150 research outputs found
Induced Matchings and the Algebraic Stability of Persistence Barcodes
We define a simple, explicit map sending a morphism of
pointwise finite dimensional persistence modules to a matching between the
barcodes of and . 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 and . 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 -interleaving
morphism between two persistence modules induces a -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
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
In 2009, Chazal et al. introduced -interleavings of persistence
modules. -interleavings induce a pseudometric on (isomorphism
classes of) persistence modules, the interleaving distance. The definitions of
-interleavings and 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,
is equal to the bottleneck distance . 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 -interleaving relation on multidimensional
persistence modules. This expresses transparently the sense in which two
-interleaved modules are algebraically similar. Third, using this
characterization, we show that when we define our persistence modules over a
prime field, satisfies a universality property. This universality result
is the central result of the paper. It says that satisfies a stability
property generalizing one which is known to satisfy, and that in
addition, if is any other pseudometric on multidimensional persistence
modules satisfying the same stability property, then . We also show
that a variant of this universality result holds for , over arbitrary
fields. Finally, we show that 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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