Persistent homology and partial similarity of shapes

Abstract

The ability to perform shape retrieval based not only on full similarity, but also partial similarity is a key property for any content-based search engine. We prove that persistence diagrams can reveal a partial similarity between two shapes by showing a common subset of points. This can be explained using the MayerVietoris formulas that we develop for ordinary, relative and extended persistent homology. An experiment outlines the potential of persistence diagrams as shape descriptors in retrieval tasks based on both full and partial similarity.

Alexander duality for functions

“…• Can our results be extended to the more general setting of a decomposition X = A ∪ B of a topological space X, as considered by Di Fabio and Landi [4,5].…”

“…The General Shore Theorem can also be proven with algebraic arguments: Di Fabio and Landi [4,5] consider decompositions X = A ∪ B, for a general topological space X, and relate the ordinary, relative, and extended (horizontal plus vertical) persistent Betti numbers of a function restricted to these sets with each other. These relationships depend on the ranks of maps between certain absolute and relative homology groups; see Theorems 3.1, 3.3, and 3.4 in [5] for details.…”

“…These relationships depend on the ranks of maps between certain absolute and relative homology groups; see Theorems 3.1, 3.3, and 3.4 in [5] for details. Applied to the case of a sphere, these maps are trivial except for the extended case in dimensions 0 and n. Investigating this case more carefully leads to the same result as (20).…”

A topology-based approach to computing neighborhood-of-interest points using the Morse complex

“…Therefore, given a correspondence of the points in their LMCs, it is possible to obtain a measure of how similar the two patterns are. We propose to compute a matching score as score(λ r , τ s ) = {(λ r , τ s ) | Pr(λ r ) ≈ Pr(τ s )} (16) where (λ r , τ s ) are corresponding cells between LMCs that have values Pr(λ r ) and Pr(τ s ) for some property on the structural patterns. The symbol denotes set cardinality so that the score counts the number of corresponding cells that agree with respect to property Pr.…”

“…Computational topology is a field gaining importance for analyzing images at qualitative, structural and abstract levels [2,5,9,15,16,29,39]. In this work, we present an approach based on the topology of functions, given by Morse complexes, to defining locally meaningful connectivity of interest points.…”