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Directed Probabilistic Watershed
Enrique Fita Sanmart́ın Sebastian Damrich Fred A. Hamprecht
HCI/IWR at Heidelberg University
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 1 / 19
Transductive semi-supervised learningalgorithm on directed graphs
s1 ? ?
? ? ?
? ? s2
=⇒Directed
ProbabilisticWatershed
s1
s2
Web graphs
Citation graphs
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Enrique Fita Sanmart́ın Directed Probabilistic Watershed 3 / 19
Probabilistic WatershedUndirected graphs
Fita Sanmart́ın et al. (2019)
=⇒ Directed Probabilistic WatershedDirected graphs
Forests =⇒ In-forests rootedat the seeds
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 4 / 19
Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning, Fita Sanmart́ın et al. (2019)
Probabilistic WatershedUndirected graphs
Fita Sanmart́ın et al. (2019)
=⇒ Directed Probabilistic WatershedDirected graphs
Forests =⇒ In-forests rootedat the seeds
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 4 / 19
Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning, Fita Sanmart́ın et al. (2019)
Framework
s1
s2
s1
s2
Directed Graph
Seeds (labeled nodes)
Edge-Costs, ce(∼affinity between nodes)
Classification
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 5 / 19
Framework
s1
s2
s1
s2
Directed Graph
Seeds (labeled nodes)
Edge-Costs, ce(∼affinity between nodes)
Classification
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 5 / 19
Framework
s1
s2
s1
s2
Directed Graph
Seeds (labeled nodes)
Edge-Costs, ce(∼affinity between nodes)
Classification
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 5 / 19
Framework
s1
s2
s1
s2
Directed Graph
Seeds (labeled nodes)
Edge-Costs, ce(∼affinity between nodes)
Classification
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 5 / 19
Definition in-tree & in-forest
(a) Directed graph
s
(b) In-tree rooted at s
s1
s2
(c) In-forest rooted at s1 and s2
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 6 / 19
Definition in-tree & in-forest
(a) Directed graph
s
(b) In-tree rooted at s
s1
s2
(c) In-forest rooted at s1 and s2
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 6 / 19
Definition in-tree & in-forest
(a) Directed graph
s
(b) In-tree rooted at s
s1
s2
(c) In-forest rooted at s1 and s2
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 6 / 19
Main question
What is the probability of samplingan in-forest such that a node ofinterest, q, belongs to a tree rootedat a certain seed?
s1
q
s2
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Main question
What is the probability of samplingan in-forest such that a node ofinterest, q, belongs to a tree rootedat a certain seed?
s1
q
s2
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Directed Probabilistic Watershed Probabilities
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 8 / 19
Directed Probabilistic Watershed Probabilities
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 9 / 19
Gibbs distribution
Gibbs distribution
Pr(F ) ∝ w(F ) := exp(− µ c(F )︸︷︷︸∑
e∈Fce
)
s1
s2
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Directed Probabilistic Watershed Probabilities
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Directed Probabilistic Watershed Probabilities
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Computation Probabilities Directed Probabilistic Watershed
Number in-forests increasesexponentially with the number
of nodes and edges=⇒ Naive approach infeasible
Directed Matrix Tree TheoremLeenheer (2019)
=⇒ Efficient computation probabilities
Linear System
L>Uxs2U = −[B>
1 ]s2
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An elementary proof of a matrix tree theorem for directed graphs, Leenheer (2019)
Computation Probabilities Directed Probabilistic Watershed
Number in-forests increasesexponentially with the number
of nodes and edges=⇒ Naive approach infeasible
Directed Matrix Tree TheoremLeenheer (2019)
=⇒ Efficient computation probabilities
Linear System
L>Uxs2U = −[B>
1 ]s2
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 13 / 19
An elementary proof of a matrix tree theorem for directed graphs, Leenheer (2019)
Comparison linear systems
Probabilistic Watershed
LUxs2U = −[B>
1 ]s2
Directed Probabilistic Watershed
L>Uxs2U = −[B>
1 ]s2
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 14 / 19
Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning, Fita Sanmart́ın et al. (2019)
Equivalence Random Walker
Probabilistic Watershed
Random Walker(Seed absorption probabibility)
Grady (2006)
∼ Probabilisic WatershedFita (2019)
Directed Probabilistic Watershed
Directed Random Walker(Seed absorption probabibility)
∼ DirectedProbabilisic Watershed
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 15 / 19
Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning, Fita Sanmart́ın et al. (2019)Random walks for image segmentation, Grady (2006)
Minimum entropy case
Gibbs distribution
Pr(F ) ∝ exp(− µc(F )
)= exp
(− µ
∑e∈F
ce)
µ→∞ =⇒ Minimum entropy =⇒ Count minimum costin-forests (in-mSF)
(a) µ = 1 (b) µ = 2 (c) µ = 20
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Minimum entropy case
Probabilistic Watershed
Power WatershedCouprie et al. (2011)
∼Minimum entropy
Probabilisic WatershedFita (2019)
Directed Probabilistic Watershed
DirectedPower Watershed
∼Minimum entropy
DirectedProbabilisic Watershed
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 17 / 19
Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning, Fita Sanmart́ın et al. (2019)Power Watershed: A Unifying Graph-Based Optimization Framework, Couprie at al. (2011)
Summary
Probabilistic WatershedFita Sanmart́ın et al. (2019)
=⇒ Directed Probabilistic Watershed
Forests =⇒ In-forestsrooted at the seeds
Count efficientlynumber forests
=⇒ Count efficientlynumber in-forests
Equivalence Random WalkerGrady (2006)
=⇒ Equivalence DirectedRandom Walker
Minimum entropyPower Watershed
Couprie et al. (2011)
=⇒ Minimum entropyDirected Power Watershed
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 18 / 19
Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning, Fita Sanmart́ın et al. (2019)Random walks for image segmentation, Grady (2006)
Power Watershed: A Unifying Graph-Based Optimization Framework, Couprie at al. (2011)
Summary
Probabilistic WatershedFita Sanmart́ın et al. (2019)
=⇒ Directed Probabilistic Watershed
Forests =⇒ In-forestsrooted at the seeds
Count efficientlynumber forests
=⇒ Count efficientlynumber in-forests
Equivalence Random WalkerGrady (2006)
=⇒ Equivalence DirectedRandom Walker
Minimum entropyPower Watershed
Couprie et al. (2011)
=⇒ Minimum entropyDirected Power Watershed
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 18 / 19
Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning, Fita Sanmart́ın et al. (2019)Random walks for image segmentation, Grady (2006)
Power Watershed: A Unifying Graph-Based Optimization Framework, Couprie at al. (2011)
Summary
Probabilistic WatershedFita Sanmart́ın et al. (2019)
=⇒ Directed Probabilistic Watershed
Forests =⇒ In-forestsrooted at the seeds
Count efficientlynumber forests
=⇒ Count efficientlynumber in-forests
Equivalence Random WalkerGrady (2006)
=⇒ Equivalence DirectedRandom Walker
Minimum entropyPower Watershed
Couprie et al. (2011)
=⇒ Minimum entropyDirected Power Watershed
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 18 / 19
Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning, Fita Sanmart́ın et al. (2019)Random walks for image segmentation, Grady (2006)
Power Watershed: A Unifying Graph-Based Optimization Framework, Couprie at al. (2011)
Summary
Probabilistic WatershedFita Sanmart́ın et al. (2019)
=⇒ Directed Probabilistic Watershed
Forests =⇒ In-forestsrooted at the seeds
Count efficientlynumber forests
=⇒ Count efficientlynumber in-forests
Equivalence Random WalkerGrady (2006)
=⇒ Equivalence DirectedRandom Walker
Minimum entropyPower Watershed
Couprie et al. (2011)
=⇒ Minimum entropyDirected Power Watershed
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 18 / 19
Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning, Fita Sanmart́ın et al. (2019)Random walks for image segmentation, Grady (2006)
Power Watershed: A Unifying Graph-Based Optimization Framework, Couprie at al. (2011)
Summary
Probabilistic WatershedFita Sanmart́ın et al. (2019)
=⇒ Directed Probabilistic Watershed
Forests =⇒ In-forestsrooted at the seeds
Count efficientlynumber forests
=⇒ Count efficientlynumber in-forests
Equivalence Random WalkerGrady (2006)
=⇒ Equivalence DirectedRandom Walker
Minimum entropyPower Watershed
Couprie et al. (2011)
=⇒ Minimum entropyDirected Power Watershed
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 18 / 19
Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning, Fita Sanmart́ın et al. (2019)Random walks for image segmentation, Grady (2006)
Power Watershed: A Unifying Graph-Based Optimization Framework, Couprie at al. (2011)
Thank you for your attention
Enrique Fita Sanmart́ın Directed Probabilistic Watershed 19 / 19
