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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