2010/5/171 Overview of graph cuts

2010/5/172 Outline Introduction S-t Graph cuts Extension to multi-label problems Compare simulated annealing and alpha- expansion algorithm

2010/5/173 Introduction Discrete energy minimization methods that can be applied to Markov Random Fields (MRF) with binary labels or multi- labels.

2010/5/174 Outline Introduction S-t Graph cuts Extensions to multi-label problems Compare simulated annealing and alpha- expansion algorithm

2010/5/175 Max flow / Min cut Flow network Maximize amount of flows from source to sink Equal to minimum capacity removed from the network that no flow can pass from the source to the sink t s Max-flow/Min-cut method ： Augmenting paths (Ford Fulkerson Algorithm)

2010/5/176 A subset of edges such that source and sink become separated G(C)=<V,E-C> the cost of a cut : Minimum cut : a cut whose cost is the least over all cuts S-t Graph Cut

2010/5/177 How to separate a graph to two class? Two pixels p1 and p2 corresponds to two class s and t. Pixels p in the Graph classify by subtracting p with two pixels p1,p2. d1=(p-p1), d2 = (p-p2) If d1 is closer zero than d2, p is class s. Absolute of d1 and d2

2010/5/178 Noise in the boundary of two class The classified graph may have the noise occurs nearing the pixel (p1+p2)/2 Adding another constrain (smoothing) to prevent this problem.

2010/5/179 energy function energy function t-links n-links Boundary term Regional term n-links t s a cut C t-link

2010/5/1710 S-t Graph cuts for optimal boundary detection n-links t s a cut C hard constraint hard constraint Minimum cost cut can be computed in polynomial time

2010/5/1711 Global minimized for binary energy function Global minimized for binary energy function Characterization of binary energies that can be globally minimized by s-t graph cuts E(f) can be minimized by s-t graph cuts t-links n-links Boundary term Regional term (regular function)

2010/5/1712 Regular F 2 functions: What Energy Functions Can Be Minimized via Graph Cuts?

2010/5/1713 Outline Introduction S-t Graph cuts Extensions to multi-label problems Compare simulated annealing and alpha- expansion algorithm

2010/5/1714 Multi way Graph cut algorithm NP-hard problem(3 or more labels) two labels can be solved via s-t cuts (Greig et. al 1989) two labels can be solved via s-t cuts (Greig et. al 1989) Two approximation algorithms (Boykov et.al 1998,2001) Basic idea : break multi-way cut computation into a sequence of binary s-t cuts. Basic idea : break multi-way cut computation into a sequence of binary s-t cuts. Alpha-expansion Alpha-expansion Each label competes with the other labels for space in the image Each label competes with the other labels for space in the image Alpha-beta swap Alpha-beta swap Define a move which allows to change pixels from alpha to beta and beta to alpha Define a move which allows to change pixels from alpha to beta and beta to alpha

2010/5/1715 other labels a Alpha-expansion move Break multi-way cut computation into a sequence of binary s-t cuts

2010/5/1716 Alpha-expansion algorithm (|L| iterations) Stop when no expansion move would decrease energy

2010/5/1717 Alpha-expansion algorithm Guaranteed approximation ratio by the algorithm: Produces a labeling f such that, where f* is the global minimum Produces a labeling f such that, where f* is the global minimum and and Prove in : efficient graph-based energy minimization methods in computer vision

2010/5/1718 alpha-expansion moves initial solution -expansion

2010/5/1719 Alpha-Beta swap algorithm Handles more general energy function

2010/5/1720 Moves αexpansionα-βswapInitial labeling

2010/5/1721 Metric Semi-metric – – If V also satisfies the triangle inequality

2010/5/1722 Alpha-expansion : Metric Alpha-expansion satisfy the regular function Alpha-beta swap Prove in: what energy functions can be minimized via graph cuts?

2010/5/1723 Different types of Interaction V V(dL) dL=Lp-Lq Potts model “discontinuity preserving” Interactions V V(dL) dL=Lp-Lq “Convex” Interactions V V(dL) dL=Lp-Lq V(dL) dL=Lp-Lq “linear” model

2010/5/1724 convex vs. discontinuity-preserving ” “linear” V truncated “linear” V

2010/5/1725 The use of Alpha-expansion and alpha-beta swap Three energy function, each with a quadratic D p. –E1 = D p + min(K,|f p -f q | 2 ) –E2 uses the Potts model –E3 = D p + min(K,|f p -f q |) E1 : semi-metric (use ) E2,E3 : metric (can use both)

2010/5/1726 Outline Introduction S-t Graph cuts Extensions to multi-label problems Compare simulated annealing and alpha- expansion algorithm

2010/5/1727 Single “one-pixel” move (Simulated annealing) Single alpha-expansion move Only one pixel change its label at a time Large number of pixels can change their labels simultaneously Computationally intensive O(2^n) (s-t cuts)

2010/5/1728 參考文獻 Graph Cuts in Vision and Graphics: Theories and Application Fast Approximate Energy Minimization via Graph Cuts, 2001 What energy functions can be minimized via graph cuts?