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Analysis of Motion

Analysis of Motion. Recovering 3-D structure from motion. Recovering 3D structure from motion. Ambiguity of 3D recovery. birds’ eye views. We need additional constraint to recover 3D structure uniquely. “rigidity constraint”. perspective projection. Image projections.

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Analysis of Motion

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  1. Analysis of Motion Recovering 3-D structure from motion

  2. Recovering 3D structure from motion

  3. Ambiguity of 3D recovery birds’ eye views We need additional constraint to recover 3D structure uniquely “rigidity constraint”

  4. perspective projection Image projections orthographic projection Z Z image plane X X image plane (X, Y, Z)  (X/Z, Y/Z) (X, Y, Z)  (X, Y)  only scaled depth  requires translation of observer relative to scene  only relative depth  requires object rotation

  5. Using the rigidity constraint Ullman: Given 3 distinct orthographic views of 4 non-coplanar points in motion, if there exists a rigid 3-D structure consistent with these views, then this structure is unique. view 1 view 2 view 3 set of equations to solve 2D positions of points in 3 views 3D structure of points caveat: depth reversals

  6. What is needed to compute a unique rigid 3D structure? orthographic projection perspective projection Sample results: correspondence * velocity field * Ullman Rigidity constraint alone is sufficient to compute 3D structure from motion, BUT…

  7. Human recovery of 3D structure from motion:  Needs extended time to obtain an accurate 3D 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 of Ullman’s incremental rigidity scheme

  8. Incremental Rigidity Scheme x depth: Z initially, Z=0 at all points (x1 y1 z1) (x3 y3 z3) y (x1' y1' ??) (x3' y3' ??) Find new 3D model that maximizes rigidity (x2 y2 z2) (x2' y2' ??) Compute new Z values that minimize change in 3D structure image

  9. Bird’s eye view: Z lij depth Lij x image current model Find new Zi that minimize Σ (Lij – lij)2/Lij3 new image

  10. Incremental rigidity scheme: results bird’s eye views new 3D model after every 10° rotation true structure computed 3D model builds 3D model incrementally over extended time

  11. Spontaneous depth reversal true structure computed 3D model

  12. Human recovery of 3D structure from motion  2-3 points sufficient  requires extended time to build up accurate percept of 3-D structure Anderson & colleagues

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