Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack theoretically sound connection popular kernel-based learning techniques, such as kernel SVMs or PCA. In this work, establish by designing multi-scale for persistence diagrams, stable summary representation topological features in data. We show that is positive definite and prove its stability with respect the 1-Wasserstein distance. Experiments on two benchmark datasets 3D...

Accurate segmentation of tubular, network-like structures, such as vessels, neurons, or roads, is relevant to many fields research. For the topology their most important characteristic; particularly preserving connectedness: in case vascular networks, missing a connected vessel entirely alters blood-flow dynamics. We introduce novel similarity measure termed centerlineDice (short clDice), which calculated on inter-section masks and (morphological) skeleta. theoretically prove that clDice...

We present an algorithm for the computation of Vietoris-Rips persistence barcodes and describe its implementation in software Ripser. The method relies on implicit representations coboundary operator filtration order simplices, avoiding explicit construction storage matrix. Moreover, it makes use apparent pairs, a simple but powerful constructing discrete gradient field from total simplices simplicial complex, which is also independent interest. Our shows substantial improvements over...

We propose a metric for Reeb graphs, called the functional distortion distance. Under this distance, graph is stable against small changes of input functions. At same time, it remains discriminative at differentiating In particular, main result that distance between two graphs bounded from below by bottleneck both ordinary and extended persistence diagrams appropriate dimensions.

Abstract We establish tight bi-Lipschitz bounds certifying quasi-universality (universality up to a constant factor) for various distances between Reeb graphs: the interleaving distance, functional distortion and contortion distance. The definition of latter distance is novel contribution, special case contour trees we also prove strict universality this Furthermore, that merge coincides with yielding all four in case.

We define a simple, explicit map sending morphism f: M → N of pointwise finite dimensional persistence modules to matching between the barcodes and N. Our main result is that, in precise sense, quality this tightly controlled by lengths longest intervals ker f coker f.

We solve the problem of minimizing number critical points among all functions on a surface within prescribed distance {\delta} from given input function. The result is achieved by establishing connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2{\delta}, constructively proving tightness lower bound stability theorem homology in dimension two for any also show that an optimal solution can be computed...

For $X$ a finite category and $F$ field, we study the additive image of functor $\operatorname{H}_0(-,F) \colon \operatorname{rep}(X, \mathbf{Top}) \to \mathbf{Vect}_F)$, or equivalently, free $\operatorname{rep}(X, \mathbf{Set}) \mathbf{Vect}_F)$. We characterize all categories for which indecomposables in coincide with indecomposable indicator representations provide examples quivers wild representation type where contains only finitely many indecomposables. Motivated by questions...

Abstract Algebraic persistence studies modules (typically, linear representations of the poset $\textbf{R}^{n}$ with $n \geq 1$) and algebraic relationships between that are interleaved. The notion $\varepsilon $-interleaving is a generalization isomorphism (recovering when = 0$), which can be used to quantify how far any two from being isomorphic. An emblematic example this kind study stability theorem, strengthens Krull–Schmidt property one-parameter (representations $\textbf{R}$) by...

Despite their favorable pharmacokinetic properties, single-chain Fv antibody fragments (scFvs) are not commonly used as therapeutics, mainly due to generally low stabilities and poor production yields. In this work, we describe the identification optimization of a human scFv scaffold, termed FW1.4, which is suitable for humanization stabilization broad variety rabbit variable domains. A motif consisting five structurally relevant framework residues that highly conserved in domains was...

We define a simple, explicit map sending morphism $f:M \rightarrow N$ of pointwise finite dimensional persistence modules to matching between the barcodes $M$ and $N$. Our main result is that, in precise sense, quality this tightly controlled by lengths longest intervals $\ker f$ $\mathop{\mathrm{coker}} f$. As an immediate corollary, we obtain new proof algebraic stability persistence, fundamental theory persistent homology. In contrast previous proofs, ours shows explicitly how...

Given a finite set of points in $\mathbb R^n$ and radius parameter, we study the \v{C}ech, Delaunay-\v{C}ech, Delaunay (or Alpha), Wrap complexes light generalized discrete Morse theory. Establishing \v{C}ech as sublevel sets functions, prove that four are simple-homotopy equivalent by sequence simplicial collapses, which explicitly described single gradient field.

Abstract We consider the setting of Reeb graphs piecewise linear functions and study distances between them that are stable, meaning which similar in supremum norm ought to have graphs. define an edit distance for prove it is stable universal, provides upper bound any other distance. In contrast, via a specific construction, we show interleaving functional distortion on not universal.

We investigate the reactivity of hexagonal boron nitride (h-BN) on a Ni(1 1 1) single crystal towards atomic hydrogen over wide exposure range. Near edge x-ray absorption fine structure and photoelectron spectroscopy (XPS) show that for low exposures hydrogenation h-BN sheet is found. In contrast, intercalation between substrate occurs high exposures. For intermediate regimes, mixture observed. From temperature-programmed desorption XPS experiments, we conclude covalently bound to rather...

We have investigated the surface chemistry of molecular solar thermal energy storage system valence isomer pair norbornadiene (NBD)/quadricyclane (QC) on Ni(111). Our multimethod approach includes UV-photoelectron spectroscopy (UPS), high-resolution X-ray photoelectron (XPS), near edge absorption fine structure (NEXAFS), and density functional theory (DFT) calculations. The NBD/QC holds potential to be utilized in future technologies due its comparably high gravimetric density, release a...

The Reeb graph is a construction that studies topological space through the lens of real valued function. It has widely been used in applications, however its use on data means it desirable and increasingly necessary to have methods for comparison graphs. Recently, several define metrics graphs presented. In this paper, we focus two: functional distortion distance interleaving distance. former based Gromov--Hausdorff distance, while latter utilizes equivalence between particular class...

Abstract The extended persistence diagram is an invariant of piecewise linear functions, which known to be stable under perturbations functions with respect the bottleneck distance as introduced by Cohen–Steiner, Edelsbrunner, and Harer. We address question universality, asks for largest possible on diagrams, showing that a more discriminative variant universal. Our result applies generally settings where diagrams are considered only up certain degree. achieve our results establishing...