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Topological Data AnalysisGeneralities and some Applications
Jaraf Mustapha
Université Internationale de Rabat
M.A.A.T
04-Mars-2017
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Sommaire
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Visual Perception
• Visual perception is the ability to interpret thesurrounding environment using light in the visible spectrumreflected by the objects in the environment.
• We believe that the various actions leading to a developedperception are carried out in five stages:
VisualizationStructuring
TransfigurationDetermination
And Classification
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Visual Perception
• Visual perception is the ability to interpret thesurrounding environment using light in the visible spectrumreflected by the objects in the environment.
• We believe that the various actions leading to a developedperception are carried out in five stages:
VisualizationStructuring
TransfigurationDetermination
And Classification
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Visual Perception
• Visual perception is the ability to interpret thesurrounding environment using light in the visible spectrumreflected by the objects in the environment.
• We believe that the various actions leading to a developedperception are carried out in five stages:
Visualization
StructuringTransfigurationDetermination
And Classification
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Visual Perception
• Visual perception is the ability to interpret thesurrounding environment using light in the visible spectrumreflected by the objects in the environment.
• We believe that the various actions leading to a developedperception are carried out in five stages:
VisualizationStructuring
TransfigurationDetermination
And Classification
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Visual Perception
• Visual perception is the ability to interpret thesurrounding environment using light in the visible spectrumreflected by the objects in the environment.
• We believe that the various actions leading to a developedperception are carried out in five stages:
VisualizationStructuring
Transfiguration
DeterminationAnd Classification
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Visual Perception
• Visual perception is the ability to interpret thesurrounding environment using light in the visible spectrumreflected by the objects in the environment.
• We believe that the various actions leading to a developedperception are carried out in five stages:
VisualizationStructuring
TransfigurationDetermination
And Classification
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Visual Perception
• Visual perception is the ability to interpret thesurrounding environment using light in the visible spectrumreflected by the objects in the environment.
• We believe that the various actions leading to a developedperception are carried out in five stages:
VisualizationStructuring
TransfigurationDetermination
And Classification
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
What these points looks like ?
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
"Data has Shape and Shape has Meaning"G. Carlson
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Sommaire
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Topological Data Analysis
• Topological Data Analysis (TDA) is an approach to theanalysis of datasets using techniques from topology.
• The initial motivation is to study the shape of data. TDAhas combined algebraic topology and other tools from puremathematics to give mathematically rigorous andquantitative study of "shape". The main tool is persistenthomology, an adaptation of homology to point cloud data.
• Persistent homology has been applied to many types ofdata across many fields. Moreover, its mathematicalfoundation is also of theoretical importance. The uniquefeatures of TDA make it a promising bridge betweentopology and geometry.
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Topological Data Analysis
• Topological Data Analysis (TDA) is an approach to theanalysis of datasets using techniques from topology.
• The initial motivation is to study the shape of data. TDAhas combined algebraic topology and other tools from puremathematics to give mathematically rigorous andquantitative study of "shape". The main tool is persistenthomology, an adaptation of homology to point cloud data.
• Persistent homology has been applied to many types ofdata across many fields. Moreover, its mathematicalfoundation is also of theoretical importance. The uniquefeatures of TDA make it a promising bridge betweentopology and geometry.
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Topological Data Analysis
• Topological Data Analysis (TDA) is an approach to theanalysis of datasets using techniques from topology.
• The initial motivation is to study the shape of data. TDAhas combined algebraic topology and other tools from puremathematics to give mathematically rigorous andquantitative study of "shape". The main tool is persistenthomology, an adaptation of homology to point cloud data.
• Persistent homology has been applied to many types ofdata across many fields. Moreover, its mathematicalfoundation is also of theoretical importance. The uniquefeatures of TDA make it a promising bridge betweentopology and geometry.
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mathematical Dependency Tree
Topological Data Analysis
Linear Algebra Set Theory
Abstract Simplicial Complex Topology
Cellular Homology
Cw-ComplexSimplicial ComplexHomology
Filtration
Persistent Homology Simplicial Homology
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Sommaire
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
• GraphA (finite, combinatorial) graph is a pair (V ,E ), where V is afinite set and E is any collection of 2-element subsets of V .
• V = {1,2,3,4,5}E = {
(1,2) ,(2,3) ,(3,4) ,(2,4) ,(4,5)}
• Simplicial ComplexA (finite, combinatorial) simplicial complex is a pair (V ,X )where V is a finite set and X is any collection of subsets of Vsuch that : Y ∈X and Y
′ ⊆Y ⇒ Y′ ∈X
• V = {1,2,3,4,5}E = {
(1,2) ,(2,3) ,(3,4) ,(2,4) ,(4,5) ,(2,3,4)}
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
• GraphA (finite, combinatorial) graph is a pair (V ,E ), where V is afinite set and E is any collection of 2-element subsets of V .
• V = {1,2,3,4,5}E = {
(1,2) ,(2,3) ,(3,4) ,(2,4) ,(4,5)}
• Simplicial ComplexA (finite, combinatorial) simplicial complex is a pair (V ,X )where V is a finite set and X is any collection of subsets of Vsuch that : Y ∈X and Y
′ ⊆Y ⇒ Y′ ∈X
• V = {1,2,3,4,5}E = {
(1,2) ,(2,3) ,(3,4) ,(2,4) ,(4,5) ,(2,3,4)}
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
• GraphA (finite, combinatorial) graph is a pair (V ,E ), where V is afinite set and E is any collection of 2-element subsets of V .
• V = {1,2,3,4,5}E = {
(1,2) ,(2,3) ,(3,4) ,(2,4) ,(4,5)}
• Simplicial ComplexA (finite, combinatorial) simplicial complex is a pair (V ,X )where V is a finite set and X is any collection of subsets of Vsuch that : Y ∈X and Y
′ ⊆Y ⇒ Y′ ∈X
• V = {1,2,3,4,5}E = {
(1,2) ,(2,3) ,(3,4) ,(2,4) ,(4,5) ,(2,3,4)}
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
• GraphA (finite, combinatorial) graph is a pair (V ,E ), where V is afinite set and E is any collection of 2-element subsets of V .
• V = {1,2,3,4,5}E = {
(1,2) ,(2,3) ,(3,4) ,(2,4) ,(4,5)}
• Simplicial ComplexA (finite, combinatorial) simplicial complex is a pair (V ,X )where V is a finite set and X is any collection of subsets of Vsuch that : Y ∈X and Y
′ ⊆Y ⇒ Y′ ∈X
• V = {1,2,3,4,5}E = {
(1,2) ,(2,3) ,(3,4) ,(2,4) ,(4,5) ,(2,3,4)}
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
RelationshipGraph
Simplicial Complex
1-skeleton Clique Complex
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Simplicial Complexes
n-simplex
• A (k +1)-tuple of points in Rn, (x0, ...,xn), where xi ∈Rn, issaid to be affinely independent if the set of vectors{−−→xjx0|1≤ j ≤ k} are linearly independent.
• An n-simplex is an ordered (n+1)-tuple of affinelyindependent point σ=< x0, ...,xn >.
0-Simplex 1-Simplex 2-Simplex 3-Simplex
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Simplicial ComplexA simplicial complex K is a finite set of simplicies such that :
• σ ∈K , τ≤σ ⇒ τ ∈K• σ,σ
′ ∈K ⇒σ∩σ′ ≤σ;σ
′
Example Non-Example
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Betti-Numbers and GraphsThe first Betti-Number of a graph G = (V,E) with n vertices, medges and k connected components is : β1 =m−n+k
Figure: Connected Graph
Connected Components of R2 \DIf G = (V,E) is a combinatorial graph, T ⊆E is a spanning treeof G, and D ⊆R2 is any realisation of an embedding of G in theplane, then the number of connected components of R2 \D is|E |− |T |
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Betti-Numbers and GraphsThe first Betti-Number of a graph G = (V,E) with n vertices, medges and k connected components is : β1 =m−n+k
Figure: Set of Vertices k = β0 = 6
β1 = 0
Connected Components of R2 \DIf G = (V,E) is a combinatorial graph, T ⊆E is a spanning treeof G, and D ⊆R2 is any realisation of an embedding of G in theplane, then the number of connected components of R2 \D is|E |− |T |
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Betti-Numbers and GraphsThe first Betti-Number of a graph G = (V,E) with n vertices, medges and k connected components is : β1 =m−n+k
Figure: Spanning Tree β0 = 1
β1 = 0
Connected Components of R2 \DIf G = (V,E) is a combinatorial graph, T ⊆E is a spanning treeof G, and D ⊆R2 \D is any realisation of an embedding of G inthe plane, then the number of connected components of R2 \Dis |E |− |T |
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Betti-Numbers and GraphsThe first Betti-Number of a graph G = (V,E) with n vertices, medges and k connected components is : β1 =m−n+k
Figure: First Component β0 = 1
β1 = 1
Connected Components of R2 \DIf G = (V,E) is a combinatorial graph, T ⊆E is a spanning treeof G, and D ⊆R2 is any realisation of an embedding of G in theplane, then the number of connected components of R2 \D is|E |− |T |
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Betti-Numbers and GraphsThe first Betti-Number of a graph G = (V,E) with n vertices, medges and k connected components is : β1 =m−n+k
Figure: Second Component β0 = 1
β1 = 2
Connected Components of R2 \DIf G = (V,E) is a combinatorial graph, T ⊆E is a spanning treeof G, and D ⊆R2 is any realisation of an embedding of G in theplane, then the number of connected components of R2 \D is|E |− |T |
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Betti-Numbers and GraphsThe first Betti-Number of a graph G = (V,E) with n vertices, medges and k connected components is : β1 =m−n+k
Figure: Third Component β0 = 1
β1 = 3
Connected Components of R2 \DIf G = (V,E) is a combinatorial graph, T ⊆E is a spanning treeof G, and D ⊆R2 is any realisation of an embedding of G in theplane, then the number of connected components of R2 \D is|E |− |T |
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Betti-Numbers and GraphsThe first Betti-Number of a graph G = (V,E) with n vertices, medges and k connected components is : β1 =m−n+k
Figure: Last Component β0 = 1
β1 = 4
Connected Components of R2 \DIf G = (V,E) is a combinatorial graph, T ⊆E is a spanning treeof G, and D ⊆R2 is any realisation of an embedding of G in theplane, then the number of connected components of R2 \D is|E |− |T |
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Homology
• Homology Theory generalizes the notion of connectivity inGraph Theory to higher dimensions.
• It’s defined by a family of groups that capture the numberof :
• Connected components
• The number of holes
• The number of cavities
• The number of such equivalent units features in largerdimension
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Homology
• Homology Theory generalizes the notion of connectivity inGraph Theory to higher dimensions.
• It’s defined by a family of groups that capture the numberof :
• Connected components
• The number of holes
• The number of cavities
• The number of such equivalent units features in largerdimension
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Homology
• Homology Theory generalizes the notion of connectivity inGraph Theory to higher dimensions.
• It’s defined by a family of groups that capture the numberof :
• Connected components
• The number of holes
• The number of cavities
• The number of such equivalent units features in largerdimension
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Homology
• Homology Theory generalizes the notion of connectivity inGraph Theory to higher dimensions.
• It’s defined by a family of groups that capture the numberof :
• Connected components
• The number of holes
• The number of cavities
• The number of such equivalent units features in largerdimension
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Homology
• Homology Theory generalizes the notion of connectivity inGraph Theory to higher dimensions.
• It’s defined by a family of groups that capture the numberof :
• Connected components
• The number of holes
• The number of cavities
• The number of such equivalent units features in largerdimension
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Homology
• Homology Theory generalizes the notion of connectivity inGraph Theory to higher dimensions.
• It’s defined by a family of groups that capture the numberof :
• Connected components
• The number of holes
• The number of cavities
• The number of such equivalent units features in largerdimension
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
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Applications
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
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TopologicalData
Analysis
JarafMustapha
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Introduction
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TopologicalData
Analysis
JarafMustapha
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Introduction
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
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TopologicalData
Analysis
JarafMustapha
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Introduction
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TopologicalData
Analysis
JarafMustapha
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Introduction
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
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Applications
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
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TopologicalData
Analysis
JarafMustapha
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Introduction
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TopologicalData
Analysis
JarafMustapha
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Introduction
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TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
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Betti-Numbers and Homology
Simplicial Complex
Chain Complex
Homology Groups
Betti Numbers
Algebraic Structure
Measure of Exactness
Dimension of Homology Groups
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Sommaire
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Robotics
TopologicalData
Analysis
JarafMustapha
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Introduction
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Applications
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Autonomous Localization
TopologicalData
Analysis
JarafMustapha
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Autonomous Localization
TopologicalData
Analysis
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Autonomous Localization
TopologicalData
Analysis
JarafMustapha
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Autonomous Localization
TopologicalData
Analysis
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Autonomous Localization
Figure: Planardomain,D
Figure: CovisibilityNetwork, N
Figure: LandmarkComplex, K
TopologicalData
Analysis
JarafMustapha
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Introduction
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Neural Coding
TopologicalData
Analysis
JarafMustapha
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Introduction
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Hippocampal Spatial MapFormation
TopologicalData
Analysis
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Place Cells and Place Fields
TopologicalData
Analysis
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Place Cells and Place Fields
TopologicalData
Analysis
JarafMustapha
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Place Cells and Place Fields
TopologicalData
Analysis
JarafMustapha
Motivation
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Place Cells and Place Fields
TopologicalData
Analysis
JarafMustapha
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Place Cells and Place Fields
TopologicalData
Analysis
JarafMustapha
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Place Cells and Place Fields
TopologicalData
Analysis
JarafMustapha
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Place Cells and Place Fields
TopologicalData
Analysis
JarafMustapha
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Place Cells and Place Fields
TopologicalData
Analysis
JarafMustapha
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Introduction
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Place Cells and Place Fields
TopologicalData
Analysis
JarafMustapha
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Introduction
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Place Cells and Place Fields
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
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Place Cells and Place Fields
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
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Neural Coding : HippocampalSpatial Map Formation
TopologicalData
Analysis
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Introduction
BasicConcepts
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"Pyramidal neurons in rodent hippocampus exhibit a geometricorganization due to their role in position coding. Each of theseneurons, called place cells, acts as a position sensor, exhibitinga high firing rate when the animal’s position lies inside the
neuron’s place field, its preferred region of the spatialenvironment" – Giusti, Clique Topology Reveals Intrinsic
Geometric Structure in Neural Correlations.
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
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ResearchFocus
Bibliography
Mouse Trajectory
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Other Application
• Facial Recognition.
• Image processing.• Musical Applications.• Clinical variation management• Financial risk modeling
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Other Application
• Facial Recognition.• Image processing.
• Musical Applications.• Clinical variation management• Financial risk modeling
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Other Application
• Facial Recognition.• Image processing.• Musical Applications.
• Clinical variation management• Financial risk modeling
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Other Application
• Facial Recognition.• Image processing.• Musical Applications.• Clinical variation management
• Financial risk modeling
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Other Application
• Facial Recognition.• Image processing.• Musical Applications.• Clinical variation management• Financial risk modeling
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Sommaire
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Computing Algebraic Topology
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Sommaire
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Bibliography• A. Hatcher. Algebraic Topology. Cambridge UniversityPress, 2002.
• Zomorodian AJ. (2005) Topology for computing.Cambridge, UK; New York: Cambridge University Press.xiii, 243 p. p.
• H. Edelsbrunner, J. Harer. Computational Topology.Duke University. 2009
• J. Derenick, A. Speranzon, R. Ghrist. HomologicalSensing for Mobile Robot Localisation. 2013.
• R. Ghrist, D. Lipsky, J. Derenick, and A. Speranzon.Topological landmark-based navigation and mapping.University of Pennsylvania, Department of Mathematics,Tech. Rep., 08 2012.
• Y.Dabaghian, F.Mémoli, L. Frank, G. Carlson. ATopological Paradigm for Hippocampal Spatial MapFormation Using Persistent Homology.Plos ComputationalBiology.2012 Aug; 8(8): e1002581.
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
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