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Geometry

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Can we use this in matching ? Answer: yes, up to 4 views ... Geek Cool: Einstein Summation. Tensor Notation. Einstein Summation: sum over repeated indices ... – PowerPoint PPT presentation

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Title: Geometry


1
Geometry Tensors
  • Frank Dellaert

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

3
Geometry of N views
  • Often more views are available
  • Can we use this in matching ?
  • Answer yes, up to 4 views

4
Trinocular Epipolar Constraints
These constraints are not independent!
5
Trinocular Epipolar Constraints Transfer
Given p and p , p can be computed as the
solution of linear equations.
1
2
3
6
Tensor Notation
7
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

8
Tensor Notation
9
Join cross product !
  • Join of two lines is a point
  • Join of two points is a line

10
Cross Product Magic
11
Homography
Maps points from image B to image A
image B
image A
xB
xA
12
3D Projective Space
13
Projective Camera
xa
3D space a
Mixed 3D to 2D tensor
image A
14
Multiview Tensors
15
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
16
4 Views Quadrifocal tensors
  • Constraint on 4 lines gt 4-valent contravariant
    tensor QABCD

17
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

18
Trifocal tensors
  • Relate three views
  • Depends on covariant view
  • Indices suggest 3-view constraints

19
line-line-line transfer
  • Two lines in views B and C determine line in view
    A
  • Obvious !

20
point-line-point
  • line in B -gt induces plane homography from C to A

21
point-point-point
  • more complex

22
Whats in a Tensor ?
23
What is Quadrifocal Tensor ?
  • Take 3 planes, calculate x
  • Project in PD
  • Must be on line lD

24
Whats an F Matrix ?
xa
xB
xA
PA
PB
25
Important to Remember
  • All tensors encapsulate incidence constraints
  • Hence invariant to projective transformations
  • Valency suggests derived constraints
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