Value-Offset Bifiltrations for Digital Images (2207.02990v1)

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.