Stanford CS223B Computer Vision, Winter 2006 Lecture 5 Stereo I

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Stanford CS223B Computer Vision, Winter
2006Lecture 5 Stereo I
Stereo
Stereo

• Professor Sebastian Thrun
• CAs Dan Maynes-Aminzade, Mitul Saha, Greg
  Corrado

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Homework 1
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Vocabulary Quiz

• Baseline
• Epipole
• Fundamental Matrix
• Essential Matrix
• Stereo Rectification

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Stereo Vision Illustration
http//www.well.com/user/jimg/stereo/stereo_list.h
tml
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Stereo Example (Stanley Robot)
Disparity map
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Stereo Example
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Stereo Vision Outline

• Basic Equations
• Epipolar Geometry
• Image Rectification
• Reconstruction
• Correspondence
• Dense and Layered Stereo
• (Active Range Imaging Techniques)

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The Two Problems of Stereo

• Correspondence (Wed)
• Reconstruction (Today)

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Pinhole Camera Model
Image plane
Focal length f
Center of projection
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Pinhole Camera Model
Image plane
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Pinhole Camera Model
Image plane
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Basic Stereo Derivations
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Basic Stereo Derivations
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What If?
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Epipolar Geometry
P
Pl
Pr
Yr
p
p
r
l
Yl
Zl
Zr
Xl
fl
fr
Ol
Or
Xr
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Epipolar Geometry
P
Pl
Pr
Epipolar Plane
Epipolar Lines
p
p
l
r
Ol
el
er
Or
Epipoles
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Epipolar Geometry

• Epipolar plane plane going through point P and
  the centers of projection (COPs) of the two
  cameras
• Epipoles The image in one camera of the COP of
  the other
• Epipolar Constraint Corresponding points must
  lie on epipolar lines

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Essential Matrix
P
Pl
Pr
p
p
r
l
Ol
el
er
Or
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Essential Matrix
P
Pl
Pr
p
p
r
l
Ol
el
er
Or
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Fundamental Matrix

• Same as Essential Matrix in Camera Pixel
  Coordinates

Pixel coordinates
Intrinsic parameters
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Intrinsic Parameters (See Chapter 2)
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Computing F The Eight-Point Algorithm

• Problem Recover F (3-3 matrix of rank 2)
• Ides Get 8 points
• Minimize
• Notice Argument linear in coefficients of F

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Computing F The Eight-Point Algorithm

• Run Singular Value Decomposition of A
• Appendix A.6, page 322-325
• See also G. Strang Linear algebra and its
  applications
• Least squares solution column of V corresponding
  to the smallest eigenvalue of A

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Computing F The Eight-Point Algorithm

• Idea Compile points into matrix A

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Computing F The Eight-Point Algorithm

• Decompose A via SVD
• Solution F is column of V corresponding to the
  smallest eigenvector of A
• In practice F will be of rank 3, not 2. Correct
  by
• SVD decomposition of F
• Set smallest eigenvalue to 0
• Reconstruct F

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Computing F The Eight-Point Algorithm

• Input n point correspondences ( n gt 8)
• Construct homogeneous system Ax 0 from
• x (f11,f12, ,f13, f21,f22,f23 f31,f32, f33)
  entries in F
• Each correspondence give one equation
• A is a nx9 matrix
• Obtain estimate F by SVD of A
• x (up to a scale) is column of V corresponding to
  the least singular value
• Enforce singularity constraint since Rank (F)
  2
• Compute SVD of F
• Set the smallest singular value to 0 D -gt D
• Correct estimate of F
• Output the estimate of the fundamental matrix
  F
• Similarly we can compute E given intrinsic
  parameters

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Recitification

• Idea Align Epipolar Lines with Scan Lines.
• Question What type transformation?

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Locating the Epipoles
P
Pl
Pr
p
p
r
l
Ol
el
er
Or
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Stereo Rectification (see Trucco)
P
Pl
Pr
Yr
p
p
l
r
Yl
Xl
Zl
Zr
T
Ol
Or
Xr

• Stereo System with Parallel Optical Axes
• Epipoles are at infinity
• Horizontal epipolar lines

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Reconstruction (3-D) Idealized
Pl
Pr
P
p
p
r
l
Ol
Or
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Reconstruction (3-D) Real
Pl
Pr
P
p
p
r
l
Ol
Or
See Trucco/Verri, pages 161-171
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Summary Stereo Vision (Class 1)

• Epipolar Geometry Corresponding points lie on
  epipolar line
• Essential/Fundamental matrix Defines this line
• Eight-Point Algorithm Recovers Fundamental
  matrix
• Rectification Epipolar lines parallel to
  scanlines
• Reconstruction Minimize quadratic distance