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

Abstract

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).

Necessary conditions for discontinuities of multidimensional persistent Betti numbers

“…Also, our results may be of help in case the computation is achieved through the use of the foliation method. Indeed, this alternative approach has revealed to be useful in the development of possible distances between multidimensional PBNs . In this context, Example suggests that, by virtue of Proposition and our new Theorem , it could be possible to track leaf by leaf the movements of cornerpoints associated with the 1‐dimensional restrictions of PBNs functions, thus avoiding to compute such restrictions from scratch every time a new leaf in the foliation is visited.…”

“…An approach to this research is the one proposed in , which is based on the foliation method : The authors show that, when k > 1, a dimensionality reduction can be used to decompose k ‐dimensional PBNs into a family of 1‐dimensional PBNs. This allows for the definition of a proven stable distance between k ‐dimensional PBNs , which can be effectively evaluated through suitable approximation techniques . Beyond stability, the foliation method has led to prove that multidimensional PBNs allow for the reconstruction of planar curves, thus providing the first advancement towards the solution of the inverse problem in persistence .…”

Recent Trends, Applications, and Perspectives in 3D Shape Similarity Assessment

“…Other classes of deformations which are relevant for applications include certain classes of non‐isometric transformations, which do not preserve the Rienmannian structure of the shape [BCFG11]. Typical examples are shape stretching, scaling and affine transformations [RBB*11, RK14].…”

“…Simple rotation‐ and translation invariant‐structures include those based on Euclidean distances (e.g. from the object centre of mass [BCF*08], other points or regions of interest [BK10a, BCFG11] including shape boundaries [MDTS09, LH13]). Extrinsic structures can be extended to cope with global scale or affine transformations.…”

PHoS: Persistent Homology for Virtual Screening

“…The stability of persistent homology implies that the matching distance d M (B, C) between fibered barcodes B and C can be computed approximately, up to arbitrary accuracy, by computing the bottleneck distance d b (M L , N L ) for a finite number of lines L [11]. The computational complexity of the best known algorithm for computing the bottleneck distance is O(n 1.5 log n) where n is the number of intervals [42] in the two barcodes.…”