RECENT ADVANCES IN MULTIDIMENSIONAL PERSISTENT TOPOLOGY Patrizio Frosini Vision Mathematics Group University of Bologna - Italy OUTLINE • Introduction and history • Multidimensional size functions • Reduction by the foliation method • Consequences: 1) STABILITY 2) LOCALIZATION OF DISCONTINUITIES • Conclusions Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 2/50 Let us start from three questions... Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 3/50 How similar are the colorings of these leaves? Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 4/50 How similar are the Riemannian structures of these manifolds? Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 5/50 How similar are the spatial positions of these wires? Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 6/50 All previous questions concern the comparison between pairs (M,ϕ), where M is a topological space and ϕ: M → IRk is a continuous function: ●LEAVES: ϕ(P)=color (r,g,b) at P ●3D MODELS: ϕ(P)=vector describing the first fundamental form at P •WIRES: ϕ(P)=(x(P),y(P),z(P)) Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 7/50 Multidimensional Persistent Topology studies the topological properties of the sublevel sets of continuous functions ϕ: M → IRk. It focuses on those properties that “persist” under change of the level. REMARK: the word multidimensional refers to the dimension of IRk, not the dimension of M! Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 8/50 From the beginning of the 90's, the study of Persistent Topology has originated different lines of research. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 9/50 Persistence of connected components (size functions): P. Frosini, A distance for similarity classes of submanifolds of a Euclidean space, Bull. Austral. Math. Soc. 42, 3 (1990), 407-416. A. Verri, C. Uras, P. Frosini, M. Ferri, On the use of size functions for shape analysis, Biological Cybernetics, 70, (1993), 99-107. C. Uras, A. Verri, Computing Size Functions from Edge Maps, International Journal of Computer Vision, 23 (1997), n. 2, 169-183. F. Dibos, P. Frosini and D. Pasquignon, The use of Size Functions for Comparison of Shapes through Differential Invariants, Journal of Mathematical Imaging and Vision, 21, (2004), n. 2, 107-118. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 10/50 Persistence in algebraic topological structures (size homotopy groups, size functor and persistent homology groups): P. Frosini, M. Mulazzani, Size homotopy groups for computation of natural size distances, Bull. Belg. Math. Soc., 6 (1999), 455-464. H. Edelsbrunner, D. Letscher and A. Zomorodian, Topological persistence and simplification, Discrete and Computational Geometry, 28 (2002), 511-533. F. Cagliari, M. Ferri and P. Pozzi, Size Functions from the categorical viewpoint, Acta Applicandae Mathematicae, 67 (2002), 225-235. G. Carlsson, A. Zomorodian, A. Collins and L. Guibas, Persistence barcodes for shapes, International Journal of Shape Modeling, 11(2) (2005), 149-187. D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Stability of persistence diagrams, Discrete and Computational Geometry. 37(1) (2007), 103-120. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 11/50 Natural pseudodistance (metric for persistence comparison) P. Donatini, P. Frosini, Natural pseudodistances between closed manifolds, Forum Mathematicum, 16 (2004), n. 5, 695-715. P. Donatini, P. Frosini, Natural pseudodistances between closed surfaces, Journal of the European Mathematical Society, 9 (2007), 331-353. P. Donatini, P. Frosini, Natural pseudodistances between closed curves, Forum Mathematicum (in press). Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 12/50 The natural pseudodistance Assume that M, N are topological spaces and ϕ : M→IRk, ψ : N→IRk are continuous functions. The natural pseudodistance between the pairs (M,ϕ), (N,ψ ) is defined to be where H(M,N) denotes the set of all homeomorphisms between M and N. Persistent topology can be also seen as a way to obtain lower bounds for the natural pseudodistance. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 13/50 JUST TWO EXAMPLES... Size functions: Theorem. The matching distance between size functions is a lower bound for the natural pseudodistance. Size homotopy groups: Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 14/50 Multidimensional persistence P. Frosini, M. Mulazzani, Size homotopy groups for computation of natural size distances, Bull. Belg. Math. Soc. 6 (1999), 455-464). G. Carlsson, A. Zomorodian, The theory of multidimensional persistence homology, Proc. 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing. ACM Press, New York, NY, 184-193. S. Biasotti, A. Cerri, P. Frosini, D. Giorgi, C. Landi, Multidimensional size functions for shape comparison, Journal of Mathematical Imaging and Vision, 32 (2008), 161-179. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 15/50 Surveys and general papers R. Ghrist, Barcodes: the persistent topology of data, Bull. Amer. Math. Soc., 45(1) (2008), 61-75. S. Biasotti, L. De Floriani, B. Falcidieno, P. Frosini, D. Giorgi, C. Landi, L. Papaleo, M. Spagnuolo, Describing shapes by geometrical-topological properties of real functions, ACM Computing Surveys, 40 (2008), n. 4, 12:1-12:87. H. Edelsbrunner, J.Harer, Persistent homology – a survey, Contemporary Mathematics, 453 (2008), 257-282. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 16/50 Let us go back to size functions Size functions describe the shape of a topological k space M with respect to a function ϕ : Μ → IR . Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 17/50 NOTE: in the following of this talk we ≤ shall write x=(x1,…, xk ) y=(y1,…, yk ) to ≤ mean xi yi for every index i. Analogously, (x1,…, xk ) < (y1,…, yk ) will mean that xi < yi for every index i. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 18/50 Definition of size function Let M be a topological space and assume that a continuous function ϕ : M →IRk is given. For each y ∈ IRk we denote by M 〈ϕ ≤ y〉 the set of all points of M at which the measuring function ϕ takes a value not greater than y. For each real vector y we say that two points P,Q are 〈ϕ ≤ y〉-connected if and only if they 〈ϕ ≤ 〉 belong to the same component in M y . Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 19/50 We call size function (of the pair (M,ϕ)) the ∆+ { ∈ 2k } → function ℓ (M,ϕ ): = (x,y) IR : x < y IN that takes each point (x,y) to the number of equivalence classes of M 〈ϕ ≤ x〉 with respect to 〈ϕ ≤ y〉-connectedness. EQUIVALENT DEFINITION ℓ(M,ϕ )(x,y) is the number of connected components of M〈ϕ ≤ y〉 that contain at least one point of M〈ϕ ≤ x〉. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 20/50 Let us make the definition of size function clear by some example in the case k=1. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 21/50 This is the size function ℓ(M,ϕ ). Example: M is the displayed curve, ϕ is the distance from C. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 22/50 More details about our example Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 23/50 More details about our example Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 24/50 More details about our example Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 25/50 More details about our example Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 26/50 When we move to the multidimensional case (i.e. to the case of measuring functions taking values in IRk instead of IR) the domain of every size function is a subset of IR2k and many new problems arise. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 27/50 ➔How can we represent size functions and persistent homology groups? Which distances can we use to compare size functions (or persistent homology groups)? ➔Are size functions and persistent homology groups stable with respect to small changes of the data? ➔Where are the discontinuities located, in the multidimensional case? Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 28/50 SOME NEW THEORETICAL RESULTS Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 29/50 REPRESENTATION BY FOLIATIONS Introduced in S. Biasotti, A. Cerri, P. Frosini, D. Giorgi, C. Landi, Multidimensional size functions for shape comparison, Journal of Mathematical Imaging and Vision, 32 (2008), 161-179. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 30/50 The theory of k-dimensional size functions can be reduced to the theory of 1-dimensional size functions. Indeed, a parameterized family of half- planes in IRk x IRk exists such that the restriction of ℓ(M,ϕ ) to each of these half- planes can be seen as a particular 1- dimensional size function. Patrizio Frosini (U. Bologna) - MULTIDIMENSIONAL PERSISTENT TOPOLOGY 31/50 k For every unit vector u=(u1,…, uk ) of IR such that ui>0 for i=1,…,k and for every k Σ vector b=(b1,…, bk ) of IR such that i bi=0, we shall say that the pair (u,b) is admissible. Given an admissible pair (u,b), π k k we define the half-plane (u,b) of IR x IR by the following parametric equations: x=su+b s<t {y=tu+b Patrizio Frosini (U.