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

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TopologicalData

Analysis

JarafMustapha

Motivation

Introduction

BasicConcepts

Applications

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TopologicalData

Analysis

JarafMustapha

Motivation

Introduction

BasicConcepts

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TopologicalData

Analysis

JarafMustapha

Motivation

Introduction

BasicConcepts

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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

Motivation

Introduction

BasicConcepts

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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

Motivation

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TopologicalData

Analysis

JarafMustapha

Motivation

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TopologicalData

Analysis

JarafMustapha

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TopologicalData

Analysis

JarafMustapha

Motivation

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TopologicalData

Analysis

JarafMustapha

Motivation

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TopologicalData

Analysis

JarafMustapha

Motivation

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TopologicalData

Analysis

JarafMustapha

Motivation

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TopologicalData

Analysis

JarafMustapha

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TopologicalData

Analysis

JarafMustapha

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TopologicalData

Analysis

JarafMustapha

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TopologicalData

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JarafMustapha

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TopologicalData

Analysis

JarafMustapha

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TopologicalData

Analysis

JarafMustapha

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TopologicalData

Analysis

JarafMustapha

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TopologicalData

Analysis

JarafMustapha

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TopologicalData

Analysis

JarafMustapha

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TopologicalData

Analysis

JarafMustapha

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TopologicalData

Analysis

JarafMustapha

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TopologicalData

Analysis

JarafMustapha

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TopologicalData

Analysis

JarafMustapha

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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

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Robotics

TopologicalData

Analysis

JarafMustapha

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Autonomous Localization

TopologicalData

Analysis

JarafMustapha

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Autonomous Localization

TopologicalData

Analysis

JarafMustapha

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Autonomous Localization

TopologicalData

Analysis

JarafMustapha

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Autonomous Localization

TopologicalData

Analysis

JarafMustapha

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Autonomous Localization

Figure: Planardomain,D

Figure: CovisibilityNetwork, N

Figure: LandmarkComplex, K

TopologicalData

Analysis

JarafMustapha

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Neural Coding

TopologicalData

Analysis

JarafMustapha

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Hippocampal Spatial MapFormation

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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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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Place Cells and Place Fields

TopologicalData

Analysis

JarafMustapha

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Place Cells and Place Fields

TopologicalData

Analysis

JarafMustapha

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Neural Coding : HippocampalSpatial Map Formation

TopologicalData

Analysis

JarafMustapha

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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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Mouse Trajectory

TopologicalData

Analysis

JarafMustapha

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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

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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