150 research outputs found

    Induced Matchings and the Algebraic Stability of Persistence Barcodes

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    We define a simple, explicit map sending a morphism f:MNf:M \rightarrow N of pointwise finite dimensional persistence modules to a matching between the barcodes of MM and NN. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of kerf\ker f and cokerf\mathop{\mathrm{coker}} f. As an immediate corollary, we obtain a new proof of the algebraic stability of persistence, a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ\delta-interleaving morphism between two persistence modules induces a δ\delta-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules, and yields a novel "single-morphism" characterization of the interleaving relation on persistence modules.Comment: Expanded journal version, to appear in Journal of Computational Geometry. Includes a proof that no definition of induced matching can be fully functorial (Proposition 5.10), and an extension of our single-morphism characterization of the interleaving relation to multidimensional persistence modules (Remark 6.7). Exposition is improved throughout. 11 Figures adde

    Topologically faithful image segmentation via induced matching of persistence barcodes

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    Image segmentation is a largely researched field where neural networks find vast applications in many facets of technology. Some of the most popular approaches to train segmentation networks employ loss functions optimizing pixel-overlap, an objective that is insufficient for many segmentation tasks. In recent years, their limitations fueled a growing interest in topology-aware methods, which aim to recover the correct topology of the segmented structures. However, so far, none of the existing approaches achieve a spatially correct matching between the topological features of ground truth and prediction. In this work, we propose the first topologically and feature-wise accurate metric and loss function for supervised image segmentation, which we term Betti matching. We show how induced matchings guarantee the spatially correct matching between barcodes in a segmentation setting. Furthermore, we propose an efficient algorithm to compute the Betti matching of images. We show that the Betti matching error is an interpretable metric to evaluate the topological correctness of segmentations, which is more sensitive than the well-established Betti number error. Moreover, the differentiability of the Betti matching loss enables its use as a loss function. It improves the topological performance of segmentation networks across six diverse datasets while preserving the volumetric performance. Our code is available in https://github.com/nstucki/Betti-matching

    The Theory of the Interleaving Distance on Multidimensional Persistence Modules

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    In 2009, Chazal et al. introduced ϵ\epsilon-interleavings of persistence modules. ϵ\epsilon-interleavings induce a pseudometric dId_I on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of ϵ\epsilon-interleavings and dId_I generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, dId_I is equal to the bottleneck distance dBd_B. This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the ϵ\epsilon-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two ϵ\epsilon-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, dId_I satisfies a universality property. This universality result is the central result of the paper. It says that dId_I satisfies a stability property generalizing one which dBd_B is known to satisfy, and that in addition, if dd is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then ddId\leq d_I. We also show that a variant of this universality result holds for dBd_B, over arbitrary fields. Finally, we show that dId_I restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in Foundations of Computational Mathematics. 36 page
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