Multidimensional Size Functions for Shape Comparison

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

“…The rest of this Section shows how the framework of 2-dimensional size functions can be reduced to the case k = 1, by a change of variable and the use of a suitable foliation, using a strategy similar to the one in [1]. The main idea is to provide a parameterized family of half-planes in R 2 × R 2 , and prove that the restriction of a 2-dimensional size function ℓ (X, ϕ) to each of these half-planes turns out to be a particular 1-dimensional size function.…”

“…In fact, data are often characterized by two or more properties; this happens for example with physical simulations, where several measurements are made about an observed phenomenon, or when data have multidimensional features, such as colors in the RGB model. These considerations have recently drawn the attention to the study of a multidimensional setting [1,2,6,14,19]. The term multidimensional, or equivalently kdimensional, is related to considering measuring functions taking value in R k , that is, ϕ : X → R k , and the subsequent extension of shape descriptors to this case.…”

“…This is due to the fact that a complete, discrete and stable representation for the Persistent Topology shape descriptors seems not to be available in the multidimensional setting, differently from what happens in the 1-dimensional situation. The arising computational difficulties have been faced following different strategies [1,5,10], but not completely solved.…”

“…As a partial solution, in [1] the authors studied the concept of k-dimensional size functions and proved that the restrictions of a k-dimensional size function to suitable subsets of its domain turn out to be 1-dimensional. This allowed the definition of a stable matching distance between k-dimensional size functions, namely the k-dimensional matching distance, building on existing results for the 1-dimensional case.…”

“…This allowed the definition of a stable matching distance between k-dimensional size functions, namely the k-dimensional matching distance, building on existing results for the 1-dimensional case. Unfortunately, [1] does not explain how to approximate the matching distance in a way to obtain a good compromise between computational cost and quality of results. Indeed, the straightforward application of the method could require a very huge number of calculations (see also [2]).…”