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CS332 Visual ProcessingDepartment of Computer ScienceWellesley College
Analysis of Motion
Recovering 3-D structurefrom motion
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Recovering 3D structure from motion
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Ambiguity of 3D recovery
birds’ eyeviews
We need additional constraintto recover 3D structure uniquely
“rigidity constraint”
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Image projectionsZ
X
perspectiveprojection
imageplane
Z
X
orthographicprojection
image plane
(X, Y, Z) (X, Y)(X, Y, Z) (X/Z, Y/Z)
• only relative depth
• requires object rotation
• only scaled depth
• requires translation ofobserver relative to scene
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Using the rigidity constraint
Ullman: Given 3 distinct orthographic viewsof 4 non-coplanar points in motion, if thereexists a rigid 3-D structure consistent withthese views, then this structure is unique.
view 1 view 2 view 3
2D positions of pointsin 3 views
set ofequations to
solve
3D structureof points
caveat: depth reversals
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What is needed to compute a unique rigid 3D structure?
5 points1 view
5 points1-1/2 views
7 points2 views
4 points3 views
orthographicprojection
perspectiveprojection
correspondence
velocity field
Sample results:
*
* Ullman
Rigidity constraint alone is sufficient tocompute 3D structure from motion, BUT…
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Human recovery of 3D structure from motion:
• Needs extended time to obtain an accurate3D structure, and derivation is not all-or-none
• Can cope with significant deviations from rigidity
• Integrates multiple sources of information
These factors motivated the design ofUllman’s incremental rigidity scheme
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Incremental Rigidity Scheme
x
y
image
(x1 y1 z1)
(x2 y2 z2)
(x3 y3 z3)
(x1' y1' ??)
(x2' y2' ??)
(x3' y3' ??)
depth: Z
initially, Z=0at all points
Find new 3D modelthat maximizes rigidity
Compute new Zvalues that minimize
change in 3Dstructure
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Bird’s eye view:de
pth
ximagecurrent model
new imageFind new Zi that minimize
Σ (Lij – lij)2/Lij3
Lij
lijZ
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Incremental rigidity scheme: results
bird’s eye views
new 3D model afterevery 10° rotation
true structure
computed 3D model
builds 3D modelincrementally over
extended time
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Spontaneous depth reversal
truestructure
computed3D model
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Human recovery of 3D structure from motion• 2-3 points sufficient
• requires extended time to build up accuratepercept of 3-D structure
Anderson & colleagues
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