H1persistent homology from 2D image Simple example

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arXiv link: https://arxiv.org/abs/2401.01160 • In our preprint Train-Free Segmentation in MRI with Cubical Persistent Homology , we adapt ideas from Topological Data Analysis to the problem of segmentation in medical images. This animation serves as a simple illustration of persistent homology on a 2D image. We build a cubical complex on a 2D greyscale image, with values between 0 and 1, and compute the persistent homology of its superlevel sets filtration. • Abstract: We describe a new general method for segmentation in MRI scans using Topological Data Analysis (TDA), offering several advantages over traditional machine learning approaches. • It works in three steps, first identifying the whole object to segment via automatic thresholding, then detecting a distinctive subset whose topology is known in advance, and finally deducing the various components of the segmentation. Although convoking classical ideas of TDA, such an algorithm has never been proposed separately from deep learning methods. To achieve this, our approach takes into account, in addition to the homology of the image, the localization of representative cycles, a piece of information that seems never to have been exploited in this context. In particular, it offers the ability to perform segmentation without the need for large annotated data sets. TDA also provides a more interpretable and stable framework for segmentation by explicitly mapping topological features to segmentation components. By adapting the geometric object to be detected, the algorithm can be adjusted to a wide range of data segmentation challenges. We carefully study the examples of glioblastoma segmentation in brain MRI, where a sphere is to be detected, as well as myocardium in cardiac MRI, involving a cylinder, and cortical plate detection in fetal brain MRI, whose 2D slices are circles. We compare our method to state-of-the-art algorithms.

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