Multidimensional Persistent Topology as a Metric Approach to Shape Comparison Patrizio Frosini 1 , 2 1 Department of Mathematics, University of Bologna, Italy 2 ARCES - Vision Mathematics Group, University of Bologna, Italy frosini@dm.unibo.it GETCO 2010 Geometric and Topological Methods in Computer Science Aalborg University, January 11-15, 2010 Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 1 / 51
The point of this talk In brief, the main message of this talk: T ECHNIQUES FOR THE STABLE COMPUTATION AND THE COMPARISON OF P ERSISTENT T OPOLOGY IN THE MULTIDIMENSIONAL SETTING ( I . E ., FOR FILTERING FUNCTIONS TAKING VALUES IN R k ) ARE AVAILABLE . Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 2 / 51
Outline A Metric Approach to Shape Comparison 1 Lower Bounds for the Natural Pseudodistance 2 New Results in the Multidimensional Setting 3 Experiments 4 Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 3 / 51
The results I am going to present refer to a collective work of the Vision Mathematics Group (Niccolò Cavazza, Andrea Cerri, Barbara Di Fabio, Massimo Ferri, Patrizio Frosini, Claudia Landi). The experimental results I shall show at the end of this talk have been obtained in a joint work with the C.N.R. IMATI Group (Silvia Biasotti, Daniela Giorgi). Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 4 / 51
A Metric Approach to Shape Comparison A Metric Approach to Shape Comparison 1 Lower Bounds for the Natural Pseudodistance 2 New Results in the Multidimensional Setting 3 Experiments 4 Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 5 / 51
A Metric Approach to Shape Comparison Shape depends on persistent perceptions Massimo and Claudia have already presented some motivations to study Persistent Topology. Just a few words to recall our approach to shape comparison: “Science is nothing but perception.” Plato “Reality is merely an illusion, albeit a very persistent one.” Albert Einstein Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 6 / 51
A Metric Approach to Shape Comparison Our formal setting As shown by Massimo and Claudia, we propose that Each perception is formalized by a pair ( X , � ϕ ) , where X is a topological space and � ϕ is a continuous function. X represents the set of observations made by the observer, while ϕ describes how each observation is interpreted by the observer. � Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 7 / 51
A Metric Approach to Shape Comparison Our formal setting Persistence is quite important. Without persistence (in space, time, with respect to the analysis level...) perception could have little sense. This remark compels us to require that X is a topological space and � ϕ is a continuous function; this ϕ describes X from the point of view of the observer. It is function � called a measuring function. Persistent Topology is used to study the stable properties of the pair ( X , � ϕ ) . Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 8 / 51
A Metric Approach to Shape Comparison Our formal setting A possible objection: sometimes we have to manage discontinuous functions (e.g., color). Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 9 / 51
A Metric Approach to Shape Comparison Our formal setting A possible objection: sometimes we have to manage discontinuous functions (e.g., color). An answer: in that case the topological space X can describe the discontinuity set, and persistence can concern the properties of this topological space with respect to a suitable measuring function. ϕ : X → R 2 and � ψ : Y → R 2 , As measuring functions we can take � where the components ϕ 1 , ϕ 2 and ψ 1 , ψ 2 represent the colors on each side of the considered discontinuity set. Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 9 / 51
A Metric Approach to Shape Comparison Our formal setting A categorical way to formalize our approach Let us consider a category C such that The objects of C are the pairs ( X , � ϕ ) where X is a compact ϕ : X → R k is a continuous function. topological space and � � � The set Hom ( X , � ϕ ) , ( Y , � ψ ) of all morphisms between the objects ( X , � ϕ ) , ( Y , � ψ ) is a subset of the set of all homeomorphisms between X and Y (possibly empty). � � If Hom ( X , � ϕ ) , ( Y , � is not empty we say that the objects ( X , � ψ ) ϕ ) , ( Y , � ψ ) are comparable. Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 10 / 51
A Metric Approach to Shape Comparison Our formal setting Do not compare apples and oranges... � � Remark: Hom ( X , � ϕ ) , ( Y , � can be empty also in case X and Y are ψ ) homeomorphic. Example: Consider a segment X = Y embedded into R 3 and consider the ϕ ( x ) and the set of observations given by measuring the color � ψ ( x ) of each point x of the segment. triple of coordinates � Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 11 / 51
A Metric Approach to Shape Comparison Our formal setting Do not compare apples and oranges... � � Remark: Hom ( X , � ϕ ) , ( Y , � can be empty also in case X and Y are ψ ) homeomorphic. Example: Consider a segment X = Y embedded into R 3 and consider the ϕ ( x ) and the set of observations given by measuring the color � ψ ( x ) of each point x of the segment. triple of coordinates � ϕ and � It does not make sense to compare the perceptions � ψ . In other words the pairs ( X , � ϕ ) and ( Y , � ψ ) are not comparable, even if X = Y . Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 11 / 51
A Metric Approach to Shape Comparison Our formal setting Do not compare apples and oranges... � � Remark: Hom ( X , � ϕ ) , ( Y , � can be empty also in case X and Y are ψ ) homeomorphic. Example: Consider a segment X = Y embedded into R 3 and consider the ϕ ( x ) and the set of observations given by measuring the color � ψ ( x ) of each point x of the segment. triple of coordinates � ϕ and � It does not make sense to compare the perceptions � ψ . In other words the pairs ( X , � ϕ ) and ( Y , � ψ ) are not comparable, even if X = Y . � � We express this fact by setting Hom ( X , � ϕ ) , ( Y , � ψ ) = ∅ . Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 11 / 51
A Metric Approach to Shape Comparison Our formal setting We can now define the following (extended) pseudometric: � � ( X , � ϕ ) , ( Y , � x ∈ X | ϕ i ( x ) − ψ i ◦ h ( x ) | ψ ) = inf max max δ i h ∈ Hom ( ( X ,� ϕ ) , ( Y ,� ψ ) ) � � if Hom ( X , � ϕ ) , ( Y , � ψ ) � = ∅ , and + ∞ otherwise. � � ( X , � ϕ ) , ( Y , � ψ ) We shall call δ the natural pseudodistance between ( X , � ϕ ) and ( Y , � ψ ) . The functional Θ( h ) = max i max x ∈ X | ϕ i ( x ) − ψ i ◦ h ( x ) | represents the “cost” of the matching between observations induced by h . The lower this cost, the better the matching between the two observations is. Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 12 / 51
A Metric Approach to Shape Comparison Our formal setting The natural pseudodistance δ measures the dissimimilarity between the perceptions expressed by the pairs ( X , � ϕ ) , ( Y , � ψ ) . Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 13 / 51
A Metric Approach to Shape Comparison Our formal setting The natural pseudodistance δ measures the dissimimilarity between the perceptions expressed by the pairs ( X , � ϕ ) , ( Y , � ψ ) . The value δ is small if and only if we can find a homeomorphism between X and Y that induces a small change of the measuring function (i.e., of the shape property we are interested to study). Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 13 / 51
A Metric Approach to Shape Comparison Our formal setting The natural pseudodistance δ measures the dissimimilarity between the perceptions expressed by the pairs ( X , � ϕ ) , ( Y , � ψ ) . The value δ is small if and only if we can find a homeomorphism between X and Y that induces a small change of the measuring function (i.e., of the shape property we are interested to study). For more information: Patrizio Frosini (Department of Mathematics) Multidimensional Persistent Topology GETCO 2010 13 / 51
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