Topological Structures in the Analysis of Images and Data Chao Chen City University of New York (CUNY) Oct. 2016 C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 1 / 43
Outline Topological Structures 1 High Dimensional Data 2 Algorithms Applications C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 2 / 43
Topological Structures global, multi-scale, independent to geometry 0 dim 1 dim 2 dim C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 3 / 43
Topological Structures of Data For a dataset, what are the components and loops of the data? TDA: detect these structures in a robust way. C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 4 / 43
Topological Structures of Data For a dataset, what are the components and loops of the data? TDA: detect these structures in a robust way. C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 4 / 43
Persistent Homology: A Robust Way to Extract Topological Structures Input: a (density) function, f Output: topological structures & their persistence C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 5 / 43
Persistent Homology: A Robust Way to Extract Topological Structures Input: a (density) function, f Output: topological structures & their persistence Def: given threshold t , the superlevel set f − 1 [ t , + ∞ ) := { x | f ( x ) ≥ t } C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 5 / 43
Persistent Homology (continued) the true structures are hidden in superlevel sets consider the whole stack of superlevel sets identify structures that often appear (high persistence) Output: persistence diagram – dots representing all structures C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 6 / 43
Persistent Homology (continued) the true structures are hidden in superlevel sets consider the whole stack of superlevel sets identify structures that often appear (high persistence) Output: persistence diagram – dots representing all structures Diagram C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 6 / 43
Why Topological Structures: Cardiac data ( Demo ) C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 7 / 43
Why Topological Structures: Cardiac data ( Demo ) Thresholding Thresholding: local evidence, minimize energy E ( y ) � E ( y ) = E v ( y v ) , y v ∈ { BG , FG } v C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 7 / 43
Why Topological Structures: Cardiac data ( Demo ) Thresholding Advanced Thresholding: local evidence, minimize energy E ( y ) � E ( y ) = E v ( y v ) , y v ∈ { BG , FG } v Advanced: pairwise local evidence � � E ( y ) = E v ( y v ) + E u , v ( y u , y v ) v ( u , v ) C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 7 / 43
Why Topology Data Analysis? Recovering missing trabeculae: [Gao, Chen , et al . IPMI’13] C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 8 / 43
Why Topology Data Analysis? Recovering missing trabeculae: [Gao, Chen , et al . IPMI’13] C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 8 / 43
Morphological Analysis Endocardial Surface [ISBI’14] C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 9 / 43
Follow-up Questions (Ongoing) Validation on a specimen Homology localization problem Bad Generator Ground Truth Simulation Good Generator C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 10 / 43
Topological Information as Constraints in Segmentation [ Chen et al . CVPR 2011] C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 11 / 43
Topological Information as Constraints in Segmentation [ Chen et al . CVPR 2011] Input Stencil 1 2 3 4 Final [Jain, Chen , et al . , Computer & Graphics, 2015] C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 11 / 43
Additional Application: Multi-Layer Stencil Creation Canvas/wall result: Website, interactive [Jain, Chen , et al . , Computer & Graphics, 2015] C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 12 / 43
Outline Topological Structures 1 High Dimensional Data 2 Algorithms Applications C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 13 / 43
Topological Structures for High Dimensional Data Plenty have been done: data centric, simplicial complex, mapper, etc. C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 14 / 43
Topological Structures for High Dimensional Data Plenty have been done: data centric, simplicial complex, mapper, etc. My focus: density function. ◮ Need a good model: high dim, flexibility, computation – graphical model C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 14 / 43
Topological Structures for High Dimensional Data Plenty have been done: data centric, simplicial complex, mapper, etc. My focus: density function. ◮ Need a good model: high dim, flexibility, computation – graphical model ◮ Locations that contribute to major topological events, critical points C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 14 / 43
Graphical Model Markov Random Field (MRF) D dimension; values/labels L = { 1 , . . . , L } configurations/labelings: X = L D = { 1 , · · · , L } D V 3 V 8 V 5 V 7 V 1 V 6 V 4 V 2 C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 15 / 43
Graphical Model Markov Random Field (MRF) D dimension; values/labels L = { 1 , . . . , L } configurations/labelings: X = L D = { 1 , · · · , L } D V 3 V 8 1 0 Binary Potentials θ ij ( x i , x j ) V 5 x j x i 0 1 1 V 7 1 1 θ ij (0 , 0) θ ij (0 , 1) 0 V 1 θ ij (1 , 0) θ ij (1 , 1) 1 1 0 0 V 6 V 4 V 2 Energy: E ( x ) = � ( i , j ) ∈E θ ij ( x i , x j ) Probability: P ( x ) = exp ( − E ( x )) / Z C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 15 / 43
What can we do with a graphical model? Previously: Computing the maximum a posteriori (MAP): argmax x ∈X P ( x ) = argmin x ∈X E ( x ) marginals, sampling, etc. P ( x ) MAP C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 16 / 43
What can we do with a graphical model? Previously: Computing the maximum a posteriori (MAP): argmax x ∈X P ( x ) = argmin x ∈X E ( x ) marginals, sampling, etc. P ( x ) MAP mode mode New Question: How about modes (local maxima)? C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 16 / 43
Why modes? A concise description of the probabilistic landscape P ( x ) MAP mode mode C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 17 / 43
Why modes? A concise description of the probabilistic landscape Multiple predictions ◮ model is not perfect, ambiguity ◮ multiple hypotheses, diverse, highly possible Other applications: biology, NLP Previous: mean-shift C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 17 / 43
Definitions Given a distance function d ( · , · ) and a scalar δ ◮ Neighborhood: N δ ( x ) = { x ′ | d ( x , x ′ ) ≤ δ } ◮ x is a mode if it has a bigger prob. than all its neighbors ◮ M δ : the set of all modes for a given scale δ C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 18 / 43
Definitions Given a distance function d ( · , · ) and a scalar δ ◮ Neighborhood: N δ ( x ) = { x ′ | d ( x , x ′ ) ≤ δ } ◮ x is a mode if it has a bigger prob. than all its neighbors ◮ M δ : the set of all modes for a given scale δ P ( x ) δ = 1 C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 18 / 43
Definitions Given a distance function d ( · , · ) and a scalar δ ◮ Neighborhood: N δ ( x ) = { x ′ | d ( x , x ′ ) ≤ δ } ◮ x is a mode if it has a bigger prob. than all its neighbors ◮ M δ : the set of all modes for a given scale δ P ( x ) δ = 4 C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 18 / 43
Definitions Given a distance function d ( · , · ) and a scalar δ ◮ Neighborhood: N δ ( x ) = { x ′ | d ( x , x ′ ) ≤ δ } ◮ x is a mode if it has a bigger prob. than all its neighbors ◮ M δ : the set of all modes for a given scale δ P ( x ) δ = 7 C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 18 / 43
Definitions D the dimension; L = { 1 , . . . , L } the label set; X = L D the domain Given a distance function d ( · , · ) and a scalar δ ◮ Neighborhood: N δ ( x ) = { x ′ | d ( x , x ′ ) ≤ δ } ◮ x is a mode if it has a bigger prob. than all its neighbors ◮ M δ : the set of all modes for a given scale δ X = M 0 ⊇ M 1 ⊇ · · · ⊇ M ∞ = { global maximum (MAP) } δ = 0 δ = 1 δ = 4 δ = 7 C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 19 / 43
Problem Problem (MModes) Given a scale δ , compute the top M elements in M δ . Challenge: exponential domain, exponential neighborhood C. Chen (CUNY) Topological Structures in the Analysis of Images and Data 20 / 43
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