Overview of graph cuts

1
Overview of graph cuts
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Outline

• Introduction
• S-t Graph cuts
• Extension to multi-label problems
• Compare simulated annealing and alpha-expansion
  algorithm

3
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

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

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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)
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S-t Graph Cut

• A subset of edges such that source and
  sink become separated
• G(C)ltV,E-Cgt
• the cost of a cut
• Minimum cut a cut whose cost is the least over
  all cuts

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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

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Noise in the boundary of two class

• The classified graph may have the noise occurs
  nearing the pixel (p1p2)/2
• Adding another constrain (smoothing) to prevent
  this problem.

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energy function
Regional term
Boundary term
n-links
t-links
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S-t Graph cuts for optimal boundary detection
Minimum cost cut can be computed in polynomial
time
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Global minimized for binary energy function
Regional term
Boundary term
t-links
n-links

• 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)
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What Energy Functions Can Be Minimized via Graph
Cuts?

• Regular F2 functions

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Outline

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

14
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(Boykov et.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

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Alpha-expansion move
Break multi-way cut computation into a sequence
of binary s-t cuts
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Alpha-expansion algorithm
Stop when no expansion move would decrease energy
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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
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alpha-expansion moves
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Alpha-Beta swap algorithm
Handles more general energy function
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Moves
aexpansion
a-ßswap
Initial labeling
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Metric

• Semi-metric
• If V also satisfies the triangle inequality

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Alpha-expansion Metric

• Alpha-expansion satisfy the regular function
• Alpha-beta swap

Prove in what energy functions can be minimized
via graph cuts?
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Different types of Interaction V
Convex Interactions V
discontinuity preserving Interactions V
V(dL)
dLLp-Lq
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convex vs. discontinuity-preserving
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The use of Alpha-expansion and alpha-beta swap

• Three energy function, each with a quadratic Dp.
• E1 Dp min(K,fp-fq2)
• E2 uses the Potts model
• E3 Dp min(K,fp-fq)
• E1 semi-metric (use )
• E2,E3 metric (can use both)

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Outline

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

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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(2n) (s-t cuts)
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????

• 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?