Reduction and Approximation of Multidimensional Persistent Homology Massimo Ferri 1 , 2 1 Dip. di Matematica, Univ. di Bologna, Italia 2 ARCES - Vision Mathematics Group, Univ. di Bologna, Italia ferri@dm.unibo.it GETCO 2010 Geometric and Topological Methods in Computer Science Aalborg University, January 11-15, 2010 Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 1 / 50
Outline Shape 1 Persistent topology 2 Distances 3 Multidimensional persistent homology 4 One-dimensional reduction 5 Ball coverings 6 Combinatorial representation 7 Conclusions 8 Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 2 / 50
Shape Shape 1 Persistent topology 2 Distances 3 Multidimensional persistent homology 4 One-dimensional reduction 5 Ball coverings 6 Combinatorial representation 7 Conclusions 8 Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 3 / 50
Shape Which object has the same shape as the circle? Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 4 / 50
Shape Shape = Geometry? Homoteties surely preserve shape Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 5 / 50
Shape Shape = Geometry? Homoteties surely preserve shape . . . but not only they do. Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 5 / 50
Shape Shape = Topology? Sometimes homeomorphisms are deceptive. Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 6 / 50
Shape Shape = Psychology? Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 7 / 50
Shape Size pairs A possible setting: persistent topology of a size pair ( X , − → ϕ ) Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 8 / 50
Shape Size pairs A possible setting: persistent topology of a size pair ( X , − → ϕ ) X is a topological space, ϕ : X → R n a continuous map, called measuring (filtering) function. Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 8 / 50
Shape Size pairs A possible setting: persistent topology of a size pair ( X , − → ϕ ) X is a topological space, ϕ : X → R n a continuous map, called measuring (filtering) function. − → ϕ provides the geometrical aspects; Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 8 / 50
Shape Size pairs A possible setting: persistent topology of a size pair ( X , − → ϕ ) X is a topological space, ϕ : X → R n a continuous map, called measuring (filtering) function. − → ϕ provides the geometrical aspects; the core idea is to study the evolution and persistence of topological features of the sublevel sets of − → ϕ ; Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 8 / 50
Shape Size pairs A possible setting: persistent topology of a size pair ( X , − → ϕ ) X is a topological space, ϕ : X → R n a continuous map, called measuring (filtering) function. − → ϕ provides the geometrical aspects; the core idea is to study the evolution and persistence of topological features of the sublevel sets of − → ϕ ; the choice of − → ϕ conveys the subjective viewpoint of the observer. Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 8 / 50
Shape Size pairs A possible setting: persistent topology of a size pair ( X , − → ϕ ) X is a topological space, ϕ : X → R n a continuous map, called measuring (filtering) function. − → ϕ provides the geometrical aspects; the core idea is to study the evolution and persistence of topological features of the sublevel sets of − → ϕ ; the choice of − → ϕ conveys the subjective viewpoint of the observer. Examples of measuring functions 1-dimensional: distance from center of mass, ordinate, curvature, . . . Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 8 / 50
Shape Size pairs A possible setting: persistent topology of a size pair ( X , − → ϕ ) X is a topological space, ϕ : X → R n a continuous map, called measuring (filtering) function. − → ϕ provides the geometrical aspects; the core idea is to study the evolution and persistence of topological features of the sublevel sets of − → ϕ ; the choice of − → ϕ conveys the subjective viewpoint of the observer. Examples of measuring functions 1-dimensional: distance from center of mass, ordinate, curvature, . . . multidimensional: color, coordinates, curvature and torsion, . . . Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 8 / 50
Shape Size pairs A possible setting: persistent topology of a size pair ( X , − → ϕ ) X is a topological space, ϕ : X → R n a continuous map, called measuring (filtering) function. − → ϕ provides the geometrical aspects; the core idea is to study the evolution and persistence of topological features of the sublevel sets of − → ϕ ; the choice of − → ϕ conveys the subjective viewpoint of the observer. Examples of measuring functions 1-dimensional: distance from center of mass, ordinate, curvature, . . . multidimensional: color, coordinates, curvature and torsion, . . . (Claudia Landi and Patrizio Frosini will further elaborate on that. In fact, what I am presenting is largely the product of team work at our Vision Mathematics group in Bologna.) Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 8 / 50
Shape Persistence A different approach to the same idea [Edelsbrunner et al. 2000]: Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 9 / 50
Shape Persistence A different approach to the same idea [Edelsbrunner et al. 2000]: Persistent topological features are the ones which persist under resolution change Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 9 / 50
Shape Persistence A different approach to the same idea [Edelsbrunner et al. 2000]: Persistent topological features are the ones which persist under resolution change Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 9 / 50
Shape Persistence A different approach to the same idea [Edelsbrunner et al. 2000]: Persistent topological features are the ones which persist under resolution change Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 9 / 50
Shape Natural pseudodistance More than by a size pair, the concept of shape is well represented by the comparison of shapes, i.e. a measure of dissimilarity. Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 10 / 50
Shape Natural pseudodistance More than by a size pair, the concept of shape is well represented by the comparison of shapes, i.e. a measure of dissimilarity. Such a measure is the natural pseudodistance d between pairs ϕ ) , ( Y , − → ( X , − → ψ ) , where X and Y are homeomorphic compact spaces. d is defined as ϕ ) , ( Y , − → ϕ ( P ) − − → � ( X , − → � P ∈ X �− → d ψ ( f ( P )) � ∞ ψ ) = inf max f where f varies among all homeomorphisms from X to Y [Frosini et al. 1999]. Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 10 / 50
Persistent topology Shape 1 Persistent topology 2 Distances 3 Multidimensional persistent homology 4 One-dimensional reduction 5 Ball coverings 6 Combinatorial representation 7 Conclusions 8 Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 11 / 50
Persistent topology Rank invariant [Carlsson et al. 2007] We define the following relation � ( ≺ ) in R n : if � u = ( u 1 , . . . , u n ) v = ( v 1 , . . . , v n ) , we write � u � � v ( � u ≺ � v ) if and only if u j ≤ v j and � ( u j < v j ) for j = 1 , . . . , n . Let also ∆ + be the set v ) ∈ R n × R n | � u ,� u ≺ � v } . { ( � Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 12 / 50
Persistent topology Rank invariant [Carlsson et al. 2007] We define the following relation � ( ≺ ) in R n : if � u = ( u 1 , . . . , u n ) v = ( v 1 , . . . , v n ) , we write � u � � v ( � u ≺ � v ) if and only if u j ≤ v j and � ( u j < v j ) for j = 1 , . . . , n . Let also ∆ + be the set v ) ∈ R n × R n | � u ,� u ≺ � v } . { ( � u � the lower level set { p ∈ X | − → f ( p ) � − → We denote by X � � f � � u } . Massimo Ferri (DM-ARCES, U. Bologna) Reduction and Approximation GETCO 2010 12 / 50
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