2022
DOI: 10.1038/s41598-022-09464-7
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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 … Show more

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Cited by 8 publications
(6 citation statements)
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“…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.…”
Section: Discussionmentioning
confidence: 99%
See 2 more Smart Citations
“…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.…”
Section: Discussionmentioning
confidence: 99%
“…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.…”
Section: Discussionmentioning
confidence: 99%
See 1 more Smart Citation
“…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.…”
Section: Methodsmentioning
confidence: 99%
“…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.…”
Section: Justification Of the Gaussian Kernelmentioning
confidence: 99%