Size homotopy groups for computation of natural size distances

Computer Graphics Forum

Bronstein

et al. 2015

123

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.

Section: Mathematical Backgroundmentioning

confidence: 99%

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

Biasotti

Computer Graphics Forum

Bronstein

et al. 2015

123

“…This generalization to vector‐valued functions is usually known as the Multidimensional Persistence Theory , where the adjective multidimensional refers to the fact that filtering functions are vector‐valued, and has no connections with the dimensionality of the space under study. The use of vector‐valued filtering functions in this context, introduced in for persistence of homotopy groups, enables the analysis of richer data structures.…”

Section: Introductionmentioning

confidence: 99%

“…Multidimensional persistence was firstly investigated in as regards homotopy groups, and by Carlsson and Zomorodian in as regards homology modules. In this context, the first stability result has been obtained for the 0th homology in : A distance between the 0th persistent Betti numbers, also called size functions , has been introduced and proven to be stable under perturbations of continuous vector‐valued filtering functions.…”

Section: Introductionmentioning

confidence: 99%

“…In [22], Chazal et al introduce the concept of persistence module and prove stability under the assumption that it is finite-dimensional. The problem of stability for scalar filtering functions is also approached in [23], where it is solved by assuming that the considered filtering functions are no more than continuous, but only for the 0th homology.Multidimensional persistence was firstly investigated in [19] as regards homotopy groups, and by Carlsson and Zomorodian in [24] as regards homology modules. In this context, the first stability result has been obtained for the 0th homology in [16]: A distance between the 0th persistent Betti numbers, also called size functions, has been introduced and proven to be stable under perturbations of continuous vector-valued filtering functions.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Betti numbers in multidimensional persistent homology are stable functions

Math Methods in App Sciences

Fabio

Ferri

et al. 2013

Self Cite

124

212

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.

“…Moreover, sometimes, the properties of data to be studied are intrinsically multidimensional, such as the coordinates of an object in a two‐dimensional or three‐dimensional image (e.g., for tracking applications), or photometric properties, which are usually taken into account in digital image segmentation. These considerations drove the attention to the concept of size homotopy group and, later on, to the theory of multidimensional persistence . Here, the term multidimensional, or equivalently k ‐dimensional, refers to the fact that data properties are described through functions taking values in

{double-struckR}^{k}

.…”

Section: Introductionmentioning

confidence: 99%

Necessary conditions for discontinuities of multidimensional persistent Betti numbers

Math Methods in App Sciences

Frosini

2014

Self Cite

Topological persistence has proven to be a promising framework for dealing with problems concerning the analysis of data. In this context, it was originally introduced by taking into account 1‐dimensional properties of data, modeled by real‐valued functions. More recently, topological persistence has been generalized to consider multidimensional properties of data, coded by vector‐valued functions. This extension enables the study of multidimensional persistent Betti numbers, which provide a representation of data based on the properties under examination. In this contribution, we establish a new link between multidimensional topological persistence and Pareto optimality, proving that discontinuities of multidimensional persistent Betti numbers are necessarily pseudocritical or special values of the considered functions. Copyright © 2014 John Wiley & Sons, Ltd.