Value-Offset Bifiltrations for Digital Images

TL;DR

This paper introduces the value-offset bifiltration for analyzing grayscale digital images using two-parameter persistent homology, providing efficient algorithms and demonstrating its effectiveness on real and noisy images.

Contribution

The paper presents the first study of bifiltrations constructed from digital images and introduces the value-offset bifiltration with efficient algorithms for its computation.

Findings

Efficient algorithms for value-offset bifiltration with respect to taxicab and Euclidean distances.

Analysis of runtime complexity and performance on sample images.

Contrast between bifiltrations from real images and random noise.

Abstract

Persistent homology, an algebraic method for discerning structure in abstract data, relies on the construction of a sequence of nested topological spaces known as a filtration. Two-parameter persistent homology allows the analysis of data simultaneously filtered by two parameters, but requires a bifiltration -- a sequence of topological spaces simultaneously indexed by two parameters. To apply two-parameter persistence to digital images, we first must consider bifiltrations constructed from digital images, which have scarcely been studied. We introduce the value-offset bifiltration for grayscale digital image data. We present efficient algorithms for computing this bifiltration with respect to the taxicab distance and for approximating it with respect to the Euclidean distance. We analyze the runtime complexity of our algorithms, demonstrate the results on sample images, and contrast the bifiltrations obtained from real images with those obtained from random noise.