Nonlocal Setting Research Articles

New normalized nonlocal hybrid level set method for image segmentation

This article introduces a new normalized nonlocal hybrid level set method for image segmentation. Due to intensity overlapping, blurred edges with complex backgrounds, simple intensity and texture information, such kind of image segmentation is still a challenging task. The proposed method uses both the region and boundary information to achieve accurate segmentation results. The region information can help to identify rough region of interest and prevent the boundary leakage problem. It makes use of normalized nonlocal comparisons between pairs of patches in each region, and a heuristic intensity model is proposed to suppress irrelevant strong edges and constrain the segmentation. The boundary information can help to detect the precise location of the target object, it makes use of the geodesic active contour model to obtain the target boundary. The corresponding variational segmentation problem is implemented by a level set formulation. We use an internal energy term for geometric active contours to penalize the deviation of the level set function from a signed distance function. At last, experimental results on synthetic images and real images are shown in the paper with promising results.

Delaunay hypersurfaces with constant nonlocal mean curvature

We study hypersurfaces of RN with constant nonlocal (or fractional) mean curvature. This is the equation associated with critical points of the fractional perimeter functional under a volume constraint. We establish the existence of a smooth branch of periodic cylinders in RN, N≥2, all of them with the same constant nonlocal mean curvature, and bifurcating from a straight cylinder. These are Delaunay type cylinders in the nonlocal setting. The proof uses the Crandall–Rabinowitz theorem applied to a quasilinear type fractional elliptic equation.

Efficient image decolorization with a multimodal contrast-preserving measure

Decolorization is widely used in monotonic displaying, black-and-white printing, single-channel image processing, and many computer vision applications. The conversion of a color image into a grayscale one suffers from data loss. This paper presents a novel parallel decolorization method which effectively preserves not only the spatially local color contrast but also the dominant non-local color contrast. A new multimodal contrast-preserving measure with a multimodal Gaussian distribution is proposed to relax the constraint of color contrast. The dominant non-local color pair set is constructed by taking advantage of a linear bounding volume hierarchy while the local color pair set is produced by removing duplicate instances of local color pairs. The whole pipeline design is highly parallel, allowing for a real-time GPU-based implementation. Experimental results and comparisons show that the proposed method can produce plausible decolorized images. A number of single-channel image processing applications, including edge detection, image segmentation, color augmentation, and image stylization, are demonstrated to verify the feasibility of the proposed decolorization method.

Free energy analysis of cell spreading

In this study we present a steady-state adaptation of the thermodynamically motivated stress fiber (SF) model of Vigliotti et al. (2015). We implement this steady-state formulation in a non-local finite element setting where we also consider global conservation of the total number of cytoskeletal proteins within the cell, global conservation of the number of binding integrins on the cell membrane, and adhesion limiting ligand density on the substrate surface. We present a number of simulations of cell spreading in which we consider a limited subset of the possible deformed spread-states assumed by the cell in order to examine the hypothesis that free energy minimization drives the process of cell spreading. Simulations suggest that cell spreading can be viewed as a competition between (i) decreasing cytoskeletal free energy due to strain induced assembly of cytoskeletal proteins into contractile SFs, and (ii) increasing elastic free energy due to stretching of the mechanically passive components of the cell. The computed minimum free energy spread area is shown to be lower for a cell on a compliant substrate than on a rigid substrate. Furthermore, a low substrate ligand density is found to limit cell spreading. The predicted dependence of cell spread area on substrate stiffness and ligand density is in agreement with the experiments of Engler et al. (2003). We also simulate the experiments of Théry et al. (2006), whereby initially circular cells deform and adhere to “V-shaped” and “Y-shaped” ligand patches. Analysis of a number of different spread states reveals that deformed configurations with the lowest free energy exhibit a SF distribution that corresponds to experimental observations, i.e. a high concentration of highly aligned SFs occurs along free edges, with lower SF concentrations in the interior of the cell. In summary, the results of this study suggest that cell spreading is driven by free energy minimization based on a competition between decreasing cytoskeletal free energy and increasing passive elastic free energy.

Efficient quantum computation in a network with probabilistic gates and logical encoding

A new approach to efficient quantum computation with probabilistic gates is proposed and analyzed in both a local and non-local setting. It combines heralded gates previously studied for atom or atom-like qubits with logical encoding from linear optical quantum computation in order to perform high fidelity quantum gates across a quantum network. The error-detecting properties of the heralded operations ensure high fidelity while the encoding makes it possible to correct for failed attempts such that deterministic and high-quality gates can be achieved. Importantly, this is robust to photon loss, which is typically the main obstacle to photonic based quantum information processing. Overall this approach opens a novel path towards quantum networks with atomic nodes and photonic links.

A sharp eigenvalue theorem for fractional elliptic equations

By using variational methods, in this paper we study a nonlinear elliptic problem defined in a bounded domain Ω ⊂ ℝ N , with smooth boundary ∂Ω, involving fractional powers of the Laplacian operator together with a suitable nonlinear term f. More precisely, we prove a characterization theorem on the existence of one weak solution for the elliptic problem $$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^{\alpha /2}}\mu = \lambda f(\mu )in\Omega ,} \\ {u > 0in\Omega ,} \\ {u = 0in\partial \Omega ,} \end{array}} \right.$$ , where α ∈ (0, 2), N > α, λ > 0 and (−Δ)α/2 denotes the nonlocal fractional Laplacian operator. Our result extends to the nonlocal setting recent theorems for ordinary and classical elliptic equations, as well as a characterization for elliptic problems on certain non-smooth domains. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary.

Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium

We consider a non-local phase transition equation set in a periodic medium and we construct solutions whose interface stays in a slab of prescribed direction and universal width. The solutions constructed also enjoy a local minimality property with respect to a suitable non-local energy functional.

Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes

We study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a Gagliardo-type fractional seminorm. We prove that minimizers of such non-differentiable functionals are locally bounded, Hölder continuous, and that they satisfy a suitable Harnack inequality. Hence, we provide an extension of celebrated results of M. Giaquinta and E. Giusti to the nonlocal setting. To do this, we introduce a particular class of fractional Sobolev functions, reminiscent of that considered by E. De Giorgi in his seminal paper of 1957. The flexibility of these classes allows us to also establish regularity of solutions to rather general nonlinear integral equations.

Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations

This paper studies nonlocal diffusion models associated with a finite nonlocal horizon parameter δ that characterizes the range of nonlocal interactions. The focus is on the variational formulation associated with Neumann type constraints and its numerical approximations. We establish the well-posedness for some variational problems associated and study their local limit as δ → 0. A main contribution is to derive a second order convergence to the local limit. We then discuss the numerical approximations including standard finite element methods and quadrature based finite difference methods. We study their convergence in the nonlocal setting and in the local limit.

Reduced basis methods for nonlocal diffusion problems with random input data

The construction, analysis, and application of reduced-basis methods for uncertainty quantification problems involving nonlocal diffusion problems with random input data is the subject of this work. Because of the lack of sparsity of discretized nonlocal models relative to analogous local partial differential equation models, the need for reduced-order modeling is much more acute in the nonlocal setting. In this effort, we develop reduced-basis approximations for nonlocal diffusion equations with affine random coefficients. Efficiency estimates of the proposed greedy reduced-basis methods are provided. Numerical examples are used to illustrate the effect varying various model parameters have on the efficiency and accuracy of the reduced-basis method relative to sparse-grid interpolation using the full finite element method. It is shown that the proposed reduced-basis approach can indeed provide substantial savings over standard sparse-grid methods.

On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results

We consider a nonlocal equation set in an unbounded domain with the epigraph property. We prove symmetry, monotonicity and rigidity results. In particular, we deal with halfspaces, coercive epigraphs and epigraphs that are flat at infinity.

Nonlocal PDEs on Graphs: From Tug-of-War Games to Unified Interpolation on Images and Point Clouds

In this paper, we introduce a new general class of partial difference operators on graphs, which interpolate between the nonlocal $$\infty $$?-Laplacian, the Laplacian, and a family of discrete gradient operators. In this context, we investigate an associated Dirichlet problem for this general class of operators and prove the existence and uniqueness of respective solutions. We show that a certain partial difference equation based on this class of operators recovers many variants of a stochastic game known as `Tug-of-War' and extends them to a nonlocal setting. Furthermore, we discuss a connection with certain nonlocal partial differential equations. Finally, we propose to use this class of operators as general framework to solve many interpolation problems in a unified manner as arising, e.g., in image and point cloud processing.

Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential

We prove existence, qualitative properties and asymptotic behavior of positive solutions to the doubly critical problem $$\begin{aligned} (-\Delta )^s u=\vartheta \frac{u}{|x|^{2s}}+u^{2_s^*-1}, \quad u\in \dot{H}^s(\mathbb {R}^N), \quad N> 2s,\quad 0<s<1. \end{aligned}$$ The technique that we use to prove the existence is based on variational arguments. The qualitative properties are obtained by using the moving plane method, in a nonlocal setting, on the whole \(\mathbb {R}^{N}\) and some comparison results. Moreover, in order to find the asymptotic behavior of solutions, we use a representation result that allows to transform the original problem into a different nonlocal problem in a weighted fractional space.

Nontrivial solutions of superlinear nonlocal problems

Abstract We study the question of the existence of infinitely many weak solutions for nonlocal equations of fractional Laplacian type with homogeneous Dirichlet boundary data, in presence of a superlinear term. Starting from the well-known Ambrosetti–Rabinowitz condition, we consider different growth assumptions on the nonlinearity, all of superlinear type. We obtain three different existence results in this setting by using the Fountain Theorem, which extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.

Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay

Abstract We are concerned with hypersurfaces of ℝ N {\mathbb{R}^{N}} with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in ℝ N {\mathbb{R}^{N}} with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or “cylinders” in ℝ 2 {\mathbb{R}^{2}} with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay-type bands in the nonlocal setting. Here we use a Lyapunov–Schmidt procedure for a quasilinear type fractional elliptic equation.

A critical fractional Laplace equation in the resonant case

In this paper we complete the study of the following non-local fractional equation involving critical nonlinearities $$ \cases (-\Delta)^s u-\lambda u=|u|^{2^*-2}u & {\text{in }} \Omega,\\ u=0 & {\text{in }} \mathbb{R}^n\setminus \Omega, \endcases $$ started in the recent papers \cite{13}, \cite{17}-\cite{19}. Here $s\in (0,1)$ is a fixed parameter, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive constant, $2^*=2n/(n-2s)$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $\RR^n$, $n> 2s$, with Lipschitz boundary. Aim of this paper is to study this critical problem in the special case when $n\not=4s$ and $\lambda$ is an eigenvalue of the operator $(-\Delta)^s$ with homogeneous Dirichlet boundary datum. In this setting we prove that this problem admits a non-trivial solution, so that with the results obtained in \cite{13}, \cite{17}-\cite{19}, we are able to show that this critical problem admits a nontrivial solution provided \roster \item"$\bullet$" $n> 4s$ and $\lambda> 0$, \item"$\bullet$" $n=4s$ and $\lambda> 0$ is different from the eigenvalues of $(-\Delta)^s$, \item"$\bullet$" $2s< n< 4s$ and $\lambda> 0$ is sufficiently large. \endroster In this way we extend completely the famous result of Brezis and Nirenberg (see \cite{4}, \cite{5}, \cite{9}, \cite{23}) for the critical Laplace equation to the non-local setting of the fractional Laplace equation.

A density property for fractional weighted Sobolev spaces

In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.

From dislocation motion to an additive velocity gradient decomposition, and some simple models of dislocation dynamics

A mathematical theory of time-dependent dislocation mechanics of unrestricted geometric and material nonlinearity is reviewed. Within a “small deformation” setting, a suite of simplified and interesting models consisting of a nonlocal Ginzburg Landau equation, a nonlocal level set equation, and a nonlocal generalized Burgers equation is derived. In the finite deformation setting, it is shown that an additive decomposition of the total velocity gradient into elastic and plastic parts emerges naturally from a micromechanical starting point that involves no notion of plastic deformation but only the elastic distortion, material velocity, dislocation density and the dislocation velocity. Moreover, a plastic spin tensor emerges naturally as well.

Superlinear nonlocal fractional problems with infinitely many solutions

In this paper we study the existence of infinitely many weak solutions for equations driven by nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. A model for these operators is given by the fractional Laplacianwhere s ∈ (0, 1) is fixed.We consider different superlinear growth assumptions on the nonlinearity, starting from the well-known Ambrosetti–Rabinowitz condition. In this framework we obtain three different results about the existence of infinitely many weak solutions for the problem under consideration, by using the Fountain Theorem. All these theorems extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.

On the Numerical Integration of Scalar Nonlocal Conservation Laws

We study a rather general class of 1D nonlocal conservation laws from a numerical point of view. First, following [F. Betancourt, R. Bürger, K.H. Karlsen and E.M. Tory, On nonlocal conservation laws modelling sedimentation. Nonlinearity 24 (2011) 855–885], we define an algorithm to numerically integrate them and prove its convergence. Then, we use this algorithm to investigate various analytical properties, obtaining evidence that usual properties of standard conservation laws fail in the nonlocal setting. Moreover, on the basis of our numerical integrations, we are led to conjecture the convergence of the nonlocal equation to the local ones, although no analytical results are, to our knowledge, available in this context.