Advances in Multidimensional Size Theory

Abstract

Size Theory was proposed in the early 90's as a geometrical/topological approach to the problem of Shape Comparison, a very lively research topic in the fields of Computer Vision and Pattern Recognition. The basic idea is to discriminate shapes by comparing shape properties that are provided by continuous functions valued in R, called measuring functions and defined on topological spaces associated to the objects to be studied. In this way, shapes can be compared by using a descriptor named size function, whose role is to capture the features described by measuring functions and represent them in a quantitative way. However, a common scenario in applications is to deal with multidimensional information. This observation has led to considering vector-valued measuring functions, and consequently the multidimensional extension of size functions, namely the k-dimensional size functions. In this work we survey some recent results about size functions in this multidimensional setting, with particular reference to the localization of their discontinuities.

Comparison of persistent homologies for vector functions: From continuous to discrete and back

“…This suggests that in order to satisfy Equation (11), it suffices to choose admissible pairs ( l, b) ∈ Ladm * 2 at a distance within ǫ/9 of each other, guaranteeing that every member of Ladm * 2 is within ǫ/18 of a tested pair. In practice, our algorithm is reminiscent of the grid algorithm shown in Section 3 of [2], in the sense that we take pairs at a distance of 1/2 N of each other with N sufficiently large.…”

“…To begin with, critical points are no longer isolated even in non-degenerate situations [17]. Although the relevant points for persistent homology of vector functions are a subset of the critical points, precisely the Pareto critical points, these are still non-isolated [11]. For example, in the case of the sphere x 2 + y 2 + z 2 = 1 with the function f = (y, z), the Pareto critical points are those in the set x = 0, y 2 + z 2 = 1, yz ≥ 0.…”

Gromov‐Hausdorff Stable Signatures for Shapes using Persistence

“…In the context of shape analysis, one typically wishes to be able to discriminate shapes under different notions of invariance. Many approaches have been proposed for the problem of (pose invariant) shape classification and recognition, including the size theory of Frosini and collaborators [Fro90, CFL06a, CFL06b], the work of Hilaga et al [HSKK01], the shape contexts [BMP02], the integral invariants of [MCH*06], the eccentricity functions of [HK03], the shape distributions of [OFCD02], the canonical forms of [EK03], and the shape DNA and global point signatures based spectral methods in [RWP05] and [Rus07], respectively. The common underlying idea revolves around the computation and comparison of certain metric invariants, or signatures , so as to ascertain whether two given data sets represent in fact the same object, up to a certain notion of invariance.…”

On the geometrical properties of the coherent matching distance in 2D persistent homology

“…Such an extension has revealed to be a spring of new interesting mathematical problems, and the search of their solutions has stimulated the introduction of new ideas. One of these ideas is represented by the so-called foliation method, consisting in foliating the domain of the ranks of persistent homology groups by means of a family of half-planes [2,6,12]. An important consequence of this approach has been the recent introduction of a multidimensional matching distance between the ranks of persistent homology groups and the proof of its stability [10], opening the way to the application of multidimensional persistent homology in shape comparison.…”