Epipolar Geometry and the Fundamental Matrix F

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Epipolar Geometry and the Fundamental Matrix F

• The Epipolar Geometry is the intrinsic projective
  geometry between 2 views and the Fundamental
  Matrix encapsulates this geometry
• x F x 0

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Epipolar geometry

• The Epipolar geometry depends only on the
  internal parameters of the cameras and the
  relative pose.
• A point X in 3 space is imaged in 2 views x and
  x
• X, x, x and the camera centre C are coplanar
  in the plane p
• The rays back-projected from x and x meet at X

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Point correspondence geometry
Fig. 8.1
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Point correspondence geometry
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Epipolar Geometry
Fig. 8.2
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Epipolar geometry
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The geometric entities involved in epipolar
geometry
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Fig 8.3
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Converging cameras
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Fig 8.4
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Motion parallel to the image plane
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Geometric derivation
Fig. 8.5
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Point transfer via a plane
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The fundamental matrix F

• x ? l
• Geometric Derivation
• Step 1 Point transfer via a plane
• There is a 2D homography Hp mapping
• each xi to xi
• Step 2 Constructing the epipolar line

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Constructing the epipolar line
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Cross products

• If a ( a1, a2 , a3)T is a 3-vector, then one
  define a corresponding skew-sysmmetric matrix as
  follows

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Cross products 2

• Matrix ax is singular and a is its null
  vector
• a x b ( a2b3 - a3b2, a3b1 - a1b3 , a1b2
  a2b1)T
• a x b ax b ( aT bx )T

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Algebraic derivation
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Algebraic derivation 2
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Example 8.2
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Example 8.2 b
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Properties of the fundamental matrix (a)
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Properties of the fundamental matrix (b)
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Summary of the Properties of the fundamental
matrix 1
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Summary of the properties of the fundamental
matrix 2
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Epipolar line homography 1
Fig. 8.6a
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Epipolar line homography 2
Fig. 8.6 b
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Epipolar line homography
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The epipolar line homography
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A pure camera motion
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Pure translation
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Fig. 8.8
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Pure translation motion
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Example of pure translation
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General camera motion
Fig. 8.9
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General camera motion
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Example of general motion
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Pure planar motion
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Retrieving the camera matricesUsing F to
determine the camera matrices of 2 views

• Projective invariance and canonical cameras
• Since the relationships l Fx and
• x F x 0 are projective relationships
• which

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Projective invariance and canonical cameras

• The camera matrix relates 3-space measurements to
  image measurements and so depends on both the
  image coordinate frame and the choice of world
  coordinate frame.
• F is unchanged by a projective transformation of
  3-space.

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Projective invariance and canonical cameras 2
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Canonical form camera matrices
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Projective ambiguity of cameras given F
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Projective ambiguity of cameras given F2
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Projective ambiguity of cameras given F3
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Canonical cameras given F
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Canonical cameras given F 2
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Canonical cameras given F 3
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Canonical cameras given F 4
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The Essential Matrix
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Normalized Coordinates
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Normalized coordinates 2
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Normalized coordinates 3
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Properties of the Essential Matrix
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Result 8.17 on Essential matrix
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Result 8.17 on Essential matrix 2
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Extraction of cameras from the Essential Matrix
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(No Transcript)
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Determine the t part of the camera matrix P
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Result 8.19
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Geometrical interpretation of the four solutions
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Geometrical interpretation of the four solutions
2
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The 4 possible solutions for calibrated
reconstruction from E