Overview of graph cuts

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# Overview of graph cuts - PowerPoint PPT Presentation

Overview of graph cuts. Outline. Introduction S-t Graph cuts Extension to multi-label problems Compare simulated annealing and alpha-expansion algorithm. Introduction. Discrete energy minimization methods that can be applied to Markov Random Fields (MRF) with binary labels or multi-labels.

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### Overview of graph cuts

Outline
• Introduction
• S-t Graph cuts
• Extension to multi-label problems
• Compare simulated annealing and alpha-expansion algorithm
Introduction
• Discrete energy minimization methods that can be applied to Markov Random Fields (MRF) with binary labels or multi-labels.
Outline
• Introduction
• S-t Graph cuts
• Extensions to multi-label problems
• Compare simulated annealing and alpha-expansion algorithm
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

s

t

Max-flow/Min-cut method ：Augmenting paths (Ford Fulkerson Algorithm)

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
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
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.

a cut C

s

t

energy function

Regional term

Boundary term

a cut C

hard

constraint

hard

constraint

s

t

S-t Graph cuts for optimal boundary detection

Minimum cost cut can be computed in polynomial time

Global minimized for binary energy function

Regional term

Boundary term

• Characterization of binary energies that can be globally minimized by s-t graph cuts

E(f) can be minimized by s-t graph cuts

(regular function)

Outline
• Introduction
• S-t Graph cuts
• Extensions to multi-label problems
• Compare simulated annealing and alpha-expansion algorithm
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 approximation algorithms(Boykovet.al 1998,2001)

Basic idea : break multi-way cut computation into a sequence of binary s-t cuts.

• Alpha-expansion

Each label competes with the other labels for space in the image

• Alpha-beta swap

Define a move which allows to change pixels from alpha to beta and beta to alpha

other labels

a

Alpha-expansion move

Break multi-way cut computation into a sequence of binary s-t cuts

(|L|iterations)

Alpha-expansion algorithm

Stop when no expansion move would decrease energy

Alpha-expansion algorithm
• Guaranteed approximation ratio by the algorithm:
• Produces a labeling f such that ,where f* is the global minimum

and

Prove in : efficient graph-based energy minimization methods in computer vision

initial solution

-expansion

-expansion

-expansion

-expansion

-expansion

-expansion

-expansion

alpha-expansion moves
Alpha-Beta swap algorithm

Handles more general energy function

Moves

Initial labeling

α-βswap

αexpansion

Metric
• Semi-metric
• If V also satisfies the triangle inequality
Alpha-expansion : Metric
• Alpha-expansion satisfy the regular function
• Alpha-beta swap

Prove in: what energy functions can be minimized via graph cuts?

V(dL)

Potts

model

“linear”

model

V(dL)

V(dL)

dL=Lp-Lq

dL=Lp-Lq

dL=Lp-Lq

Different types of Interaction V

“Convex”

Interactions V

“discontinuity preserving”

Interactions V

V(dL)

dL=Lp-Lq

The use of Alpha-expansion and alpha-beta swap
• Three energy function, each with a quadratic Dp.
• E1 = Dp + min(K,|fp-fq|2)
• E2 uses the Potts model
• E3 = Dp + min(K,|fp-fq|)
• E1 : semi-metric (use )
• E2,E3 : metric (can use both)
Outline
• Introduction
• S-t Graph cuts
• Extensions to multi-label problems
• Compare simulated annealing and alpha-expansion algorithm

Single “one-pixel” move (Simulated annealing)

Single alpha-expansion move

Large number of pixels can change their labels simultaneously

Only one pixel change its label at a time

Computationally intensive O(2^n)

(s-t cuts)

• 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?