A multi-parameter persistence framework for mathematical morphology

Abstract

The field of mathematical morphology offers well-studied techniques for image processing and is applicable for studies ranging from materials science to ecological pattern formation. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis. We demonstrate that morphological operations naturally form a multiparameter filtration and that persistent homology can then be used to extract information about both topology and geometry in the images as well as to automate methods for optimizing the study and rendering of structure in images. For illustration, we develop an automated approach that utilizes this framework to denoise binary, grayscale, and color images with salt and pepper and larger spatial scale noise. We measure our example unsupervised denoising approach to state-of-the-art supervised, deep learning methods to show that our results are comparable.

Automated Earthwork Detection Using Topological Persistence

“…Although we worked with persistent homology using the Ripser package, it may be desirable to use more sophisticated TDA packages in future applications: see Otter et al (2017) for a partial comparison of available software. In particular, applications of multi-parameter persistence, such as the techniques used in Chung et al (2022) and Vipond et al (2021) for image data, could provide enough robustness to more accurately identify partially-degraded berms and stock ponds.…”

“…In particular, applications of multi‐parameter persistence, such as the techniques used in Chung et al. (2022) and Vipond et al. (2021) for image data, could provide enough robustness to more accurately identify partially‐degraded berms and stock ponds.…”

“…In hydrology, persistence has been used to analyze fracture networks (Suzuki et al, 2021), subsurface pore structure (e.g., Herring et al, 2019;Moon et al, 2019), Kolmogorov flow (Kramar et al, 2016), hurricane patterns (Tymochko et al, 2020), cloud development in radar maps (Corcoran, 2019b;Corcoran & Jones, 2018), and to detect critical points in flood early warning systems (Syed Musa et al, 2021). Persistence has also been developed to quantify morphology of 2d and 3d shapes (Chung et al, 2022), for example, to estimate the structural integrity of tunnels (Zhang et al, 2020) and identify morphological differences between leaves of different plant species (Li et al, 2018). In landscape analysis, persistence has been proposed as a method to reduce noise in rasters (Corcoran, 2019a) and to identify landslides (Syzdykbayev et al, 2020b).…”

Fabric defect detection using cartoon–texture image decomposition model and visual saliency method

“…The basic shape characteristics of the original image can be maintained, and irrelevant structural features can be removed in image processing. 41 Thus, mathematical morphology is used to further process the visual saliency map to obtain the final binary segmentation result in this article.…”

Feature detection and hypothesis testing for extremely noisy nanoparticle images using topological data analysis

“…Examples include treating the locations of black pixels as embedded into Euclidean space, and then applying a Vietoris-Rips or density-based filtration (Garin and Tauzin, 2019;Turkes et al, 2021). As mentioned earlier, these methods require the choice of a threshold and hence multi-parameter persistent homology may be a more appropriate tool-see Chung, Day and Hu (2022). In sum, our hope is that this study can be find adoption in the microscopy community and also be used as point of departure for future studies at the intersection of image processing, statistics, and TDA.…”