Natural Pseudo-Distance and Optimal Matching between Reduced Size Functions

Abstract

This paper studies the properties of a new lower bound for the natural pseudodistance. The natural pseudo-distance is a dissimilarity measure between shapes, where a shape is viewed as a topological space endowed with a real-valued continuous function. Measuring dissimilarity amounts to minimizing the change in the functions due to the application of homeomorphisms between topological spaces, with respect to the L ∞ -norm. In order to obtain the lower bound, a suitable metric between size functions, called matching distance, is introduced. It compares size functions by solving an optimal matching problem between countable point sets. The matching distance is shown to be resistant to perturbations, implying that it is always smaller than the natural pseudo-distance. We also prove that the lower bound so obtained is sharp and cannot be improved by any other distance between size functions.

A new algorithm for computing the 2-dimensional matching distance between size functions

“…The combinatorial representation of size functions using cornerpoints implies that size functions can be compared via a suitable distance between formal series, namely the matching distance, see details in [12]. Roughly speaking, The matching between the two formal series, realizing the matching distance between the two size functions.…”

“…Indeed, the following Matching Stability Theorem has been proven [11,12] (see also [9]): Remark 2.3. The hypothesis that X and Y are homeomorphic is not so restrictive.…”

Multi-scale Approximation of the Matching Distance for Shape Retrieval

“…the max-norm) produces just a small changing in the associated persistence diagram w.r.t. the matching distance [7,9]. This result will be useful later.…”

Advances in Multidimensional Size Theory

“…Indeed, Size Theory has been widely developed in this setting (Biasotti et al, 2008b), proving that each 1-dimensional size function admits a compact representation as a formal series of points and lines of R 2 (Frosini and Landi, 2001). As a consequence of this peculiar structure, a suitable matching distance between 1-dimensional size functions can be easily introduced, showing the stability of these descriptors with respect to such a distance (d'Amico et al, 2003;2010). All these properties make the concept of 1-dimensional size function central in the approach to the k-dimensional case proposed in Biasotti et al (2008a).…”

“…Part of the qualitative information contained in (M , ϕ) is then quantitatively stored in the associated size function ℓ (M ,ϕ) , describing some topological attributes that persist in the sublevel sets of M induced by ϕ. Following this approach, comparing two shapes can be reduced to the simpler comparison of the associated size functions, making use of a suitable distance as, e.g., the matching distance (d'Amico et al, 2003;2010). In the context of Algebraic Topology, an analogous notion to the one of size function has been developed under the name of size homotopy group (Frosini and Mulazzani, 1999).…”