Locating topological structures in digital images via local homology

TL;DR

This work addresses the challenge of locating local topological features, such as holes, within digital images by moving beyond global persistent homology barcodes. It introduces a local-system framework built from triads (X, X1, X2) and short filtrations to translate local topological information into global sections and merging numbers, enabling localization of holes in binary images. The authors formalize the connection between local merging metrics and barcode occurrences, and demonstrate a practical hole localization workflow using sliding windows to generate heatmaps that pinpoint hole positions and sizes. The approach offers a theoretically grounded, computationally efficient alternative to exhaustive subimage analysis, with potential applications to image-based pore analysis and extensions to point-cloud data and crystalline materials.

Abstract

Topological data analysis (TDA) is a rising branch in modern applied mathematics. It extracts topological structures as features of a given space and uses these features to analyze digital data. Persistent homology, one of the central tools in TDA, defines persistence barcodes to measure the changes in local topologies among deformations of topological spaces. Although local spatial changes characterize barcodes, it is hard to detect the locations of corresponding structures of barcodes due to computational limitations. The paper provides an efficient and concise way to divide the underlying space and applies the local homology of the divided system to approximate the locations of local holes in the based space. We also demonstrate this local homology framework on digital images.

Figures (7)

• Figure 1: Two filtrations of 2-dimensional black pixels; that is, $f_0^{-1}(0) \subseteq f_1^{-1}(0) \subseteq f_2^{-1}(0) \subseteq f_3^{-1}(0) \subseteq f_4^{-1}(0)$ and $g_0^{-1}(0) \subseteq g_1^{-1}(0) \subseteq g_2^{-1}(0) \subseteq g_3^{-1}(0) \subseteq g_4^{-1}(0)$. Although images $f$ and $g$ share the same $1$-dimensional homology space $\mathbb{Z}_2$, the persistent homologies of these two images depict different lifespans. Indeed, the $1$-dimensional hole in (a)-(e) has the barcode $(0,2)$ while the hole in (f)-(j) has $(0,4)$.
• Figure 2: First row: a $6 \times 6$ image domain $P$ in $\mathbb{Z}^2$, a grayscale image $g: P \rightarrow \{ 0,1,2,3\}$, and a binary image $f: P \rightarrow \{ 0,1 \}$. Figures (c) and (d) are two different representations for the image $f$. In a binary image $f$ as in (d), pixels with a value of $0$ represent the black pixels of the image. Second row: a filtration of binary images made by image $g$ and thresholds $0, 1, 2,$ and $3$. Third row: the white-black pixel representations of images in the second row.
• Figure 3: Two local systems $(X,X_1,X_2)$ of topological spaces. Let $\Gamma_0$ denote the global section space of the sheaf structure $H_0(X_1) \xrightarrow{} H_0(X) \xleftarrow{} H_0(X_2)$. Then we have the following information. The first row: $H_0(X_1) \simeq \mathbb{Z}_2$, $H_0(X_2) \simeq \mathbb{Z}_2^5$, $H_0(X_1 \cup X_2) \simeq \mathbb{Z}_2^6$, $H_0(X) \simeq \mathbb{Z}_2^2$, and $\Gamma_0 \simeq \mathbb{Z}_2^4$. The second row: $H_0(X_1) \simeq \mathbb{Z}_2^4$, $H_0(X_2) \simeq \mathbb{Z}_2^2$, $H_0(X_1 \cup X_2) \simeq \mathbb{Z}_2^6$, $H_0(X) \simeq \mathbb{Z}_2$, and $\Gamma_0 \simeq \mathbb{Z}_2^5$.
• Figure 4: An illustration of the construction of a local system in a 2D binary image. In this example, we have $m_0(X_1;X_2) = 3$, $o_0(X_1;X_2) = 0$, $m_1(X_1;X_2) = 0$, and $o_1(X_1;X_2) = 1$.
• Figure 5: An illustration of bounding regions of the hole structure. (a) A binary image that contains a single 1-dimensional loop structure. (b) A rectangular bounding box formed by yellow and brown pixels. (c) A more compact bounding region formed by red and purple pixels.

Theorems & Definitions (29)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1: Greenberg, (9.2)
• Proposition 2: Greenberg
• Proposition 3: Greenberg
• Proposition 4: Corollary (15.5), Greenberg
• Definition 4: Greenberg
• Definition 5: EdelsbrunnerHarerbook2010
• Definition 6: EdelsbrunnerHarerbook2010