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GraphCut -based Optimisation for Computer Vision

GraphCut -based Optimisation for Computer Vision. Ľubor Ladický. Overview. Motivation Min-Cut / Max-Flow (Graph Cut) Algorithm Markov and Conditional Random Fields Random Field Optimisation using Graph Cuts Submodular vs. Non-Submodular Problems Pairwise vs. Higher Order Problems

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GraphCut -based Optimisation for Computer Vision

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  1. GraphCut-based Optimisation for Computer Vision Ľubor Ladický

  2. Overview • Motivation • Min-Cut / Max-Flow (Graph Cut) Algorithm • Markov and Conditional Random Fields • Random Field Optimisation using Graph Cuts • Submodular vs. Non-Submodular Problems • Pairwise vs. Higher Order Problems • 2-Label vs. Multi-Label Problems • Recent Advances in Random Field Optimisation • Conclusions

  3. Overview • Motivation • Min-Cut / Max-Flow (Graph Cut) Algorithm • Markov and Conditional Random Fields • Random Field Optimisation using Graph Cuts • Submodular vs. Non-Submodular Problems • Pairwise vs. Higher Order Problems • 2-Label vs. Multi-Label Problems • Recent Advances in Random Field Optimisation • Conclusions

  4. Image Labelling Problems Assign a label to each image pixel Geometry Estimation Image Denoising Object Segmentation Depth Estimation Sky Building Tree Grass

  5. Image Labelling Problems • Labellings highly structured Unprobable labelling Impossible labelling Possible labelling

  6. Image Labelling Problems • Labellings highly structured • Labels highly correlated with very complex dependencies • Neighbouring pixels tend to take the same label • Low number of connected components • Classes present may be seen in one image • Geometric / Location consistency • Planarity in depth estimation • … many others (task dependent)

  7. Image Labelling Problems • Labelling highly structured • Labels highly correlated with very complex dependencies • Independent label estimation too hard

  8. Image Labelling Problems • Labelling highly structured • Labels highly correlated with very complex dependencies • Independent label estimation too hard • Whole labelling should be formulated as one optimisation problem

  9. Image Labelling Problems • Labelling highly structured • Labels highly correlated with very complex dependencies • Independent label estimation too hard • Whole labelling should be formulated as one optimisation problem • Number of pixels up to millions • Hard to train complex dependencies • Optimisation problem is hard to infer

  10. Image Labelling Problems • Labelling highly structured • Labels highly correlated with very complex dependencies • Independent label estimation too hard • Whole labelling should be formulated as one optimisation problem • Number of pixels up to millions • Hard to train complex dependencies • Optimisation problem is hard to infer Vision is hard !

  11. Image Labelling Problems • You can • either • Change the subject from the Computer Vision to the History of Renaissance Art Vision is hard !

  12. Image Labelling Problems • You can • either • Change the subject from the Computer Vision to the History of Renaissance Art • or • Learn everything about Random Fields and Graph Cuts Vision is hard !

  13. Foreground / Background Estimation Rother et al. SIGGRAPH04

  14. Foreground / Background Estimation Data term Smoothness term Data term Estimated using FG / BG colour models Smoothness term where Intensity dependent smoothness

  15. Foreground / Background Estimation Data term Smoothness term

  16. Foreground / Background Estimation Data term Smoothness term • How to solve this optimisation problem?

  17. Foreground / Background Estimation Data term Smoothness term • How to solve this optimisation problem? • Transform into min-cut / max-flow problem • Solve it using min-cut / max-flow algorithm

  18. Overview • Motivation • Min-Cut / Max-Flow (Graph Cut) Algorithm • Markov and Conditional Random Fields • Random Field Optimisation using Graph Cuts • Submodular vs. Non-Submodular Problems • Pairwise vs. Higher Order Problems • 2-Label vs. Multi-Label Problems • Applications • Conclusions

  19. Max-Flow Problem source 9 5 4 2 2 5 3 2 3 1 1 5 6 3 2 5 6 8 sink • Task : • Maximize the flow from the sink to the source such that • 1) The flow it conserved for each node • 2) The flow for each pipe does not exceed the capacity

  20. Max-Flow Problem source 9 5 4 2 2 5 3 2 3 1 1 5 6 3 2 5 6 8 sink

  21. Max-Flow Problem flow from the source source flow from node i to node j 9 5 capacity 4 2 2 set of edges 5 3 2 3 1 1 5 6 3 2 set of nodes conservation of flow 5 6 8 sink

  22. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 9 5 4 2 2 5 3 2 3 1 1 5 6 3 2 5 6 8 sink

  23. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 9 5 4 2 2 5 3 2 3 1 1 5 6 3 2 5 6 8 sink

  24. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 9 5 4 2 2 5 3 2 3 1 1 5 6 3 2 5 6 8 sink flow = 3

  25. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 9 5-3 4 2+3 2 5 3-3 2 3 1 1 6-3 5 +3 3 2 5 6 8-3 sink flow = 3

  26. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 9 2 4 5 2 5 2 3 1 1 3 5 3 3 2 5 6 5 sink flow = 3

  27. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 9 2 4 5 2 5 2 3 1 1 3 5 3 3 2 5 6 5 sink flow = 3

  28. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 9 2 4 5 2 5 2 3 1 1 3 5 3 3 2 5 6 5 sink flow = 6

  29. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 9-3 2 4 5 2 5 2 +3 3-3 1 1 3 5 3 3-3 2 +3 5 6 5-3 sink flow = 6

  30. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 6 2 4 5 2 5 2 3 1 1 3 5 3 3 2 5 6 2 sink flow = 6

  31. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 6 2 4 5 2 5 2 3 1 1 3 5 3 3 2 5 6 2 sink flow = 6

  32. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 6 2 4 5 2 5 2 3 1 1 3 5 3 3 2 5 6 2 sink flow = 11

  33. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 6-5 2 4 5 2+5 5-5 2 3 1+5 1 3 5-5 3 3 2 5-5 6 2 sink flow = 11

  34. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 1 2 4 5 7 2 3 1 6 3 3 3 2 6 2 sink flow = 11

  35. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 1 2 4 5 7 2 3 1 6 3 3 3 2 6 2 sink flow = 11

  36. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 1 2 4 5 7 2 3 1 6 3 3 3 2 6 2 sink flow = 13

  37. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 1 2-2 4 5 7 2-2 3 +2 1 6 3 3 +2 3 2-2 6 2-2 sink flow = 13

  38. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 1 4 5 7 3 2 1 6 3 3 2 3 6 sink flow = 13

  39. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 1 4 5 7 3 2 1 6 3 3 2 3 6 sink flow = 13

  40. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 1 4 5 7 3 2 1 6 3 3 2 3 6 sink flow = 15

  41. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 1 4-2 5 7 3 2 1 3-2 6 3+2 2-2 3 +2 6-2 sink flow = 15

  42. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 1 2 5 7 3 2 1 6 1 5 3 2 4 sink flow = 15

  43. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 1 2 5 7 3 2 1 6 1 5 3 2 4 sink flow = 15

  44. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph (“subtract” the flow) Find the path in the residual graph End 1 2 5 7 3 2 1 6 1 5 3 2 4 sink flow = 15

  45. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Why is the solution globally optimal ? 1 2 5 7 3 2 1 6 1 5 3 2 4 sink flow = 15

  46. Max-Flow Problem S source Ford & Fulkerson algorithm (1956) Why is the solution globally optimal ? 1. Let S be the set of reachable nodes in the residual graph 1 2 5 7 3 2 1 6 1 5 3 2 4 sink flow = 15

  47. Max-Flow Problem S source Ford & Fulkerson algorithm (1956) Why is the solution globally optimal ? 1. Let S be the set of reachable nodes in the residual graph 2. The flow from S to V - S equals to the sum of capacities from S to V – S 9 5 4 2 2 5 3 2 3 1 1 5 6 3 2 5 6 8 sink

  48. Max-Flow Problem A source Ford & Fulkerson algorithm (1956) Why is the solution globally optimal ? 1. Let S be the set of reachable nodes in the residual graph 2. The flow from S to V - S equals to the sum of capacities from S to V – S 3. The flow from any A to V - A is upper bounded by the sum of capacities from A to V – A 9 5 4 2 2 5 3 2 3 1 1 5 6 3 2 5 6 8 sink

  49. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Why is the solution globally optimal ? 1. Let S be the set of reachable nodes in the residual graph 2. The flow from S to V - S equals to the sum of capacities from S to V – S 3. The flow from any A to V - A is upper bounded by the sum of capacities from A to V – A 4. The solution is globally optimal 5/5 8/9 2/4 0/2 0/2 5/5 3/3 2/2 3/3 0/1 0/1 5/6 5/5 3/3 0/2 5/5 2/6 8/8 sink flow = 15 Individual flows obtained by summing up all paths

  50. Max-Flow Problem source Ford & Fulkerson algorithm (1956) Why is the solution globally optimal ? 1. Let S be the set of reachable nodes in the residual graph 2. The flow from S to V - S equals to the sum of capacities from S to V – S 3. The flow from any A to V - A is upper bounded by the sum of capacities from A to V – A 4. The solution is globally optimal 5/5 8/9 2/4 0/2 0/2 5/5 3/3 2/2 3/3 0/1 0/1 5/6 5/5 3/3 0/2 5/5 2/6 8/8 sink flow = 15

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