Algorithms as long as MAP estimation in Markov R in addition to om Fields Vladimir Kolmogorov Univer

Algorithms as long as MAP estimation in Markov R in addition to om Fields Vladimir Kolmogorov Univer www.phwiki.com

Algorithms as long as MAP estimation in Markov R in addition to om Fields Vladimir Kolmogorov Univer

Labancz-Bleasdale, Melisa, Contributing Writer has reference to this Academic Journal, PHwiki organized this Journal Algorithms as long as MAP estimation in Markov R in addition to om Fields Vladimir Kolmogorov University College London Tutorial at GDR (Optimisation Discrète, Graph Cuts et Analyse d’Images) Paris, 29 November 2005 Note: these slides contain animation Energy function p q unary terms (data) pairwise terms (coherence) – xp are discrete variables ( as long as example, xp{0,1}) – qp( ) are unary potentials – qpq( , ) are pairwise potentials Minimisation algorithms Min Cut / Max Flow [Ford&Fulkerson ‘56] [Grieg, Porteous, Seheult ‘89] : non-iterative (binary variables) [Boykov, Veksler, Zabih ‘99] : iterative – alpha-expansion, alpha-beta swap, (multi-valued variables) + If applicable, gives very accurate results – Can be applied to a restricted class of functions BP – Max-product Belief Propagation [Pearl ‘86] + Can be applied to any energy function – In vision results are usually worse than that of graph cuts – Does not always converge TRW – Max-product Tree-reweighted Message Passing [Wainwright, Jaakkola, Willsky ‘02] , [Kolmogorov ‘05] + Can be applied to any energy function + For stereo finds lower energy than graph cuts + Convergence guarantees as long as the algorithm in [Kolmogorov ’05]

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Main idea: LP relaxation Goal: Minimize energy E(x) under constraints xp{0,1} In general, NP-hard problem! Relax discreteness constraints: allow xp[0,1] Results in linear program. Can be solved in polynomial time! Energy function with discrete variables LP relaxation tight not tight Solving LP relaxation Too large as long as general purpose LP solvers (e.g. interior point methods) Solve dual problem instead of primal: Formulate lower bound on the energy Maximize this bound When done, solves primal problem (LP relaxation) Two different ways to as long as mulate lower bound Via posi as long as ms: leads to maxflow algorithm Via convex combination of trees: leads to tree-reweighted message passing Lower bound on the energy function Energy function with discrete variables LP relaxation Notation in addition to Preliminaries

Energy function – visualisation node p edge (p,q) node q label 0 label 1 Energy function – visualisation node p edge (p,q) node q label 0 label 1 0 Reparameterisation 0 0 4 2 5 -1 -1

Reparameterisation Definition. is a reparameterisation of if they define the same energy: 0 0 3 2 5 Maxflow, BP in addition to TRW per as long as m reparameterisations 1 Part I: Lower bound via posi as long as ms ( maxflow algorithm) Lower bound via posi as long as ms [Hammer, Hansen, Simeone’84] non-negative maximize

Outline of part I Maximisation algorithm Consider functions of binary variables only Maximising lower bound as long as submodular functions Definition of submodular functions Overview of min cut/max flow Reduction to max flow Global minimum of the energy Maximising lower bound as long as non-submodular functions Reduction to max flow More complicated graph Part of optimal solution Submodular functions of binary variables Definition: E is submodular if every pairwise term satisfies Can be converted to “canonical as long as m”: 2 1 2 3 4 1 0 0 0 5 zero cost Overview of min cut/max flow

Min Cut problem source sink 2 1 1 2 3 4 5 Directed weighted graph Min Cut problem sink 2 1 1 2 3 4 5 S = {source, node 1} T = {sink, node 2, node 3} Cut: source Min Cut problem Task: Compute cut with minimum cost sink 2 1 1 2 3 4 5 S = {source, node 1} T = {sink, node 2, node 3} Cut: Cost(S,T) = 1 + 1 = 2 source

Maxflow algorithm sink 2 1 1 2 3 4 5 source value(flow)=0 Maxflow algorithm sink 2 1 1 2 3 4 5 value(flow)=0 source Maxflow algorithm sink 1 1 0 3 3 4 4 value(flow)=1 source

Maxflow algorithm sink 1 1 0 3 3 4 4 value(flow)=1 source Maxflow algorithm sink 1 0 0 3 4 3 3 value(flow)=2 source Maxflow algorithm sink 1 0 0 3 4 3 3 value(flow)=2 source

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Maxflow algorithm value(flow)=2 sink 1 0 0 3 4 3 3 source Maximising lower bound as long as submodular functions: Reduction to maxflow Maxflow algorithm in addition to reparameterisation 2 1 2 3 4 1 0 0 0 5 sink 2 1 1 2 3 4 5 source value(flow)=0 0

Maxflow algorithm in addition to reparameterisation sink 2 1 1 2 3 4 5 value(flow)=0 2 1 2 3 4 1 0 0 0 5 0 source Maxflow algorithm in addition to reparameterisation sink 1 1 0 3 3 4 4 value(flow)=1 1 0 3 3 4 1 0 0 0 4 1 source Maxflow algorithm in addition to reparameterisation sink 1 1 0 3 3 4 4 value(flow)=1 1 0 3 3 4 1 0 0 0 4 1 source

Summary MAP estimation algorithms are based on LP relaxation Maximize lower bound Two ways to as long as mulate lower bound Via posi as long as ms: leads to maxflow algorithm Polynomial time solution But: applicable as long as restricted energies (e.g. binary variables) Submodular functions: global minimum Non-submodular functions: part of optimal solution Via convex combination of trees: leads to TRW algorithm Convergence in the limit ( as long as TRW-S) Applicable to arbitrary energy function Graph cuts vs. TRW: Accuracy: similar Generality: TRW is more general Speed: as long as stereo TRW is currently 2-5 times slower. But: 3 vs. 50 years of research! More suitable as long as parallel implementation (GPU Hardware) Discrete vs. continuous functionals Continuous as long as mulation (Geodesic active contours) Maxflow algorithm Global minimum, polynomial-time Metrication artefacts Level sets Numerical stability Geometrically motivated Invariant under rotation Discrete as long as mulation (Graph cuts) Geo-cuts Continuous functional Construct graph such that as long as smooth contours C Class of continuous functionals [Boykov&Kolmogorov’03], [Kolmogorov&Boykov’05]: Geometric length/area (e.g. Riemannian) Flux of a given vector field Regional term

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