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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.

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outline
Outline
  • Introduction
  • S-t Graph cuts
  • Extension to multi-label problems
  • Compare simulated annealing and alpha-expansion algorithm
introduction
Introduction
  • Discrete energy minimization methods that can be applied to Markov Random Fields (MRF) with binary labels or multi-labels.
outline1
Outline
  • Introduction
  • S-t Graph cuts
  • Extensions to multi-label problems
  • Compare simulated annealing and alpha-expansion algorithm
max flow min cut
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)

s t graph cut
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
how to separate a graph to two class
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
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.
energy function

a cut C

n-links

t-link

s

t-link

t

energy function

Regional term

Boundary term

n-links

t-links

s t graph cuts for optimal boundary detection

a cut C

hard

constraint

n-links

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
Global minimized for binary energy function

Regional term

Boundary term

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

t-links

n-links

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

(regular function)

outline2
Outline
  • Introduction
  • S-t Graph cuts
  • Extensions to multi-label problems
  • Compare simulated annealing and alpha-expansion algorithm
multi way graph cut 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

alpha expansion move

other labels

a

Alpha-expansion move

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

alpha expansion algorithm

(|L|iterations)

Alpha-expansion algorithm

Stop when no expansion move would decrease energy

alpha expansion algorithm1
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

alpha expansion moves

initial solution

-expansion

-expansion

-expansion

-expansion

-expansion

-expansion

-expansion

alpha-expansion moves
alpha beta swap algorithm
Alpha-Beta swap algorithm

Handles more general energy function

moves
Moves

Initial labeling

α-βswap

αexpansion

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

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

different types of interaction v

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
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)
outline3
Outline
  • Introduction
  • S-t Graph cuts
  • Extensions to multi-label problems
  • Compare simulated annealing and alpha-expansion algorithm
slide27

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)

slide28
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?