Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector-valued functions, called filtering functions.As is well known, in the case of scalar-valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, i.e. the dimensions of the images of the homomorphisms induced by the inclusions of lower level sets into each other. Whenever such inclusions exist for lower level sets of vector-valued filtering functions, we can consider the multidimensional analogue of persistent Betti numbers. Varying the lower level sets, we get that persistent Betti numbers can be seen as functions taking pairs of vectors to the set of non-negative integers. In this paper we prove stability of multidimensional persistent Betti numbers. More precisely, we prove that small changes of the vector-valued filtering functions imply only small changes of persistent Betti numbers functions. This result can be obtained by assuming the filtering functions to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence. In order to obtain our stability theorem, some other new results are proved for continuous filtering functions. They concern the finiteness of persistent Betti numbers for vector-valued filtering functions, and the representation via persistence diagrams of persistent Betti numbers, as well as their stability, in the case of scalarvalued filtering functions. Finally, from the stability of multidimensional persistent Betti numbers we obtain a lower bound for the natural pseudo-distance.

Differential topology, and specifically Morse theory, provide a suitable setting for formalizing and solving several problems related to shape analysis. The fundamental idea behind Morse theory is that of combining the topological exploration of a shape with quantitative measurement of geometrical properties provided by a real function defined on the shape. The added value of approaches based on Morse theory is in the possibility of adopting different functions as shape descriptors according to the properties and invariants that one wishes to analyze. In this sense, Morse theory allows one to construct a general framework for shape characterization, parametrized with respect to the mapping function used, and possibly the space associated with the shape. The mapping function plays the role of a lens through which we look at the properties of the shape, and different functions provide different insights. In the last decade, an increasing number of methods that are rooted in Morse theory and make use of properties of real-valued functions for describing shapes have been proposed in the literature. The methods proposed range from approaches which use the configuration of contours for encoding topographic surfaces to more recent work on size theory and persistent homology. All these have been developed over the years with a specific target domain and it is not trivial to systematize this work and understand the links, similarities, and differences among the different methods. Moreover, different terms have been used to denote the same mathematical constructs, which often overwhelm the understanding of the underlying common framework. The aim of this survey is to provide a clear vision of what has been developed so far, focusing on methods that make use of theoretical frameworks that are developed for classes of real functions rather than for a single function, even if they are applied in a restricted manner. The term geometrical-topological used in the title is meant to underline that both levels of information content are relevant for the applications of shape descriptions: geometrical, or metrical, properties and attributes are crucial for characterizing specific instances of features, while topological properties are necessary to abstract and classify shapes according to invariant aspects of their geometry. The approaches surveyed will be discussed in detail, with respect to theory, computation, and application. Several properties of the shape descriptors will be analyzed and compared. We believe this is a crucial step to exploit fully the potential of such approaches in many applications, as well as to identify important areas of future research.

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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.

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The Forman's discrete Morse theory appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for one-dimensional filtrations. This paper is perhaps the first attempt in the direction of extending such algorithms to multidimensional filtrations. Initial framework related to Morse matchings for the multidimensional setting is proposed, and a matching algorithm given by King, Knudson, and Mramor is extended in this direction. The correctness of the algorithm is proved, and its complexity analyzed. The algorithm is used for establishing a reduction of a simplicial complex to a smaller but not necessarily optimal cellular complex. First experiments with filtrations of triangular meshes are presented.

In this survey we illustrate how the matching distance between reduced size functions can be applied for shape comparison.We assume that each shape can be thought of as a compact connected manifold with a real continuous function defined on it, that is a pair (M, ϕ : M → R), called size pair. In some sense, the function ϕ focuses on the properties and the invariance of the problem at hand. In this context, matching two size pairs (M, ϕ) and (N , ψ) means looking for a homeomorphism between M and N that minimizes the difference of values taken by ϕ and ψ on the two manifolds. Measuring the dissimilarity between two shapes amounts to the difficult task of computing the value δ = inf f maxP ∈M |ϕ(P ) − ψ(f (P ))| where f varies among all the homeomorphisms from M to N .From another point of view, shapes can be described by reduced size functions associated with size pairs. The matching distance between reduced size functions allows for a robust to perturbations comparison of shapes.The link between reduced size functions and the dissimilarity measure δ is established by a theorem stating that the matching distance provides an easily computable lower bound for δ.Throughout this paper we illustrate this approach to shape comparison by means of examples and experiments.

In content-based image retrieval a major problem is the presence of noisy shapes. It is well known that persistent Betti numbers area shape descriptor that admits a dissimilarity distance, the matchingdistance, stable under continuous shape deformations. In this paper wefocus on the problem of dealing with noise that changes the topologyof the studied objects. We present a general method to turn persistentBetti numbers into stable descriptors also in the presence of topologicalchanges. Retrieval tests on the Kimia-99 database show the effectivenessof the method