2025/09/03 by Dae-Woong Lee, Lee, Dae-Woong, P. Christopher Staecker +1
Computer Science · #54H30 #55P99 #Algebraic Topology (math.AT) #Digital Image Processing Techniques #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Medical Image Segmentation Techniques #Primary 68U03 #Secondary 55P15 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2509.03023
openalex publication_date 2025/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the properties of digital homotopy in the context of digital pictures (X,κ, κ), where X\subsetneq \Zn is a finite set, κ is an adjacency relation on X, and κ is an adjacency relation on the complement of X. In particular we focus on homotopy equivalence between digital pictures in \Z2. We define a numerical homotopy-type invariant for digital pictures in \Z2 called the outer perimeter, which is a basic tool for distinguishing homotopy types of digital pictures. When a digital picture has no holes, we show that it is homotopy equivalent to its rc-convex hull, obtained by ``filling in the gaps'' of any row or column. We show that a digital picture (X,ci,cj) is homotopy equivalent to only finitely many other digital pictures (Y,ci,cj). At the end of the paper, we raise a conjecture on the largest digital picture of the same homotopy-type of a given digital picture.