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.

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Size Theory has proven to be a useful geometrical/topological approach to shape analysis and comparison. Originally introduced by considering 1-dimensional properties of shapes, described by means of real-valued functions, it has been subsequently generalized to take into account multidimensional properties coded by functions valued in R k .In the context of Size Theory, this generalization has led to introduce a shape descriptor called k-dimensional size function, and a distance to compare size functions, namely the k-dimensional matching distance. This paper proposes a novel computational framework to deal with the 2-dimensional case of Size Theory. More precisely, some new theoretical results about approximating the 2-dimensional matching distance are presented, leading to the formulation of an algorithm for its computation (up to an arbitrary error threshold).

The recent introduction of 3D shape analysis frameworks able to quantify the deformation of a shape into another in terms of the variation of real functions yields a new interpretation of the 3D shape similarity assessment and opens new perspectives. Indeed, while the classical approaches to similarity mainly quantify it as a numerical score, map‐based methods also define (dense) shape correspondences. After presenting in detail the theoretical foundations underlying these approaches, we classify them by looking at their most salient features, including the kind of structure and invariance properties they capture, as well as the distances and the output modalities according to which the similarity between shapes is assessed and returned. We also review the usage of these methods in a number of 3D shape application domains, ranging from matching and retrieval to annotation and segmentation. Finally, the most promising directions for future research developments are discussed.