Geometry

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Geometry Tensors

• Frank Dellaert

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Outline

• N-Views and Trifocal Tensor
• Tensor Notation
• Multiview Constraints
• 2-view
• 3-view
• 4-view

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Geometry of N views

• Often more views are available
• Can we use this in matching ?
• Answer yes, up to 4 views

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Trinocular Epipolar Constraints
These constraints are not independent!
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Trinocular Epipolar Constraints Transfer
Given p and p , p can be computed as the
solution of linear equations.
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Tensor Notation
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Tensor Notation

• Different way of writing vectors, matrices
• In general tensors
• row and column replaced by contravariant and
  covariant indices
• Index name indicates the space
• Geek Cool Einstein Summation

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Tensor Notation
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Join cross product !

• Join of two lines is a point
• Join of two points is a line

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Cross Product Magic
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Homography
Maps points from image B to image A
image B
image A
xB
xA
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3D Projective Space
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Projective Camera
xa
3D space a
Mixed 3D to 2D tensor
image A
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Multiview Tensors
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2 Views Fundamental Matrix
F maps points in view A to lines in view B gt
Tensor with two covariant indices FAB
2 Cameras have 22 DOF, 15 DOF ambiguity 7 DOF
in Fundamental Matrix FAB
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4 Views Quadrifocal tensors

• Constraint on 4 lines gt 4-valent contravariant
  tensor QABCD

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Quadrifocal DOF

• 4 Cameras 44 parameters
• 15 DOF ambiguity
• Total DOF 44-1529
• Q Tensor 3333 81 entries
• Q must have 81-29 52 algebraic constraints

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Trifocal tensors

• Relate three views
• Depends on covariant view
• Indices suggest 3-view constraints

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line-line-line transfer

• Two lines in views B and C determine line in view
  A
• Obvious !

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point-line-point

• line in B -gt induces plane homography from C to A

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point-point-point

• more complex

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Whats in a Tensor ?
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What is Quadrifocal Tensor ?

• Take 3 planes, calculate x

• Project in PD

• Must be on line lD

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Whats an F Matrix ?
xa
xB
xA
PA
PB
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Important to Remember

• All tensors encapsulate incidence constraints
• Hence invariant to projective transformations
• Valency suggests derived constraints