Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation...
Transcript of Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation...
![Page 1: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/1.jpg)
Topological Data AnalysisGeneralities and some Applications
Jaraf Mustapha
Université Internationale de Rabat
M.A.A.T
04-Mars-2017
![Page 2: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/2.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
![Page 3: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/3.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Sommaire
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
![Page 4: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/4.jpg)
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
![Page 5: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/5.jpg)
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
![Page 6: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/6.jpg)
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
![Page 7: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/7.jpg)
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
![Page 8: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/8.jpg)
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
![Page 9: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/9.jpg)
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
![Page 10: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/10.jpg)
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
![Page 11: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/11.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
![Page 12: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/12.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
![Page 13: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/13.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
![Page 14: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/14.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
![Page 15: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/15.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
![Page 16: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/16.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
![Page 17: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/17.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
![Page 18: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/18.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
![Page 19: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/19.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
![Page 20: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/20.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
![Page 21: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/21.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
![Page 22: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/22.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Persistence of vision
![Page 23: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/23.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
What these points looks like ?
![Page 24: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/24.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
"Data has Shape and Shape has Meaning"G. Carlson
![Page 25: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/25.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Sommaire
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
![Page 26: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/26.jpg)
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.
![Page 27: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/27.jpg)
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.
![Page 28: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/28.jpg)
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.
![Page 29: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/29.jpg)
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
![Page 30: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/30.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Sommaire
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
![Page 31: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/31.jpg)
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)}
![Page 32: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/32.jpg)
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)}
![Page 33: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/33.jpg)
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)}
![Page 34: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/34.jpg)
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)}
![Page 35: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/35.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
RelationshipGraph
Simplicial Complex
1-skeleton Clique Complex
![Page 36: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/36.jpg)
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
![Page 37: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/37.jpg)
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
![Page 38: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/38.jpg)
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 |
![Page 39: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/39.jpg)
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 |
![Page 40: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/40.jpg)
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 |
![Page 41: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/41.jpg)
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 |
![Page 42: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/42.jpg)
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 |
![Page 43: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/43.jpg)
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 |
![Page 44: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/44.jpg)
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 |
![Page 45: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/45.jpg)
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
![Page 46: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/46.jpg)
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
![Page 47: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/47.jpg)
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
![Page 48: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/48.jpg)
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
![Page 49: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/49.jpg)
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
![Page 50: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/50.jpg)
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
![Page 51: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/51.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 52: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/52.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 53: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/53.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 54: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/54.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 55: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/55.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 56: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/56.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 57: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/57.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 58: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/58.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 59: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/59.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 60: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/60.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 61: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/61.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 62: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/62.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 63: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/63.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 64: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/64.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 65: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/65.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 66: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/66.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 67: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/67.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 68: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/68.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 69: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/69.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 70: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/70.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 71: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/71.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 72: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/72.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 73: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/73.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 74: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/74.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 75: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/75.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 76: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/76.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 77: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/77.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 78: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/78.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 79: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/79.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 80: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/80.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 81: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/81.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 82: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/82.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
![Page 83: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/83.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Betti-Numbers and Homology
Simplicial Complex
Chain Complex
Homology Groups
Betti Numbers
Algebraic Structure
Measure of Exactness
Dimension of Homology Groups
![Page 84: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/84.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Sommaire
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
![Page 85: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/85.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Robotics
![Page 86: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/86.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Autonomous Localization
![Page 87: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/87.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Autonomous Localization
![Page 88: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/88.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Autonomous Localization
![Page 89: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/89.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Autonomous Localization
![Page 90: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/90.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Autonomous Localization
Figure: Planardomain,D
Figure: CovisibilityNetwork, N
Figure: LandmarkComplex, K
![Page 91: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/91.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Neural Coding
![Page 92: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/92.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Hippocampal Spatial MapFormation
![Page 93: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/93.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Place Cells and Place Fields
![Page 94: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/94.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Place Cells and Place Fields
![Page 95: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/95.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Place Cells and Place Fields
![Page 96: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/96.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Place Cells and Place Fields
![Page 97: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/97.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Place Cells and Place Fields
![Page 98: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/98.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Place Cells and Place Fields
![Page 99: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/99.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Place Cells and Place Fields
![Page 100: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/100.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Place Cells and Place Fields
![Page 101: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/101.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Place Cells and Place Fields
![Page 102: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/102.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Place Cells and Place Fields
![Page 103: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/103.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Place Cells and Place Fields
![Page 104: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/104.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Neural Coding : HippocampalSpatial Map Formation
![Page 105: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/105.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
"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.
![Page 106: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/106.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
![Page 107: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/107.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
![Page 108: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/108.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
![Page 109: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/109.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
![Page 110: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/110.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
![Page 111: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/111.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
![Page 112: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/112.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Mouse Trajectory
![Page 113: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/113.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Other Application
• Facial Recognition.
• Image processing.• Musical Applications.• Clinical variation management• Financial risk modeling
![Page 114: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/114.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Other Application
• Facial Recognition.• Image processing.
• Musical Applications.• Clinical variation management• Financial risk modeling
![Page 115: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/115.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Other Application
• Facial Recognition.• Image processing.• Musical Applications.
• Clinical variation management• Financial risk modeling
![Page 116: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/116.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Other Application
• Facial Recognition.• Image processing.• Musical Applications.• Clinical variation management
• Financial risk modeling
![Page 117: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/117.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Other Application
• Facial Recognition.• Image processing.• Musical Applications.• Clinical variation management• Financial risk modeling
![Page 118: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/118.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Sommaire
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
![Page 119: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/119.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Computing Algebraic Topology
![Page 120: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/120.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Sommaire
1 Motivation
2 Introduction
3 Basic Concepts
4 Applications
5 Research Focus
6 Bibliography
![Page 121: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/121.jpg)
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.
![Page 122: Topological Data Analysis - algtop.net · Topological Data Analysis Jaraf Mustapha Motivation Introduction Basic Concepts Applications Research Focus Bibliography Sommaire 1 Motivation](https://reader031.fdocuments.in/reader031/viewer/2022030615/5adfd70f7f8b9afd1a8d3c0c/html5/thumbnails/122.jpg)
TopologicalData
Analysis
JarafMustapha
Motivation
Introduction
BasicConcepts
Applications
ResearchFocus
Bibliography
Thanks you for your attention