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Parametric Max-Flow Algorithms for Total Variation Minimization

Parametric Max-Flow Algorithms for Total Variation Minimization. W.Yin (Rice University) joint with D.Goldfarb (Columbia), Y.Zhang (Rice), Y.Wang (Rice) 06-01-2007, UCLA Math, Host: S.Osher. TexPoint fonts used in EMF:. Image Processing. filter black box. Noise removal filter.

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Parametric Max-Flow Algorithms for Total Variation Minimization

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  1. Parametric Max-Flow Algorithms for Total Variation Minimization W.Yin (Rice University) joint with D.Goldfarb (Columbia), Y.Zhang (Rice), Y.Wang (Rice) 06-01-2007, UCLA Math, Host: S.Osher TexPoint fonts used in EMF:

  2. Image Processing filter black box

  3. Noise removal filter

  4. Texture removal filter

  5. Variational image processing Treat an image f as a function

  6. Output u as a minimizer of certain functional

  7. Computational methods • 1. PDE-based Gradient descent: • low memory usage • slow convergence • 2. SOCP / interior-point method: • high memory usage • better convergence SOCP: Goldfarb-Yin 05’

  8. 3. Network flows methods: • low memory usage • very fast • not as general

  9. Max flow approach outline: applicable to anisotropic TV(u) – i.e., l1 norm • Decompose f into K level sets

  10. Max flow approach outline: applicable to anisotropic TV(u) – i.e., l1 norm • Decompose f into K level sets • For each Fl, obtain Ul by solving a max-flow prob • Construct a minimizer u from the minimizers Ul Chan-Esedoglu 05’, Yin-Goldfarb-Osher 06’

  11. Requires monotonicity of Yin-Goldfarb-Osher 05’, Darbon-Sigelle 05’, Allard 06’

  12. Questions: • How do we solve the binary problems? • How many binary problems do we solve? Next, a short introduction to the max-flow/min-cut problem…

  13. s t A capacitated network • A Network is a graph G with nodes and edges: • Special nodes s (source)andt (sink) • Edges carry flow • Each edge (i,j) has a maximum capacity ci,j

  14. s t A capacitated network • A Network is a graph G with nodes and edges: • Special nodes s (source)andt (sink) • Edges carry flow • Each edge (i,j) has a maximum capacity ci,j • An s-t cut (S,T) is a 2-partition of V such that s in S, t in T • Cut value:the total s-t cap. across the cut=3+7+11=21

  15. s t A capacitated network • A Network is a graph G with nodes and edges: • Special nodes s (source)andt (sink) • Edges carry flow • Each edge (i,j) has a maximum capacity ci,j • An s-t cut (S,T) is a 2-partition of V such that s in S, t in T • A min s-t cut is one that gives the minimum cut value • Cut value:the total s-t cap. across the cut=15+3=18

  16. s t A capacitated network • A Network is a graph G with nodes and edges: • Special nodes s (source)andt (sink) • Edges carry flow • Each edge (i,j) has a maximum capacity ci,j • An s-t cut (S,T) is a 2-partition of V such that s in S, t in T • A min s-t cut is one that gives the minimum cut value • Important fact: Finding a min-cut = finding a max-flow • Cut value:the total s-t cap. across the cut=15+3=18

  17. Max flow problem

  18. Max flow problem Min cut problem (dual of above)

  19. s

  20. s t Combining 1,2,3 gives a min cut formulation!

  21. s 1 1 1 1 1 1 t

  22. s 1 1 1 1 1 1 t

  23. s 1 1 1 1 1 1 t

  24. s 1 1 1 1 1 1 t

  25. s 1 1 1 1 1 1 t

  26. Isotropic TV v.s. Anisotropic TV Watersnake: Nguyen-Worring-van den Boomgaard 03’

  27. s t

  28. Questions: • How to minimize the binary energy? Answered. • How many binary problems do we solve?

  29. Is it good to work with each level instead of the entire cake? • = finding a minimum cut of a capacitated network • For a 8-bit image, there are 28=256 levels • For a 16-bit image, there are 216=65536 levels • Answer depends on • how fast we can solve each • how many we do need to solve

  30. Outline: Further steps • Decompose f into K level sets Fi • For each Fi, obtain Ui • Uimin-cut of a network (Graph-Cut) • min-cutmax-flow • (For TV/L1) Combine K networks (para. max flow) • (For ROF) Reduce Kmax-flows to logK max-flows (e.g., K=216=65536, logK=16) • Construct a minimizer u from the minimizers Ui

  31. t Max flow / min cut algorithms • Preflow push (Goldberg-Tarjan) • Best complexity: O(nmlog(n2/m)) • Boykov-Kolmogrov, push on path • Uses approximate shortest path • Not strongly polynomial • Very fast on graph with small neighborhoods • Parametric max flow (Gallo et al.) • Complexity same as preflow push: O(nmlog(n2/m)), if # of levels is O(n) • Arcs out of source have increasing capacities • Arcs into sink have decreasing capacities s network Preflow: Goldberg-Tarjan, B-K: Boykov-Kolmogrov 04’ Parametric: Gallo, Grigoriadis, Tarjan 89’, Hochbaum 01’

  32. Important questions left out • How many functions can be minimized on network? Answers: • # is limited, but much more than it appears to be • Theory is related to pseudo-boolean polynomials, but is not complete • Approach can be combined with others.

  33. Important questions left out • What are the applications of the imaging models Answers: • Applications found in • Face recognition, image registration, medical imaging • Theories and computations extended to high-dimensional and higher-codimensional data analysis

  34. S T u v Thank you. Questions? Network flow no leaks!

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