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

Bootstrap Multigrid for the Laplace-Beltrami Eigenvalue Problem

This paper introduces bootstrap two-grid and multigrid finite element approximations to the Laplace-Beltrami (surface Laplacian) eigen-problem on a closed surface. The proposed multigrid method is suitable for recovering eigenvalues having large multiplicity, computing interior eigenvalues, and approximating the shifted indefinite eigen-problem. Convergence analysis is carried out for a simplified two-grid algorithm and numerical experiments are presented to illustrate the basic components and ideas behind the overall bootstrap multigrid approach

The enriched Crouzeix-Raviart elements are equivalent to the Raviart-Thomas elements

For both the Poisson model problem and the Stokes problem in any dimension, this paper proves that the enriched Crouzeix-Raviart elements are actually identical to the first order Raviart-Thomas elements in the sense that they produce the same discrete stresses. This result improves the previous result in literature which, for two dimensions, states that the piecewise constant projection of the stress by the first order Raviart-Thomas element is equal to that by the Crouzeix-Raviart element. For the eigenvalue problem of Laplace operator, this paper proves that the error of the enriched Crouzeix-Raviart element is equivalent to that of the Raviart-Thomas element up to higher order terms