11 Multi-view Geometry - web.media.mit.edu
Transcript of 11 Multi-view Geometry - web.media.mit.edu
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Multi-view Geometry
Readings: FP10
March 11, 2006
Multi-view Geometry and 3D• We have 2 eyes , thus we see 3D
• Using multiple views allows inference of the 3rd
dimension
How to see in 3D(Using geometry…)
• Find corresponding features• Triangulate and reconstruct depth
Correspondence
Last Lecture:
Now:
Outline Multi-view Geometry
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Multi-view Geometry Multi-view Geometry
Multi-view Geometry Multi-view Geometry
Multi-view Geometry Stereo Constraints
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Stereo Constraints Epipolar Line
Epipolar Constraint Epipolar Constraint
P
p p’
O O’
From Geometry to Algebra
P
p p’
O O’
P
O’O
p p’
p p’ ( )[ ] 0=+×⋅ p'ttp R
( )[ ] 0=×⋅ p'tp R
[ ] 0=×⋅ '' pOOO'Op
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P
O’O
p p’
p p’ ( )[ ] 0=+×⋅ p'ttp R
( )[ ] 0=×⋅ p'tp R
[ ] 0=×⋅ '' pOOO'Op
Matrix Form
Review: Matrix Form of the Cross Product Review: Matrix Form of the Cross Product
Review: Matrix Form of the Cross Product Matrix Form
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Matrix Form Essential Matrix
Epipolar Line Constraint Essential Matrix – Instantaneous case• For small motion given translation and rotation
velocity:
ε
Focus of Expansion for Translating Camera
FOE for Translating Camera
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What if calibration is unknown? Fundamental Matrix
Estimating the Fundamental Matrix Estimating the Fundamental Matrix
8 Point Algorithm 8 Point Algorithm
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Improved 8 Point Algorithm Normalizing the Input Data
Stereo Constraints
Given p1, p2, in the left and right image where is p3?
p1 p2p3
?
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Three Essential Matrices
p1 p2p3
?
Trinocular Epipolar Geometry
Trinocular Epipolar Geometry
p1 is at the intersection of the epipolar lines associated with p2 and p3
Given p2 and p2, p1 is determined!
Recall: Epipolar Line Constraint
0=lpT
Three Essential Matrices
02121 =pp εT01313 =pp εT
From lecture 3:
M
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C
From lecture 3:
1zyx
W
W
W
From lecture 3:
Trifocal Line Constraint Trifocal Line Constraint
Trifocal Line Constraint
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The Trifocal Tensor Trifocal Line Constraint
Line Transfer Uncalibrated Case
Quadrifocal Geometry
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Trifocal Constraint with Noise Project• Final project may be:
– An original implementation of a new or published idea
• A project proposal not longer than two pages must be submitted by April 1.
ProjectMarch 24: - Project Previews / Brainstorming3-5 minute presentation describing
– Specific Project Idea– Your research, or thesis proposal
(if it relates to vision)– Paper you are interested in and may form the basis
of a project