.
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• Scene reconstruction and ego-motion using
  omnidirectional cameras

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Two view perspectivethe essential matrix

• Recall that two images p1, p2 of the same space
  point X satisfy the bilinear constraint
• where E is a 33 rank 2 matrix independent of X,
  (Tsai-Huang and Longuet-Higgins)

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Two views

• Assume p1 and p2 are the catadioptric
  projections of X

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Two views
There exists an essential matrix Esuch
that ________
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Two views

• However there exist Lorentz group elements K1
  K2 such that

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Catadioptric fundamental matrix
F

• i.e. the lifted image points satisfy a bilinear
  epipolar constraint!!!
• F is the 44 catadioptric fundamental matrix
• The kernel of F is the kernel of K.

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Reconstruction algorithm much simpler than in
perspective !

• Recover camera parameters with kernel computation
  and intersection
• 2. Recover rotation and translation
• 3. Reconstruct environment or produce novel views.

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Epipolar circles