Lipschitz-Killing Curvatures Research Articles

Mean Lipschitz–Killing curvatures for homogeneous random fractals

Homogeneous random fractals form a probabilistic extension of self-similar sets with more dependencies than in random recursive constructions. For such random fractals we consider mean values of the Lipschitz–Killing curvatures of their parallel sets for small parallel radii. Under the uniform strong open set condition and some further geometric assumptions, we show that rescaled limits of these mean values exist as the parallel radius tends to 0 . Moreover, integral representations are derived for these limits which extend those known in the deterministic case.

Curvature measures of pseudo-Riemannian manifolds

Abstract The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric ( 0 , 2 ) {(0,2)} -tensors. More precisely, we construct a family of generalized curvature measures attached to such manifolds, extending the Riemannian Lipschitz–Killing curvature measures introduced by Federer. We then show that they behave naturally under isometric immersions, in particular they do not depend on the ambient signature. Consequently, we extend Theorema Egregium to surfaces equipped with a generic metric of changing signature, and more generally, establish the existence as distributions of intrinsically defined Lipschitz–Killing curvatures for such manifolds of arbitrary dimension. This includes in particular the scalar curvature and the Chern–Gauss–Bonnet integrand. Finally, we deduce a Chern–Gauss–Bonnet theorem for pseudo-Riemannian manifolds with generic boundary.

Random Lipschitz–Killing curvatures: Reduction Principles, Integration by Parts and Wiener chaos

In this survey we collect some recent results regarding the Lipschitz–Killing curvatures (LKCs) of the excursion sets of random eigenfunctions on the two-dimensional standard flat torus (arithmetic random waves) and on the two-dimensional unit sphere (random spherical harmonics). In particular, the aim of the present survey is to highlight the key role of integration by parts formulae in order to have an extremely neat expression for the random LKCs. Indeed, the main tool to study local geometric functionals of random waves on manifold is to exploit their Wiener chaos decomposition and show that (often), in the so-called high-energy limit, a single chaotic component dominates their behavior. Moreover, reduction principles show that the dominant Wiener chaotic component of LKCs of random waves’ excursion sets at threshold level u ≠ 0 u\ne 0 is proportional to the integral of H 2 ( f ) H_2(f) , f f being the random field of interest and H 2 H_2 the second Hermite polynomial. This will be shown via integration by parts formulae.

On the excursion area of perturbed Gaussian fields

We investigate Lipschitz-Killing curvatures for excursion sets of random fields on ℝ2under a very specific perturbation, namely a small spatial-invariant random perturbation with zero mean. An expansion formula for mean curvatures is derived when the magnitude of the perturbation vanishes, which recovers the Gaussian Kinematic Formula at the limit by contiguity of the model. We develop an asymptotic study of the perturbed excursion area behaviour that leads to a quantitative non-Gaussian limit theorem, in Wasserstein distance, for fixed small perturbations and growing domain. When letting both the perturbation vanish and the domain grow, a standard Central Limit Theorem follows. Taking advantage of these results, we propose an estimator for the perturbation variance which turns out to be asymptotically normal and unbiased, allowing to make inference through sparse information on the field.

Lipschitz-Killing curvatures of excursion sets for two-dimensional random fields

In the present paper we study three geometrical characteristics for the excursion sets of a two-dimensional stationary isotropic random field. First, we show that these characteristics can be estimated without bias if the considered field satisfies a kinematic formula, this is for instance the case of fields given by a function of smooth Gaussian fields or of some shot noise fields. By using the proposed estimators of these geometric characteristics, we describe some inference procedures for the estimation of the parameters of the field. An extensive simulation study illustrates the performances of each estimator. Then, we use the Euler characteristic estimator to build a test to determine whether a given field is Gaussian or not, when compared to various alternatives. The test is based on a sparse information, i.e., the excursion sets for two different levels of the field to be tested. Finally, the proposed test is adapted to an applied case, synthesized 2D digital mammograms.

A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions

We establish here a quantitative central limit theorem (in Wasserstein distance) for the Euler–Poincaré characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler–Poincaré characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level $u$ is fully degenerate, that is, the Euler–Poincaré characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. We discuss also a possible unifying framework for the Lipschitz–Killing curvatures of the excursion sets for Gaussian spherical harmonics.

Kinematic formula for heterogeneous Gaussian related fields

We provide a generalization of the Gaussian Kinematic Formula (GKF) in Taylor et al. (2006) for multivariate, heterogeneous Gaussian-related fields. The fields under consideration, f=F∘y, are non-Gaussian fields built out of smooth, independent Gaussian fields y=(y1,y2,..,yK) with heterogeneity in distribution amongst the individual building blocks. Our motivation comes from potential applications in the analysis of Cosmological Data (CMB). Specifically, future CMB experiments will be focusing on polarization data, typically modeled as isotropic vector-valued Gaussian related fields with independent, but non-identically distributed Gaussian building blocks; this necessitates such a generalization. Extending results (Taylor et al., 2006) to these more general Gaussian relatives with distributional heterogeneity, we present a generalized Gaussian Kinematic Formula (GKF). The GKF in this paper decouples the expected Euler characteristic of excursion sets into Lipschitz Killing Curvatures (LKCs) of the underlying manifold and certain Gaussian Minkowski Functionals (GMFs). These GMFs arise from Gaussian volume expansions of ellipsoidal tubes as opposed to the usual tubes in the Euclidean volume of tube formulae. The GMFs form a main contribution of this work that identifies this tubular structure and a corresponding volume of tubes expansion in which the GMFs appear.

Whitney stratifications and the continuity of local Lipschitz–Killing curvatures

In the paper we prove that the local Lipschitz–Killing curvatures of a definable set in a polynomially bounded o-minimal structure are continuous along the strata of a Whitney stratification. Moreover, if the stratification is (w)-regular the local Lipschitz–Killing curvatures are locally Lipschitz in any o-minimal structure.

Central Limit Theorem for Lipschitz–Killing Curvatures of Excursion Sets of Gaussian Random Fields

Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, asymptotics of nonlinear functionals of Gaussian random fields have been studied [see Berman (Sojourns and extremes of stochastic processes, Wadsworth & Brooks, Monterey, 1991), Kratz and Leon (Extremes 3(1):57–86, 2000), Kratz and Leon (J Theor Probab 14(3):639–672, 2001), Meshenmoser and Shashkin (Stat Probab Lett 81(6):642–646, 2011), Pham (Stoch Proc Appl 123(6):2158–2174, 2013), Spodarev (Chapter in modern stochastics and applications, volume 90 of the series Springer optimization and its applications, pp 221–241, 2013) for a sample of works in such settings], the most recent addition being (Adler and Naitzat in Stoch Proc Appl 2016; Estrade and Leon in Ann Probab 2016) where a central limit theorem (CLT) for Euler integral and Euler–Poincare characteristic, respectively, of the excursions set of a Gaussian random field is proven under some conditions. In this paper, we obtain a CLT for some global geometric functionals, called the Lipschitz–Killing curvatures of excursion sets of Gaussian random fields, in an appropriate setting.

A central limit theorem for Lipschitz–Killing curvatures of Gaussian excursions

This paper studies the excursion set of a real stationary isotropic Gaussian random field above a fixed level. We show that the standardized Lipschitz–Killing curvatures of the intersection of the excursion set with a window converges in distribution to a normal distribution as the window grows to the d-dimensional Euclidean space. Moreover a lower bound for the asymptotic variance is derived.

The Intrinsic geometry of some random manifolds

We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the embedded manifolds also converge to deterministic limits. Particularly interesting examples of these functionals are given by the Lipschitz-Killing curvatures, for which we also prove unbiasedness, using the Gaussian kinematic formula.

CLT for Lipschitz-Killing Curvatures of Excursion Sets of Gaussian Random Fields

Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, limit theorems have been proven for various geometric functionals of excursion sets/sojourn times (see [4, 13, 14, 18, 22, 25] for a sample of works in such settings). The most recent addition being [6] where a central limit theorem (CLT) for Euler-Poincare characteristic of the excursions set of a Gaussian random field is proven under appropriate conditions. In this paper, we obtain a CLT for some global geometric functionals, called the Lipschitz-Killing curvatures of excursion sets of Gaussian random fields in an appropriate setting.

Euler obstruction and Lipschitz–Killing curvatures

Applying a local Gauss–Bonnet formula for closed subanalytic sets to the complex analytic case, we obtain characterizations of the Euler obstruction of a complex analytic germ in terms of the Lipschitz–Killing curvatures and the Chern forms of its regular part. We also prove analogous results for the global Euler obstruction. As a corollary, we give a positive answer to a question of Fu on the Euler obstruction and the Gauss–Bonnet measure.

High-frequency asymptotics for Lipschitz–Killing curvatures of excursion sets on the sphere

In this paper, we shall be concerned with geometric functionals and excursion probabilities for some nonlinear transforms evaluated on Fourier components of spherical random fields. In particular, we consider both random spherical harmonics and their smoothed averages, which can be viewed as random wavelet coefficients in the continuous case. For such fields, we consider smoothed polynomial transforms; we focus on the geometry of their excursion sets, and we study their asymptotic behaviour, in the high-frequency sense. We focus on the analysis of Euler-Poincar\'{e} characteristics, which can be exploited to derive extremely accurate estimates for excursion probabilities. The present analysis is motivated by the investigation of asymmetries and anisotropies in cosmological data. The statistics we focus on are also suitable to deal with spherical random fields which can only be partially observed, the canonical example being provided by the masking effect of the Milky Way on Cosmic Microwave Background (CMB) radiation data.

Generalized parallel <i>p<sub>i</sub></i>-equidistant ruled surfaces

In this paper, parallel pi-equidistant ruled surfaces in 3-dimensional Euclidean space E3, [6], were generalized to n-dimensional Euclidean space En. Then mean curvatures, Lipschitz–Killing curvatures, Gauss curvatures, scalar normal curvatures, Riemannian curvatures, Ricci curvatures, scalar curvatures of (m + 1)-dimensional parallel pi-equidistant ruled surfaces were calculated and some relations between these curvatures were found. Also, examples related to the parallel p3-equidistant ruled surfaces in the E3 are given.

Lipschitz-Killing Curvatures of the Excursion Sets of Skew Student's t Random Fields

In many applications related with geostatistics, biological and medical imaging, material science, and engineering surfaces, the real observations have asymmetric and heavy-tailed multivariate distributions. These observations are spatially correlated and they could be modeled by the skew random fields. However, several statistical analysis problems require studying the integral geometry of these random fields in order to detect the local changes between the different samples, via the expected Euler-Poincaré characteristic and the intrinsic volumes (Lipschitz-Killing curvatures) of the excursion set obtained at a given threshold. This article is interested in a class of skew random fields, namely, skew Student's t random field. The goal is to derive an explicit formula of Lipschitz-Killing curvatures and the expected Euler-Poincaré characteristic of the skew Student's t excursion sets on a compact subset S of ℝ2 by extending previous results reported in the literature. The motivation comes from the need to model the roughness of some engineering surfaces in order to detect the local changes of the surface peaks/valleys during a specific physical phenomenon. The analytical and empirical Euler-Poincaré characteristics are compared in order to test the skew Student's t random field on the real surface. Simulation results are also presented in the article for illustration and validation.

Set-level threshold-free tests on the intrinsic volumes of SPMs

Conventionally, set-level inference on statistical parametric maps (SPMs) is based on the topological features of an excursion set above some threshold—for example, the number of clusters or Euler characteristic. The expected Euler characteristic—under the null hypothesis—can be predicted from an intrinsic measure or volume of the SPM, such as the resel counts or the Lipschitz–Killing curvatures (LKC). We propose a new approach that performs a null hypothesis omnibus test on an SPM, by testing whether its intrinsic volume (described by LKC coefficients) is different from the volume of the underlying residual fields: intuitively, whether the number of peaks in the statistical field (testing for signal) and the residual fields (noise) are consistent or not. Crucially, this new test requires no arbitrary feature-defining threshold but is nevertheless sensitive to distributed or spatially extended patterns. We show the similarities between our approach and conventional topological inference—in terms of false positive rate control and sensitivity to treatment effects—in two and three dimensional simulations. The test consistently improves on classical approaches for moderate (> 20) degrees of freedom. We also demonstrate the application to real data and illustrate the comparison of the expected and observed Euler characteristics over the complete threshold range.

Fat Triangulations, Curvature and Quasiconformal Mappings

Department of Mathematics and Computer Science, The Open University of Israel, 1 University Rd.,Raanana 43107, Israel* Author to whom correspondence should be addressed; E-Mail: semil@tx.technion.ac.il;Tel.: +972-4-8292896; Fax: +972-4-8294799.Received: 9 April 2012; in revised form: 31 May 2012 / Accepted: 11 June 2012 /Published: 4 July 2012Abstract: We investigate the interplay between the existence of fat triangulations,PLapproximations of Lipschitz–Killing curvatures and the existence of quasiconformalmappings. In particular we prove that if there exists a quasiconformal mapping betweentwo PLor smooth n-manifolds, then their Lipschitz–Killing curvatures are bilipschitzequivalent. An extension to the case of almost Riemannian manifolds, of a previous existenceresult of quasimeromorphic mappings on manifolds due to the first author is also given.Keywords: fat triangulation; Lipschitz–Killing curvatures; quasimeromorphic mapping1. IntroductionThe present note is largely motivated by our theorem below, itself a continuation and generalizationof previous results of Martio and Srebro [1], Tukia [2] and Peltonen [3]:Theorem 1.1 ([4]) Let M

Euler characteristic and Lipschitz–Killing curvatures of closed semi-algebraic sets

We prove a formula that relates the Euler–Poincaré characteristic of a closed semi-algebraic set to its Lipschitz–Killing curvatures.

Lipschitz-Killing curvatures of self-similar random fractals

For a large class of self-similar random sets F F in R d \mathbb {R}^d , geometric parameters C k ( F ) C_k(F) , k = 0 , … , d k=0,\ldots ,d , are introduced. They arise as a.s. (average or essential) limits of the volume C d ( F ( ε ) ) C_d(F(\varepsilon )) , the surface area C d − 1 ( F ( ε ) ) C_{d-1}(F(\varepsilon )) and the integrals of general mean curvatures over the unit normal bundles C k ( F ( ε ) ) C_k(F(\varepsilon )) of the parallel sets F ( ε ) F(\varepsilon ) of distance ε \varepsilon rescaled by ε D − k \varepsilon ^{D-k} as ε → 0 \varepsilon \rightarrow 0 . Here D D equals the a.s. Hausdorff dimension of F F . The corresponding results for the expectations are also proved.