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 Fred A. Hamprecht Sebastian Damrich HCI/IWR at Heidelberg University Enrique Fita Sanmartin Probabilistic Watershed 1 / 12

What do we do? Enrique Fita Sanmartin Probabilistic Watershed 2 / 12

We count forests! 1 2 3 Enrique Fita Sanmartin Probabilistic Watershed 3 / 12

Framework Graph Enrique Fita Sanmartin Probabilistic Watershed 4 / 12

Framework s 2 Graph Seeds (labeled nodes) s 1 Enrique Fita Sanmartin Probabilistic Watershed 4 / 12

Framework 0.00 0.05 s 2 1.61 0.70 0.22 Graph Seeds (labeled nodes) 1.20 0.51 Edge-Costs ∼ affinity between nodes 0.92 0.10 0.43 s 1 0.16 0.36 Enrique Fita Sanmartin Probabilistic Watershed 4 / 12

Framework 0.00 0.05 s 2 0.22 1.61 0.70 Graph Seeds (labeled nodes) 1.20 0.51 Edge-Costs ∼ affinity between nodes 0.92 Forest 0.10 0.43 s 1 0.16 0.36 Enrique Fita Sanmartin Probabilistic Watershed 4 / 12

Forests s 2 s 2 0.00 0.05 s 2 0.22 1.61 0.70 s 1 s 1 1.20 0.51 s 2 s 2 0.92 0.10 0.43 s 1 s 1 s 1 0.16 0.36 s 2 s 2 Watershed forest / minimum cost Spanning Forest ( mSF ) s 1 s 1 Enrique Fita Sanmartin Probabilistic Watershed 5 / 12

Counting Forests Enrique Fita Sanmartin Probabilistic Watershed 6 / 12

Probabilistic Watershed exp ( − µ c ( f )) w ( f ) Pr( f ) = � = � − µ c ( f ′ ) � w ( f ′ ) s 2 � exp f ′∈F s 2 f ′∈F s 2 s 1 s 1 s 1 0.68 0.77 1 . 00 1 mSF 0.21 0 . 52 0.72 s 2 s 2 s 2 s 2 0 . 00 0.30 0.45 0 s 1 s 1 s 1 s 1 � w ( f ) F s 1 f ∈F s 1 � � Pr( f ) = w Probabilistic s 2 , q s 2 , q � → Pr( q ∼ s 2 ) := � = F s 1 . Watershed � � w w ( f ) s 2 f ∈F s 1 s 2 , q f ∈F s 1 s 2 Enrique Fita Sanmartin Probabilistic Watershed 7 / 12

How do we count the forests? s 2 192 s 1 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 1 1 � | V | ✶✶ ⊤ � � � � = det( L [ v ] ) . w ( T ) := w ( t ) = w ( e ) = | V | det L + t ∈T t ∈T e ∈ E t Enrique Fita Sanmartin Probabilistic Watershed 8 / 12

How do we count the forests? s 2 288 s 1 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 ( F v u ) the sum of the weights of the 2-trees spanning forests separating u and v , then w ( F v u ) = w ( T ) r eff uv . Enrique Fita Sanmartin Probabilistic Watershed 8 / 12

Probabilistic Watershed=Random Walker[Grady, 2006] = + Probabilistic Watershed Random Walker [Grady, 2006] Enrique Fita Sanmartin Probabilistic Watershed 9 / 12

Power Watershed new interpretation Enrique Fita Sanmartin Probabilistic Watershed 10 / 12

Summary exp ( − µ c ( f )) w ( f ) Pr( f ) = � = w ( f ′ ) � � − µ c ( f ′ ) � s 2 exp f ′∈F s 2 f ′∈F s 2 Probabilistic Watershed=Random s 1 s 1 Walker [Grady, 2006]. New interpretations of the Power s 1 Watershed [Couprie et al., 2011]. Technique to count forests. 0.68 0.77 1 . 00 1 mSF 0.21 0 . 52 0.72 s 2 s 2 s 2 s 2 0 . 00 0.30 0.45 0 s 1 s 1 s 1 s 1 Poster Room East Exhibition Hall B + C #81 10:45 AM – 12:45 PM Enrique Fita Sanmartin Probabilistic Watershed 11 / 12

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¨ osung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Str¨ ome gef¨ uhrt wird. Annalen der Physik , 148:497–508. Enrique Fita Sanmartin Probabilistic Watershed 12 / 12