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Minimal Surfaces for Stereo

Minimal Surfaces for Stereo. Chris Buehler, Steven J. Gortler, Michael F. Cohen, Leonard McMillan MIT, Harvard Microsoft Research, MIT. Motivation. Optimization based stereo over greed based No early commitment Enforce interactions: each pixel sees unique item

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Minimal Surfaces for Stereo

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  1. Minimal Surfaces for Stereo Chris Buehler, Steven J. Gortler, Michael F. Cohen, Leonard McMillan MIT, Harvard Microsoft Research, MIT

  2. Motivation • Optimization based stereo over greed based • No early commitment • Enforce interactions: each pixel sees unique item • Penalize interactions: non-smoothness

  3. Stereo by Optimization • Early algorithms: dynamic programming • (Baker ‘81, Belumeur & Mumford ‘92…) • Don’t generalize beyond 2 camera, single scanline

  4. Stereo by Optimization • Recent Algorithms: iterative a-expansion • (… Kolmogorov & Zabih ‘01) • very general • NP-Complete • Local opt found quickly in practice • Recent algorithms: MIN-CUT • (Roy & Cox ‘96, Ishikawa & Geiger ‘98) • Polynomial time global optimum • New interpretation to such methods

  5. Contributions • Stereo as a discrete minimal surface problem • Algorithms: Polynomial time globally optimal surface • Using MIN-CUT (Sullivan ‘90) • Build from shortest path • Applications to stereo vision • Rederive previous MIN-CUT stereo approaches • New 3-camera stereo formulation (Ayache ‘88)

  6. Planar Graph Shortest Path • Given: an embedded planar graph • faces, edges, vertices

  7. Planar Graph Shortest Path • A non negative cost on each edge 57

  8. Planar Graph Shortest Path • Two boundary points on the exterior of the complex

  9. Planar Graph Shortest Path • Find minimal curve: (collection of edges) with given boundary

  10. Selected Match Selected Occlusion Camera Left Camera Right Planar Graph For stereo

  11. Algorithms • Classic: Dijkstra’s • Works even for non-planar graphs • Wacky: use duality • But this will generalize to higher dimension

  12. Duality

  13. Duality • face vertex • edgecross edge • - same cost 57

  14. Duality • Split exterior

  15. Sink Source Source Duality • Add source and sink

  16. Sink Source Cuts • Cuts of dual graph = partitions of dual verts • Cost = sum of dual edges spanning the partition • MIN-CUT can be found in polynomial time

  17. Sink Source Cuts • Claim: Primalization of MIN-CUT will be shortest path

  18. Sink Sink Source Source Why this works • Cuts of dual graph = partitions of dual verts

  19. Sink Sink Source Source Why this works • Partition of dual verts = partition of primal faces

  20. Sink Sink Sink Source Source Source Why this works • Partition of primal faces = primal path

  21. Sink Sink Sink Source Source Source Why this works • Cuts in dual correspond to paths in primal • MIN-CUT in dual corresponds to shortest path in primal

  22. Same idea works for surfaces!

  23. Increasing the dimension Planar graph: verts, edges, faces cost on edges boundary: 2 points on exterior sol: min path Spacial compex: verts, edges, faces, cells cost on faces boundary: loop on exterior sol: min surface

  24. Increasing the dimension Planar graph: verts, edges, faces boundary: 2 points on exterior sol: min path Spacial compex: verts, edges, faces, cells cost on faces boundary: loop on exterior sol: min surface

  25. Increasing the dimension Planar graph: verts, edges, faces boundary: 2 points on exterior sol: min path Spacial compex: verts, edges, faces, cells cost on faces boundary: loop on exterior sol: min surface

  26. Sink Source Dual construction for min surf • face vertex • edgecross edge • cell vertex • face cross edge MIN-CUT primalizes to min surf

  27. Checkpoint • Solve for minimal paths and surfaces • MIN-CUT on dual graph • Apply these algorithms to stereo vision

  28. Flatland Stereo Geometric interpretation of Cox et al. 96 pixel Camera Left Camera Right

  29. Flatland Stereo Geometric interpretation of Cox et al. 96 pixel Camera Left Camera Right

  30. Flatland Stereo Cost: unmatched/discontinuity, β Camera Left Camera Right

  31. Flatland Stereo Cost: correspondence quality Camera Left Camera Right

  32. Camera Left Camera Right Flatland Stereo

  33. Match Unmatched Camera Left Camera Right Flatland Stereo Uniqueness & monotonicity solution is directed path

  34. Camera Left Camera Right Flatland Stereo Note: unmatched pixels also function as discontinuities Occlusion, discontinuity Match

  35. Flatland to Fatland Camera Left Camera Right

  36. Flatland to Fatland Camera Left Camera Right

  37. 2 cameras, 3d

  38. 2 cameras, 3d

  39. One Cuboid Among Many Solve for minimal surface

  40. Geometric interpretation IG98

  41. Three Camera Rectification (Ayache ‘88)

  42. Three Camera

  43. Three Camera

  44. Three Camera

  45. Three Camera

  46. One cuboid

  47. Dual graph of one cuboid

  48. One Cuboid Among Many Solve for minimal surface

  49. More divisions of middle cell

  50. More expressive decomposition

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