Nonlocal Helmholtz Decompositions and Connections to Classical Counterparts

In recent years nonlocal models have been successfully introduced in a variety of applications, such as dynamic fracture, nonlocal diffusion, flocking, and image processing. Thus, the development of a nonlocal calculus theory, together with the study of nonlocal operators has become the focus of many theoretical investigations. Our work focuses on a Helmholtz decomposition in the nonlocal (integral) framework. In the classical (differential) setting the Helmholtz decomposition states that we can decompose a three dimensional vector field as a sum of an irrotational function and a solenoidal function. We will define new nonlocal gradient and curl operators that allow us to create a similar nonlocal decomposition

Optimising Spatial and Tonal Data for PDE-based Inpainting

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods

Image Restoration Using Joint Statistical Modeling in Space-Transform Domain

This paper presents a novel strategy for high-fidelity image restoration by characterizing both local smoothness and nonlocal self-similarity of natural images in a unified statistical manner. The main contributions are three-folds. First, from the perspective of image statistics, a joint statistical modeling (JSM) in an adaptive hybrid space-transform domain is established, which offers a powerful mechanism of combining local smoothness and nonlocal self-similarity simultaneously to ensure a more reliable and robust estimation. Second, a new form of minimization functional for solving image inverse problem is formulated using JSM under regularization-based framework. Finally, in order to make JSM tractable and robust, a new Split-Bregman based algorithm is developed to efficiently solve the above severely underdetermined inverse problem associated with theoretical proof of convergence. Extensive experiments on image inpainting, image deblurring and mixed Gaussian plus salt-and-pepper noise removal applications verify the effectiveness of the proposed algorithm.Comment: 14 pages, 18 figures, 7 Tables, to be published in IEEE Transactions on Circuits System and Video Technology (TCSVT). High resolution pdf version and Code can be found at: http://idm.pku.edu.cn/staff/zhangjian/IRJSM

Dirichlet problems for several nonlocal operators via variational and topological methods

The main topic of the thesis is the study of elliptic differential equations with fractional order driven by nonlocal operators, as the fractional p-Laplacian, the fractional Laplacian for p=2, the general nonlocal operator and its anisotropic version. Recently, great attention has been focused on the study of fractional and nonlocal operators of elliptic type, both for pure mathematical research and in view of concrete real-world applications. This type of operators arises in a quite natural way in many different contexts, such as, among others, game theory, image processing, optimization, phase transition, anomalous diffusion, crystal dislocation, water waves, population dynamics and geophysical fluid dynamics. The main reason is that nonlocal operators are the infinitesimal generators of Lévy-type stochastic processes. Such processes extend the concept of Brownian motion, where the infinitesimal generator is the Laplace operator, and may contain jump discontinuities. Our aim is to show existence and multiplicity results for nonlinear elliptic Dirichlet problems, driven by a nonlocal operator, by applying variational and topological methods. Such methods usually exploit the special form of the nonlinearities entering the problem, for instance its symmetries, and offer complementary information. They are powerful tools to show the existence of multiple solutions and establish qualitative results on these solutions, for instance information regarding their location. The topological and variational approach provides not just existence of a solution, usually several solutions, but allow to achieve relevant knowledge about the behavior and properties of the solutions, which is extremely useful because generally the problems cannot be effectively solved, so the precise expression of the solutions is unknown. As a specific example of property of a solution that we look for is the sign of the solution, for example to be able to determine whether it is positive, or negative, or nodal (i.e., sign changing)