Denoising 3D images: robustness of persistent homology measures

Persistent Homology (PH) is a powerful tool for analyzing the complex topological structures of 3D porous media, such as interconnected pores and channels. However, experimental noise often introduces a massive number of short-lived topological features, which complicates computational analysis and obscures the underlying signal. This paper investigates how to effectively denoise 3D grayscale images of porous media while preserving their essential topological properties.

The authors evaluate the robustness of various topological measures by applying two denoising approaches—Gaussian convolution and a machine learning-based method—to three types of synthetic 3D datasets: Fourier-generated structures, sphere-based PuMA models, and cellular Worley-noise structures. The researchers systematically add controlled Gaussian noise to these datasets and then quantify the discrepancy between the original and denoised images using metrics such as bottleneck distance, Wasserstein distance, persistence landscapes, and persistence images.

The study highlights that the effectiveness of denoising is highly dependent on the chosen topological measure. While the bottleneck stability theorem provides a theoretical guarantee for the robustness of bottleneck distances, other vectorizations of PH are more susceptible to noise-induced perturbations. The results indicate that there is an optimal range for the denoising parameter (e.g., the smoothing scale in Gaussian convolution) that minimizes the difference between the original and denoised topological signatures. The machine learning approach offers a promising alternative by potentially reducing the need for manual parameter tuning, though its performance varies across different material structures.

Accurate characterization of porous media is critical for industrial and environmental applications, such as CO2 sequestration and oil recovery, where fluid flow is dictated by internal pore geometry. By providing a rigorous framework for assessing how denoising affects topological analysis, this work helps researchers select appropriate preprocessing steps to ensure that their topological insights are both computationally feasible and physically meaningful.