Mix-GENEO: A Flexible Filtration for Multiparameter Persistent Homology Detects Digital Images

TL;DR

The experiment results demonstrate the bifiltrations have ability to detect geometric and topological differences of digital images and are superior to 1-parameter filtrations including lower-star filtration and upper-star filtration.

Abstract

Two important tasks in the field of Topological Data Analysis are building practical multifiltrations on objects and using TDA to detect the geometry. Motivated by the tasks, we build multiparameter filtrations by operators on images named multi-GENEO, multi-DGENEO and mix-GENEO, and we prove the stability of both the interleaving distance and multiparameter persistence landscape of multi-GENEO with respect to the pseudometric on bounded functions. We also give the estimations of upper bound for multi-DGENEO and mix-GENEO. In practical applications, we regard image as a discrete function space, and then we build multifiltrations on the discrete function space. Finally, we construct comparable experiment on MNIST dataset to demonstrate our bifiltrations are superior to 1-parameter filtrations including lower-star filtration and upper-star filtration. For instance, 6 and 9 can be distinguished by our bifiltrations, while they cannot be distinguished by 1-parameter filtrations. The experiment results demonstrate our bifiltrations have ability to detect geometric and topological differences of digital images.

Figures (9)

• Figure 1: Persistence diagram $H_{0}$ and $H_{1}$ generated by lower-star filtration on the digit 3.
• Figure 2: Multiparameter persistence module $H_{0}$ and $H_{1}$ generated by Mix-GENEO filtration on the digit 3.
• Figure 3: The solid dots represent vertices that have already appeared. There is one edge with two endpoints in the left figure and there are two 2-simplices colored in yellow.
• Figure 4: Bifiltration Example. The figure records the birth coordinates of vertices, edges and faces. The vertices and edges in the birth coordinates are colored in orange, the faces in the birth coordinates are colored in yellow, the rest are colored in blue.
• Figure 5: $\mathscr{B}_{H(X)^{0}}$ reprensents the $H_{0}$ multiparameter persistence module, $\mathscr{B}_{H(X)^{1}}$ reprensents the $H_{0}$ multiparameter persistence module. One can see 1-loop in $\mathscr{B}_{H(X)^{1}}$ only birth at the coordinate (9,8) and persist to the coordinate (9,9).

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Example 1
• Theorem 1: Micheal-2015
• Theorem 2: Oli-2020
• Definition 3: zomorodian2005topology
• Theorem 3
• proof
• Lemma 4
• proof