More on single-view geometry class 10

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More on single-view geometryclass 10

• Multiple View Geometry
• Comp 290-089
• Marc Pollefeys

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Content

• Background Projective geometry (2D, 3D),
  Parameter estimation, Algorithm evaluation.
• Single View Camera model, Calibration, Single
  View Geometry.
• Two Views Epipolar Geometry, 3D reconstruction,
  Computing F, Computing structure, Plane and
  homographies.
• Three Views Trifocal Tensor, Computing T.
• More Views N-Linearities, Multiple view
  reconstruction, Bundle adjustment,
  auto-calibration, Dynamic SfM, Cheirality, Duality

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Multiple View Geometry course schedule(subject
to change)
Jan. 7, 9 Intro motivation Projective 2D Geometry
Jan. 14, 16 (no class) Projective 2D Geometry
Jan. 21, 23 Projective 3D Geometry (no class)
Jan. 28, 30 Parameter Estimation Parameter Estimation
Feb. 4, 6 Algorithm Evaluation Camera Models
Feb. 11, 13 Camera Calibration Single View Geometry
Feb. 18, 20 Epipolar Geometry 3D reconstruction
Feb. 25, 27 Fund. Matrix Comp. Structure Comp.
Mar. 4, 6 Planes Homographies Trifocal Tensor
Mar. 18, 20 Three View Reconstruction Multiple View Geometry
Mar. 25, 27 MultipleView Reconstruction Bundle adjustment
Apr. 1, 3 Auto-Calibration Papers
Apr. 8, 10 Dynamic SfM Papers
Apr. 15, 17 Cheirality Papers
Apr. 22, 24 Duality Project Demos
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Single view geometry
Camera model Camera calibration Single view
geom.
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Gold Standard algorithm

• Objective
• Given n6 2D to 2D point correspondences
  Xi?xi, determine the Maximum Likelyhood
  Estimation of P
• Algorithm
• Linear solution
• Normalization
• DLT
• Minimization of geometric error using the
  linear estimate as a starting point minimize the
  geometric error
• Denormalization

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More Single-View Geometry

• Projective cameras and
• planes, lines, conics and quadrics.
• Camera calibration and vanishing points,
  calibrating conic and the IAC

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Action of projective camera on planes
The most general transformation that can occur
between a scene plane and an image plane under
perspective imaging is a plane projective
transformation (affine camera-affine
transformation)
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Action of projective camera on lines
forward projection
back-projection
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Action of projective camera on conics
back-projection to cone
example
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Images of smooth surfaces
The contour generator G is the set of points X on
S at which rays are tangent to the surface. The
corresponding apparent contour g is the set of
points x which are the image of X, i.e. g is the
image of G
The contour generator G depends only on position
of projection center, g depends also on rest of P
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Action of projective camera on quadrics
back-projection to cone
The plane of G for a quadric Q is camera center C
is given by PQC (follows from pole-polar
relation)
The cone with vertex V and tangent to the quadric
Q is
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The importance of the camera center
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Moving the image plane (zooming)
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Camera rotation
conjugate rotation
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Synthetic view

1. Compute the homography that warps some a
   rectangle to the correct aspect ratio
2. warp the image

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Planar homography mosaicing
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close-up interlacing can be important
problem!
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Planar homography mosaicing more examples
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Projective (reduced) notation
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Moving the camera center
motion parallax
epipolar line
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What does calibration give?
An image l defines a plane through the camera
center with normal nKTl measured in the
cameras Euclidean frame
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The image of the absolute conic
mapping between p8 to an image is given by the
planar homogaphy xHd, with HKR
image of the absolute conic (IAC)

1. IAC depends only on intrinsics
2. angle between two rays
3. DIACwKKT
4. w ? K (cholesky factorisation)
5. image of circular points

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A simple calibration device

• compute H for each square
• (corners ? (0,0),(1,0),(0,1),(1,1))
• compute the imaged circular points H(1,i,0)T
• fit a conic to 6 circular points
• compute K from w through cholesky factorization

( Zhangs calibration method)
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Orthogonality pole-polar w.r.t. IAC
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The calibrating conic
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Vanishing points
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Vanishing lines
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Orthogonality relation
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Calibration from vanishing points and lines
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Calibration from vanishing points and lines
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Next class Two-view geometryEpipolar geometry