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

2 Various Optimization Approaches for Imaging Problems Image Processing filter black box

3 Various Optimization Approaches for Imaging Problems

4 Noise removal filter

5 Various Optimization Approaches for Imaging Problems Texture removal filter

6 Various Optimization Approaches for Imaging Problems Variational image processing Treat an image f as a function

7 Various Optimization Approaches for Imaging Problems Output u as a minimizer of certain functional

8 Various Optimization Approaches for Imaging Problems 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’

9 Various Optimization Approaches for Imaging Problems 3. Network flows methods: low memory usage very fast not as general

10 Various Optimization Approaches for Imaging Problems Max flow approach outline: 1.Decompose f into K level sets applicable to anisotropic TV(u) – i.e., l 1 norm

11 Various Optimization Approaches for Imaging Problems Max flow approach outline: 1.Decompose f into K level sets 2.For each F l, obtain U l by solving a max-flow prob 3.Construct a minimizer u from the minimizers U l applicable to anisotropic TV(u) – i.e., l 1 norm Chan-Esedoglu 05’, Yin-Goldfarb-Osher 06’

12 Various Optimization Approaches for Imaging Problems

13 Various Optimization Approaches for Imaging Problems Requires monotonicity of Yin-Goldfarb-Osher 05’, Darbon-Sigelle 05’, Allard 06’

14 Various Optimization Approaches for Imaging Problems Questions: 1.How do we solve the binary problems? 2.How many binary problems do we solve? Next, a short introduction to the max-flow/min-cut problem…

15 Various Optimization Approaches for Imaging Problems A Network is a graph G with nodes and edges: Special nodes s (source) and t (sink) Edges carry flow Each edge (i,j) has a maximum capacity c i,j A capacitated network t s

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

17 Various Optimization Approaches for Imaging Problems A Network is a graph G with nodes and edges: Special nodes s (source) and t (sink) Edges carry flow Each edge (i,j) has a maximum capacity c i,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 A capacitated network t s Cut value: the total s-t cap. across the cut=15+3=18

18 Various Optimization Approaches for Imaging Problems A Network is a graph G with nodes and edges: Special nodes s (source) and t (sink) Edges carry flow Each edge (i,j) has a maximum capacity c i,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 A capacitated network t s Cut value: the total s-t cap. across the cut=15+3=18

19 Various Optimization Approaches for Imaging Problems Max flow problem

20 Various Optimization Approaches for Imaging Problems Max flow problem Min cut problem (dual of above)

21 Various Optimization Approaches for Imaging Problems

22 Various Optimization Approaches for Imaging Problems

23 Various Optimization Approaches for Imaging Problems

24 Various Optimization Approaches for Imaging Problems s

25 Various Optimization Approaches for Imaging Problems s t Combining 1,2,3 gives a min cut formulation!

26 Various Optimization Approaches for Imaging Problems

27 Various Optimization Approaches for Imaging Problems t s

28 Various Optimization Approaches for Imaging Problems t s

29 Various Optimization Approaches for Imaging Problems t s

30 Various Optimization Approaches for Imaging Problems t s

31 Various Optimization Approaches for Imaging Problems t s

32 Various Optimization Approaches for Imaging Problems

33 Various Optimization Approaches for Imaging Problems Isotropic TV v.s. Anisotropic TV Watersnake: Nguyen-Worring-van den Boomgaard 03’

34 Various Optimization Approaches for Imaging Problems

35 Various Optimization Approaches for Imaging Problems Questions: 1.How to minimize the binary energy? Answered. 2.How many binary problems do we solve?

36 Various Optimization Approaches for Imaging Problems 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 2 8 =256 levels For a 16-bit image, there are 2 16 =65536 levels Answer depends on 1.how fast we can solve each 2.how many we do need to solve

37 Various Optimization Approaches for Imaging Problems Outline: Further steps 1.Decompose f into K level sets F i 2.For each F i, obtain U i 1. U i min-cut of a network (Graph-Cut) 2. min-cut max-flow 3.(For TV/L 1 ) Combine K networks (para. max flow) 4.(For ROF) Reduce K max-flows to log K max-flows (e.g., K=2 16 =65536, logK=16) 3.Construct a minimizer u from the minimizers U i

38 Various Optimization Approaches for Imaging Problems Divide and conquer (Darbon & Sigelle) G1G1 G1G1 G2G2 G1G1 G1G1 G2G2 G0G0 G1G1 G2G2 Divide and Conquer: Darbon-Sigelle 06’

39 Various Optimization Approaches for Imaging Problems Max flow / min cut algorithms Preflow push (Goldberg-Tarjan) –Best complexity: O(nmlog(n 2 /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(n 2 /m)), if # of levels is O(n) –Arcs out of source have increasing capacities –Arcs into sink have decreasing capacities s t network Preflow: Goldberg-Tarjan, B-K: Boykov-Kolmogrov 04’ Parametric: Gallo, Grigoriadis, Tarjan 89’, Hochbaum 01’

40 Various Optimization Approaches for Imaging Problems ModelNameSizebest λtotal time TV/L 1 Barbara (8-bit) 512× TV/L 1 Barbara (8-bit) 1024× ROFBarbara (8-bit) 512× ROFBarbara (8-bit) 1024× ROFBarbara (16-bit) 1024× Max-flow (Matlab/C++) numerical results Laptop - CPU: Pentium Duo 2.0GHz, Memory: 1.5 GB

41 Various Optimization Approaches for Imaging Problems Important questions left out 1.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.

42 Various Optimization Approaches for Imaging Problems Important questions left out 2.What are the applications of the imaging models Answers: 1.Applications found in Face recognition, image registration, medical imaging 2.Theories and computations extended to high-dimensional and higher-codimensional data analysis

43 Various Optimization Approaches for Imaging Problems Thank you. Questions? S T u v Network flow no leaks!