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Improved Moves for Truncated Convex Models

This study explores improved moves for truncated convex models in order to achieve accurate and efficient Maximum A Posteriori (MAP) estimation. The research focuses on convex models with random variables and edges defining the neighborhood structure. The text outlines move space exploration, graph construction, analysis, and results related to solving the problem efficiently and accurately. It highlights the challenges faced in move-making literature and proposes solutions to address them.

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Improved Moves for Truncated Convex Models

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  1. Improved Moves for Truncated Convex Models M. Pawan Kumar Philip Torr

  2. Aim Efficient, accurate MAP for truncated convex models V1 V2 … … … … … … … … … … … … … … … … … Vn Random Variables V = { V1, V2, …, Vn} Edges E define neighbourhood

  3. Aim Accurate, efficient MAP for truncated convex models lk ab;ik ab;ik = wab min{ d(i-k), M } li wab is non-negative d(.) is convex Vb b;k a;i Va Truncated Linear Truncated Quadratic ab;ik ab;ik i-k i-k

  4. Motivation Low-level Vision min{ |i-k|, M} • Smoothly varying regions Boykov, Veksler & Zabih 1998 • Sharp edges between regions Well-researched !!

  5. Things We Know • NP-hard problem - Can only get approximation • Best possible integrality gap - LP relaxation Manokaran et al., 2008 • Solve using TRW-S, DD, PP Slower than graph-cuts • Use Range Move - Veksler, 2007 None of the guarantees of LP

  6. Real Motivation Gaps in Move-Making Literature Chekuri et al., 2001 2 2 + √2 O(√M) Multiplicative Bounds

  7. Real Motivation Gaps in Move-Making Literature Boykov, Veksler and Zabih, 1999 2 2 2 + √2 2M O(√M) - Multiplicative Bounds

  8. Real Motivation Gaps in Move-Making Literature Gupta and Tardos, 2000 2 2 2 + √2 4 O(√M) - Multiplicative Bounds

  9. Real Motivation Gaps in Move-Making Literature Komodakis and Tziritas, 2005 2 2 2 + √2 4 O(√M) 2M Multiplicative Bounds

  10. Real Motivation Gaps in Move-Making Literature 2 2 2 + √2 2 + √2 O(√M) O(√M) Multiplicative Bounds

  11. Outline • Move Space • Graph Construction • Sketch of the Analysis • Results

  12. Move Space • Initialize the labelling • Choose interval I of L’ labels • Each variable can • Retain old label • Choose a label from I • Choose best labelling Va Vb Iterate over intervals

  13. Outline • Move Space • Graph Construction • Sketch of the Analysis • Results

  14. Two Problems • Choose interval I of L’ labels • Each variable can • Retain old label • Choose a label from I • Choose best labelling Large L’ => Non-submodular Non-submodular Va Vb

  15. First Problem Submodular problem Va Vb Ishikawa, 2003; Veksler, 2007

  16. First Problem Non-submodular Problem Va Vb

  17. First Problem Submodular problem Va Vb Veksler, 2007

  18. First Problem am+1 bm+1 am+2 bm+2 am+2 bm+2 an bn t Va Vb

  19. First Problem am+1 bm+1 am+2 bm+2 am+2 bm+2 an bn t Va Vb

  20. First Problem am+1 bm+1 am+2 bm+2 am+2 bm+2 an bn t Va Vb

  21. First Problem am+1 bm+1 am+2 bm+2 am+2 bm+2 an bn t Va Vb

  22. First Problem am+1 bm+1 am+2 bm+2 am+2 bm+2 an bn t Va Vb Model unary potentials exactly

  23. First Problem am+1 bm+1 am+2 bm+2 am+2 bm+2 an bn t Va Vb Similarly for Vb

  24. First Problem am+1 bm+1 am+2 bm+2 am+2 bm+2 an bn t Va Vb Model convex pairwise costs

  25. First Problem Wanted to model ab;ik = wab min{ d(i-k), M } For all li, lk  I Have modelled ab;ik = wab d(i-k) For all li, lk  I Va Vb Overestimated pairwise potentials

  26. Second Problem • Choose interval I of L’ labels • Each variable can • Retain old label • Choose a label from I • Choose best labelling Non-submodular problem !! Va Vb

  27. Second Problem - Case 1 s ∞ ∞ am+1 bm+1 am+2 bm+2 an bn t Va Vb Both previous labels lie in interval

  28. Second Problem - Case 1 s ∞ ∞ am+1 bm+1 am+2 bm+2 an bn t Va Vb wab d(i-k)

  29. Second Problem - Case 2 s ∞ ub am+1 bm+1 am+2 bm+2 an bn t Va Vb Only previous label of Va lies in interval

  30. Second Problem - Case 2 s ∞ ub M am+1 bm+1 am+2 bm+2 an bn t Va Vb ub : unary potential of previous label of Vb

  31. Second Problem - Case 2 s ∞ ub M am+1 bm+1 am+2 bm+2 an bn t Va Vb wab d(i-k)

  32. Second Problem - Case 2 s ∞ ub M am+1 bm+1 am+2 bm+2 an bn t Va Vb wab ( d(i-m-1) + M )

  33. Second Problem - Case 3 am+1 bm+1 am+2 bm+2 an bn t Va Vb Only previous label of Vb lies in interval

  34. Second Problem - Case 3 s ∞ ua M am+1 bm+1 am+2 bm+2 an bn t Va Vb ua : unary potential of previous label of Va

  35. Second Problem - Case 4 am+1 bm+1 am+2 bm+2 an bn t Va Vb Both previous labels do not lie in interval

  36. Second Problem - Case 4 s ua ub Pab M M am+1 bm+1 ab am+2 bm+2 an bn t Va Vb Pab : pairwise potential for previous labels

  37. Second Problem - Case 4 s ua ub Pab M M am+1 bm+1 ab am+2 bm+2 an bn t Va Vb wab d(i-k)

  38. Second Problem - Case 4 s ua ub Pab M M am+1 bm+1 ab am+2 bm+2 an bn t Va Vb wab ( d(i-m-1) + M )

  39. Second Problem - Case 4 s ua ub Pab M M am+1 bm+1 ab am+2 bm+2 an bn t Va Vb Pab

  40. Graph Construction Find st-MINCUT. Retain old labelling if energy increases. am+1 bm+1 am+2 bm+2 an bn t Va Vb ITERATE

  41. Outline • Move Space • Graph Construction • Sketch of the Analysis • Results

  42. Va Vb Va Vb Previous labelling f’(.) Global Optimum f*(.) Analysis Va Vb Current labelling f(.) QC ≤ Q’C QP

  43. Va Vb Previous labelling f’(.) Analysis Va Vb Va Vb Current labelling f(.) Partially Optimal f’’(.) Q’0 ≤ QC ≤ Q’C

  44. Va Vb Previous labelling f’(.) Analysis Va Vb Va Vb Current labelling f(.) Partially Optimal f’’(.) QP- Q’0 ≥ QP - Q’C

  45. Analysis Va Vb Va Vb Va Vb Current labelling f(.) Partially Optimal f’’(.) Local Optimal f’(.) QP- Q’0 ≤ 0 ≤ 0 QP - Q’C

  46. Analysis Va Vb Va Vb Va Vb Current labelling f(.) Partially Optimal f’’(.) Local Optimal f’(.) QP- Q’0 ≤ 0 Take expectation over all intervals

  47. QP ≤ 2 + max 2M , L’ L’ M Q* QP ≤ O(√M) Q* Analysis Truncated Linear Gupta and Tardos, 2000 L’ = M 4 L’ = √2M 2 + √2 Truncated Quadratic L’ = √M

  48. Outline • Move Space • Graph Construction • Sketch of the Analysis • Results

  49. Synthetic Data - Truncated Linear Energy Time (sec) Faster than TRW-S Comparable to Range Moves With LP Relaxation guarantees

  50. Synthetic Data - Truncated Quadratic Energy Time (sec) Faster than TRW-S Comparable to Range Moves With LP Relaxation guarantees

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