Probabilistic Watershed Sampling all spanning forests for seeded - - PowerPoint PPT Presentation

Probabilistic Watershed

Sampling all spanning forests for seeded segmentation and semi-supervised learning

Enrique Fita Sanmart´ ın Sebastian Damrich Fred A. Hamprecht

HCI/IWR at Heidelberg University

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What do we do?

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We count forests! 1 2 3

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Framework

Graph

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Framework

s1

s2

Graph Seeds (labeled nodes)

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Framework

s1

s2

0.16 0.10 0.36 0.43 0.92 1.20 1.61 0.51 0.70 0.22 0.00 0.05

Graph Seeds (labeled nodes) Edge-Costs∼affinity between nodes

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Framework

s1 s2

0.22 0.00 0.05 0.92 1.20 1.61 0.51 0.70 0.16 0.10 0.36 0.43

Graph Seeds (labeled nodes) Edge-Costs∼affinity between nodes Forest

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Forests

s1 s2

0.22 0.00 0.05 0.92 1.20 1.61 0.51 0.70 0.16 0.10 0.36 0.43

Watershed forest/ minimum cost Spanning Forest (mSF)

s1 s2 s1 s2 s1 s2 s1 s2 s1 s2 s1 s2 Enrique Fita Sanmartin Probabilistic Watershed 5 / 12

Counting Forests

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Probabilistic Watershed

Pr(f ) = exp (−µc(f ))

• f ′∈Fs2

s1

exp

• −µc(f ′)

= w(f )

• f ′∈Fs2

s1

w(f ′) s1 s2 s1 s2 s1 s2 s1 s2 mSF s1 s2 0.00 0.30 0.45 0.21 0.52 0.72 0.68 0.77 1.00 1

Probabilistic Watershed → Pr(q ∼ s2) :=

• f ∈Fs1

s2,q

Pr(f ) = w

• Fs1

s2,q

• w
• Fs1

s2

=

• f ∈Fs1

s2,q

w (f )

• f ∈Fs1

s2

w (f ) .

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How do we count the forests?

192

s1 s2

Matrix Tree Theorem [Kirchhoff, 1847]

Let G = (V , E, w) an edge-weighted multigraph, w(T ),the sum of the weights of the spanning trees of G, ✶ is a column vector of 1’s, L is the Laplacian matrix and L[v] is the Laplacian matrix after removing an abitrary row and column v, then w(T ) :=

• t∈T

w(t) =

• t∈T
• e∈Et

w(e) = 1 |V | det

• L +

1 |V | ✶✶⊤ = det(L[v]).

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How do we count the forests?

288

s1 s2

Modification Matrix Tree Theorem

Let G = (V , E, w) be an undirected edge-weighted connected graph, r eff

uv the effective resistance

distance between u, v ∈ V arbitrary vertices and w(Fv

u ) the sum of the weights of the 2-trees spanning

forests separating u and v, then w(Fv

u ) = w(T )r eff uv .

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Probabilistic Watershed=Random Walker[Grady, 2006]

+

Random Walker [Grady, 2006]

=

Probabilistic Watershed

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Power Watershed new interpretation

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Summary

Probabilistic Watershed=Random Walker [Grady, 2006]. New interpretations of the Power Watershed [Couprie et al., 2011]. Technique to count forests.

Pr(f ) = exp (−µc(f ))

• f ′∈Fs2

s1

exp

• −µc(f ′)

= w(f )

• f ′∈Fs2

s1

w(f ′) s1 s2 s1 s2 s1 s2 s1 s2 mSF s1 s2 0.00 0.30 0.45 0.21 0.52 0.72 0.68 0.77 1.00 1

Poster

Room East Exhibition Hall B + C #81 10:45 AM – 12:45 PM

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References

Couprie, C., Grady, L., Najman, L., and Talbot, H. (2011). Power watershed: A unifying graph-based optimization framework. IEEE Transactions on Pattern Analysis and Machine Intelligence. Grady, L. (2006). Random walks for image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence. Kirchhoff, G. (1847). ¨ Uber die Aufl¨

• sung der Gleichungen, auf welche man bei der Untersuchung der linearen

Vertheilung galvanischer Str¨

• me gef¨

uhrt wird. Annalen der Physik, 148:497–508.

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