A Seeded Image Segmentation Framework Unifying Graph Cuts and Random Walker Which Yields A New Algorithm

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A Seeded Image Segmentation Framework Unifying Graph Cuts and Random Walker Which Yields A New Algorithm . Ali Kemal Sinop * Computer Science Department Carnegie Mellon University Leo Grady Department of Imaging and Visualization Siemens Corporate Research.

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### A Seeded Image Segmentation Framework Unifying Graph Cuts and Random Walker Which Yields A New Algorithm

Ali Kemal Sinop *

Computer Science Department

Carnegie Mellon University

Department of Imaging and Visualization

Siemens Corporate Research

* work done while author was at Siemens Corporate Research

Outline
• Review of seeded segmentation – Graph Cuts and Random Walker
• Our generalized framework
• The q = ∞ case
• Comparison
• Conclusion
Seeded Segmentation Review

Interactive segmentation

• Four parts:
• Input foreground/background pixels (seeds) from the user
• Use image content to establish affinity (metric) relationships between pixels
• Perform energy minimization over the space of functions defined on pixels
• Assign a foreground/background label to each pixel corresponding to the value of the function at that pixel

abstraction

Seeded Segmentation Review – Graph Cuts

Graph cuts

Abstract image to a weighted graph

Compute min-cut/max-flow

44 image

44 weighted graph

3

1

2

2

1

4

S

T

5

6

1

abstraction

3

1V

1

2

2

1

4

5

6

1

Seeded Segmentation Review – Random Walker

Random Walker

Abstract image to a weighted graph

44 image

44 weighted graph

Compute probability that a random walker arrives at seed

Random walk view

Outline
• Review of seeded segmentation – Graph Cuts and Random Walker
• Our generalized framework
• The q = ∞ case
• Comparison
• Conclusion

Choice of q determines solution properties:

q = 1 Graph Cuts

q = 2 Random Walker

q =∞ ?

Generalized seeded segmentation framework

‘Algorithm A’

Generalized seeded segmentation: q = 1

Graph Cuts

Note: If x is binary, energy represents cut size

Unary terms implicit

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1

2

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r

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y

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;

2

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Generalized seeded segmentation: q = 2

Random Walker

Solution to random walk problem equivalent to minimization of the Dirichlet integral

with appropriate boundary conditions.

The solution is given by a harmonic function, i.e., a function

satisfying

1

1

T

T

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0

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¢

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2

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Generalized seeded segmentation: q = 2

Random Walker

Energy functional:

Subject to boundary conditions at seed locations

Euler-Lagrange:

Outline
• Review of seeded segmentation – Graph Cuts and Random Walker
• Our generalized framework
• The q = ∞ case
• Comparison
• Conclusion
The q = ∞ case

q = ∞

How to optimize?

The q = ∞ case

Problem: Uniqueness

Multiple solutions minimize functional

Solution:

minimizes the (q = 2) energy

Outline
• Review of seeded segmentation – Graph Cuts and Random Walker
• Our generalized framework
• The q = ∞ case
• Comparison
• Conclusion
Comparison - Theoretical

Metrication

q = ∞

q = 1 (Graph Cuts)

q = 2 (Random Walker)

Comparison - Quantitative

Stability relationship

Comparison - Qualitative

q = 1

(Graph Cuts)

q = 2

(Random Walker)

q = ∞

Outline
• Review of seeded segmentation – Graph Cuts and Random Walker
• Our generalized framework
• The q = ∞ case
• Comparison
• Conclusion
Conclusion

1) Graph Cuts and Random Walker algorithms may be seen as minimizing the same functional with respect to an L1 or L2 norm, respectively

2) The L∞ case was previously unexplored, may be optimized efficiently and produces “tight” segmentations with minimum sensitivity to seed number

L∞ paper: