Elsevier

Pattern Recognition Letters

Volume 33, Issue 11, 1 August 2012, Pages 1445-1450
Pattern Recognition Letters

Persistent homology and partial similarity of shapes

https://doi.org/10.1016/j.patrec.2011.11.003Get rights and content

Abstract

Persistent homology provides shapes descriptors called persistence diagrams. We use persistence diagrams to address the problem of shape comparison based on partial similarity. We show that two shapes having a common sub-part in general present a common persistence sub-diagram. Hence, the partial Hausdorff distance between persistence diagrams measures partial similarity between shapes. The approach is supported by experiments on 2D and 3D data sets.

Introduction

Distinguishing and recognizing deformable shapes is an important problem, encountered in numerous pattern recognition applications of computer vision and computer graphics. A major problem in the analysis of non-rigid shapes is finding similarity of deformable shapes which have only partial similarity, i.e., have similar as well as dissimilar parts.

The problem of partial comparison is widely discussed in (Bronstein et al., 2008b) for 2D shapes, and in (Biasotti et al., 2006b) for 3D shapes. While we refer readers to these papers for an account of methods dealing with the partial similarity problem, we underline that the most common approach in the literature is based on dividing shapes into parts and applying a global comparison method to the parts as separate objects. Instead, the authors of the aforementioned papers tackle the problem by a different strategy. Although the first one is based on the Gromov-Hausdorff distance between domains of R2, and the second one is based on a structural descriptor such as the Reeb graph of surfaces of R3, the common underlying idea is to overcome the problem that there is no objective way to define a part, by considering a trade-off between the full similarity measure between sub-parts and the importance of these sub-parts. Importantly, all the possible sub-parts are considered instead of favoring a specific one.

The aim of this paper is to show that the same paradigm allows us to cope with the partial similarity problem using shape descriptors based on persistent homology.

Persistent homology is an algebraic tool for measuring topological features of spaces and functions. It allows for a multi-scale analysis of topological data. The scale at which a feature is significant is measured by its persistence. Motivated by the problem of describing and recognizing deformable shapes, persistence of 0th homology, also known as a size function, has been studied for years first in computer vision (Frosini, 1992, Verri et al., 1993) and later in computer graphics (Biasotti et al., 2006a). Persistence of higher homology was originally introduced to study alpha-shapes and later applied to pattern recognition (Carlsson et al., 2005). Using persistent homology, we obtain a shape descriptor in terms of a multiset of points of the plane, called a persistence diagram (or a barcode). Comparison of persistence diagrams by a distance such as the Hausdorff distance gives a stable methodology to assess shape similarity (Cohen-Steiner et al., 2007, Frosini and Landi, 1997).

In order to cope with the partial similarity problem using persistence, we begin showing the relationship existing between the persistent homology of two sub-parts A and B, and that of X = A  B. In other words, we provide Mayer–Vietoris formulas for persistence. A Mayer–Vietoris formula for ordinary persistence has been obtained in (Di Fabio and Landi, 2011). Similar formulas also for relative and extended persistence are a novel contribution of this paper.

Next, we study how this reflects on persistence diagrams, i.e. our shape descriptors. The Mayer–Vietoris formula yields relationships among the persistence diagrams of X, A, B, indicating that the presence of A in X can be revealed by the presence of a common subset of points in the persistence diagrams of A and X (analogously for B and X).

Based on this idea, we propose to deal with the partial similarity problem by computing a distance between significant sub-diagrams of persistence diagrams. A distance such as the partial Hausdorff pseudo-distance can be used to impose a significance threshold to sub-diagrams, and to measure the similarity between all significant sub-diagrams. It automatically gives as output the similarity measure between the two most similar significant sub-diagrams.

This idea is effectively used in the final experiments, dealing with the detection of both full and partial similarity of shapes in the Mythological Creatures 2D and 3D datasets.

Section snippets

Background on shape analysis by persistence

Persistence involves analyzing a shape S by choosing a topological space X to represent it, and a function φ:XR to define a family of subspaces Xu=φ-1((-,u]),uR, nested by inclusion, i.e. a filtration of X. The map φ is chosen according to the shape properties of interest (e.g., height, distance from center of mass, curvature). Along the filtration, topological features such as connected components or tunnels can appear or disappear. For example, Fig. 1 shows the shape of a chair represented

Mayer–Vietoris formulas for persistent homology

In this section we give Mayer–Vietoris formulas for ordinary, relative and extended persistent homology. Given a triad (X, A, B) with X = A  B, a Mayer–Vietoris formula is a relationship among the ranks of the homology groups of X, A, B and C = A  B. It is obtainable from the Mayer–Vietoris sequenceHk+1(X)Hk(C)Hk(A)Hk(B)Hk(X),when this sequence is exact. We recall that the homomorphism Hk+1(X)  Hk(C) maps [z] to [(z|A)], the homomorphism Hk(C)  Hk(A)  Hk(B) maps [z] to ([z], [−z]), and the

Detection of sub-part similarity of shapes by persistence

Following Bronstein et al. (2008b), our model for assessment of sub-part similarity is based on two criteria: first, since there is no objective way to divide a shape into parts, all the possible partitions of the shapes should be considered, instead of favoring a specific one; second, since different parts have different importance, the parts must be significant. Therefore, given two shapes S and Q, the partial similarity dP between S and Q can be expressed as dP(S,Q)=min(S,Q)dF(S,Q),

Experiment and discussion

In order to evaluate the potential of persistent homology in partial comparison, we performed two tests on different Mythological Creatures data sets, containing shapes of horses, humans, centaurs, seahorses. Each shape differs by an articulation and/or additional parts. We assume as ground truth that a man and a centaur are dissimilar in the sense of a full similarity criterion, yet, parts of these shapes (the upper part of the centaur and the upper part of the man) are similar. Likewise, a

References (13)

  • S. Biasotti et al.

    Sub-part correspondence by structural descriptors of 3D shapes

    Computer-Aided Design

    (2006)
  • Biasotti, S., Giorgi, D., Spagnuolo, M., Falcidieno, B., 2006a. Size functions for 3D shape retrieval, in: SGP’06:...
  • A. Bronstein et al.

    Numerical Geometry of Non-Rigid Shapes

    (2008)
  • A.M. Bronstein et al.

    Analysis of two-dimensional non-rigid shapes

    Internat. J. Comput. Vision

    (2008)
  • G. Carlsson et al.

    Persistence barcodes for shapes

    Internat. J. Shape Model.

    (2005)
  • D. Cohen-Steiner et al.

    Stability of persistence diagrams

    Discrete Comput. Geom.

    (2007)
There are more references available in the full text version of this article.

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