Multiple-view Reconstruction from Points and Lines

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Multiple-view Reconstruction from Points and Lines

• Yi Ma
• ECE Department, CSL, Beckman
• University of Illinois at Urbana-Champaign

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Problem formulation
Input Corresponding images (of features) in
multiple images. Output Camera motion, camera
calibration, object structure.
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Projection model point features
Homogeneous coordinates of a 3-D point
Homogeneous coordinates of its 2-D image
Projection of a 3-D point to an image plane
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Image of a line feature
Homogeneous representation of a 3-D line
Homogeneous representation of its 2-D co-image
Projection of a 3-D line to an image plane
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Incidence relations among features
Multiview constraints are nothing but incidence
relations at play!
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Traditional multifocal constraints
For images of the same 3-D point
(leading to the conventional approach)
Multilinear constraints among 2, 3, 4-wise views
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Rank conditions for point feature
WLOG, choose camera frame 1 as the reference
Multiple-View Matrix
Lemma Rank Condition for Point Features
Let
then and are linearly dependent.
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Rank conditions vs. multifocal constraints

• These constraints are only necessary but NOT
  sufficient!
• However, there is NO further relationship among
  quadruple wise
• views. Quadrilinear constraints hence are
  redundant!

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Multiple-view matrix line vs. point
Point Features
Line Features
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Rank conditions line vs. point
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Global multiple-view analysis examples
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A family of intersecting lines
each can randomly take the image of any of the
lines
Nonlinear constraints among up to four views
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.
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Universal rank condition
Theorem The Universal Rank Condition for images
of a point on a line
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Instances with mixed features
Examples
Case 1 a line reference
Case 2 a point reference

• All previously known constraints are the
  theorems instances.
• Degenerate configurations if and only if a drop
  of rank.

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Generalization restriction to a plane
Homogeneous representation of a 3-D plane
Corollary Coplanar Features
Rank conditions on the new extended remain
exactly the same!
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Generalization restriction to a plane
Given that a point and line features lie on a
plane in 3-D space
GENERALIZATION Multiple View Matrix Coplanar
Features
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Multiple-view structure and motion recovery
Given images of points
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SVD based 4-step algorithm for SFM
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Utilizing all incidence relations
Three edges intersect at each vertex.
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Example simulations
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Example simulations
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Example experiments
Errors in all right angles lt 1o
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Summary

• Incidence relations ltgt rank conditions
• Rank conditions gt multiple-view factorization
• Rank conditions implies all multi-focal
  constraints
• Rank conditions for points, lines, planes, and
• (symmetric) structures.