Frosini, Patrizio and Jablonski, Grzegorz
(2014)
Combining persistent homology and invariance groups for shape comparison.
Full text available as:
Abstract
In many applications concerning the comparison of data expressed by R^m-valued functions defined on a topological space X, the invariance with respect to a given group G of self-homeomorphisms of X is required. While persistent homology is quite efficient in the topological and qualitative comparison of this kind of data when the invariance group G is the group Homeo(X) of all self-
homeomorphisms of X, this theory is not tailored to manage the case in which G is a proper subgroup of Homeo(X), and its invariance appears too general for several tasks. This paper proposes a way to adapt persistent homology in order
to get invariance just with respect to a given group of self-homeomorphisms of X.
The main idea consists in a dual approach, based on considering the set of all G-invariant non-expanding operators defined on the space of the admissible filtering
functions on X. Some theoretical results concerning this approach are proven and two experiments are presented. An experiment illustrates the application of the proposed technique to compare 1D-signals, when the invariance is expressed by the group of affinities, the group of orientation-preserving affinities, the group of
isometries, the group of translations and the identity group. Another experiment shows how our technique can be used for image comparison.
Abstract
In many applications concerning the comparison of data expressed by R^m-valued functions defined on a topological space X, the invariance with respect to a given group G of self-homeomorphisms of X is required. While persistent homology is quite efficient in the topological and qualitative comparison of this kind of data when the invariance group G is the group Homeo(X) of all self-
homeomorphisms of X, this theory is not tailored to manage the case in which G is a proper subgroup of Homeo(X), and its invariance appears too general for several tasks. This paper proposes a way to adapt persistent homology in order
to get invariance just with respect to a given group of self-homeomorphisms of X.
The main idea consists in a dual approach, based on considering the set of all G-invariant non-expanding operators defined on the space of the admissible filtering
functions on X. Some theoretical results concerning this approach are proven and two experiments are presented. An experiment illustrates the application of the proposed technique to compare 1D-signals, when the invariance is expressed by the group of affinities, the group of orientation-preserving affinities, the group of
isometries, the group of translations and the identity group. Another experiment shows how our technique can be used for image comparison.
Document type
Article
Creators
Keywords
Natural pseudo-distance, filtering function, group action, persistent homology group, shape comparison
Subjects
DOI
Deposit date
28 Jan 2015 08:49
Last modified
03 Mar 2015 10:23
URI
Other metadata
Document type
Article
Creators
Keywords
Natural pseudo-distance, filtering function, group action, persistent homology group, shape comparison
Subjects
DOI
Deposit date
28 Jan 2015 08:49
Last modified
03 Mar 2015 10:23
URI
Available versions of this document
-
Combining persistent homology and invariance groups for shape comparison. (deposited 28 Jan 2015 08:49)
[Currently displayed]
Downloads
Downloads
Staff only: