Image Segmentation with Eigenfunctions of an Anisotropic Diffusion Operator

We propose the eigenvalue problem of an anisotropic diffusion operator for image segmentation. The diffusion matrix is defined based on the input image. The eigenfunctions and the projection of the input image in some eigenspace capture key features of the input image. An important property of the model is that for many input images, the first few eigenfunctions are close to being piecewise constant, which makes them useful as the basis for a variety of applications such as image segmentation and edge detection. The eigenvalue problem is shown to be related to the algebraic eigenvalue problems resulting from several commonly used discrete spectral clustering models. The relation provides a better understanding and helps developing more efficient numerical implementation and rigorous numerical analysis for discrete spectral segmentation methods. The new continuous model is also different from energy-minimization methods such as geodesic active contour in that no initial guess is required for in the current model. The multi-scale feature is a natural consequence of the anisotropic diffusion operator so there is no need to solve the eigenvalue problem at multiple levels. A numerical implementation based on a finite element method with an anisotropic mesh adaptation strategy is presented. It is shown that the numerical scheme gives much more accurate results on eigenfunctions than uniform meshes. Several interesting features of the model are examined in numerical examples and possible applications are discussed

An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.Comment: 34 page

A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods

We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the method (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme

Modeling anisotropic diffusion using a departure from isotropy approach

There are a large number of finite volume solvers available for solution of isotropic diffusion equation. This article presents an approach of adapting these solvers to solve anisotropic diffusion equations. The formulation works by decomposing the diffusive flux into a component associated with isotropic diffusion and another component associated with departure from isotropic diffusion. This results in an isotropic diffusion equation with additional terms to account for the anisotropic effect. These additional terms are treated using a deferred correction approach and coupled via an iterative procedure. The presented approach is validated against various diffusion problems in anisotropic media with known analytical or numerical solutions. Although demonstrated for two-dimensional problems, extension of the present approach to three-dimensional problems is straight forward. Other than the finite volume method, this approach can be applied to any discretization method