The Impact of Changes in Resolution on the Persistent Homology of Images

TL;DR

This work addresses how image resolution affects topological summaries obtained from persistent homology by bounding the bottleneck distance $d_B$ between persistence diagrams derived from input functions and their digital approximations. It introduces leash and reach-based geometric bounds for distance transforms, including bounds in terms of $l_X(\sqrt{d} r)$, and density-based bounds, and demonstrates both theoretically and via case studies that persistence diagrams converge with finer resolution while remaining tractable at coarser scales. The main contributions are explicit $d_B$ bounds, corollaries under reach constraints, and practical downsampling strategies demonstrated on synthetic data and porous-material CT images, including percolation insights from $\mathrm{PD}_0$ diagrams. The results guide practitioners in balancing acquisition cost, computation, and accuracy in topological data analysis of images, with implications for material science and imaging pipelines.

Abstract

Digital images enable quantitative analysis of material properties at micro and macro length scales, but choosing an appropriate resolution when acquiring the image is challenging. A high resolution means longer image acquisition and larger data requirements for a given sample, but if the resolution is too low, significant information may be lost. This paper studies the impact of changes in resolution on persistent homology, a tool from topological data analysis that provides a signature of structure in an image across all length scales. Given prior information about a function, the geometry of an object, or its density distribution at a given resolution, we provide methods to select the coarsest resolution yielding results within an acceptable tolerance. We present numerical case studies for an illustrative synthetic example and samples from porous materials where the theoretical bounds are unknown.

Figures (14)

• Figure 1: An object $X$ in blue, with a voxel grid of side $r$ overlayed in gray. There is no $t$ that makes $X(r,t)$ have the same homology as $X$. If $t$ is close to $1$, the handle-shaped part of $X$ is lost. If $t$ is small, then the narrow annulus in the middle will be filled in.
• Figure 2: Illustration of the $\textrm{leash}_{A}(s)$: The ball of radius $s$ gets stuck inside part of $A$ but with a long enough leash the dog can reach every part of $A$.
• Figure 3: Diagram of our model for the digital approximation of a solid object $X$, and its geometric characterisation using signed Euclidean distance transforms and persistent homology. The top row is a simplified version of CT-imaging and segmentation. In gray: In the proof of Thm. \ref{['thm:main']}, instead of comparing the continuous distance transform $d^{\mp}_{X}$ to the discrete distance transform $D_r$ directly, we compare both to the continuous distance transform $d^{\mp}_{X(r,t)}$ of the discrete object.
• Figure 4: Illustration of the variables appearing in the proof of Lemma \ref{['lemma:DSEDTvsCSEDT']}.
• Figure 5: The four different suprema in the bound of Lemma \ref{['lemma:generalizeCS-E-H']} can all have different values. In the proof of Theorem \ref{['thm:main']} we show that suprema 1 and 3 are bounded by the two-sided leash $l_{X}(\sqrt{d}r)$, while suprema 2 and 4 are bounded by the voxel diameter $\sqrt{d}r$.

Theorems & Definitions (36)

• Proposition 3.1
• proof
• Corollary 3.2
• Corollary 3.3
• proof
• Corollary 3.4
• Corollary 3.5
• proof
• Theorem 4.1
• Lemma 4.2