Angela Handlovičová

Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution

Summary. We introduce linear semi-implicit complementary volume numerical scheme for solving level set like nonlinear degenerate diffusion equations arising in image processing and curve evolution problems. We study discretization of image selective smoothing equation of mean curvature flow type given by Alvarez, Lions and Morel ([3]). Solution of the level set equation of Osher and Sethian ([26], \[30]) is also included in the study. We prove

Variational Numerical Methods for Solving Nonlinear Diffusion Equations Arising in Image Processing

L_\infty

Gradient Schemes for Image Processing

and

Adaptive cell-centered finite volume method for diffusion equations on a consistent quadtree grid

W^{1,1}

Evaluation of Gradient Norms on a Consistent Quadtree Grid in 2D

estimates for the proposed scheme and give existence of its (generalized) solution in every discrete time-scale step. Efficiency of the scheme is given by its linearity and stability. Preconditioned iterative solvers are used for computing arising linear systems. We present computational results related to image processing and plane curve evolution.

Gradient Evaluation on a Quadtree Based Finite Volume Grid

In this paper we give a general, robust, and efficient approach for numerical solutions of partial differential equations (PDEs) arising in image processing and computer vision. The well-established variational computational techniques, namely, finite element, finite volume, and complementary volume methods, are introduced on a common base to solve nonlinear problems in image multiscale analysis. Since they are based on principles like minimization of energy (finite element method) or conservation laws (finite and complemetary volume methods), they have strong physical backgrounds. They allow clear and physically meaningful derivation of difference equations that are local and easy to implement. The variational methods are combined with semi-implicit discretization in scale, which gives favorable stability and efficiency properties of computations. We show here L∞-stability without any restrictions on scale steps. Our approach leads finally to solving linear systems in every discrete scale level, which can be done efficiently by fast preconditioned iterative solvers. We discuss such computational schemes for the regularized (in the sense of F. Catte et al., SIAM J. Numer. Anal.129, 1992, 182?193) Perona?Malik anisotropic diffusion equation (P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intell.12, 1990, 629?639) and for nonlinear degenerate diffusion equation of mean curvature flow type studied by L. Alvarez et al. (SIAM J. Numer. Anal.129, 1992, 845?866).

Non-diffusive numerical scheme for regularized mean curvature flow level set equation in image processing

We present a gradient scheme (which happens to be similar to the MPFA finite volume O-scheme) for the approximation to the solution of the Perona-Malik model regularized by a time delay and to the solution of the nonlinear tensor anisotropic diffusion equation. Numerical examples showing properties of the method and applications in image filtering are discussed.

Numerical solution of parabolic equations related to level set formulation of mean curvature flow

Models applied in image processing are often described by nonlinear PDEs in which a good approximation of gradient plays an important role especially in such cases where irregular finite volume grids are used. In image processing, such a situation can occur during a coarsening based on quadtree grids. We present a construction of a deformed quadtree grid in which the connection of representative points of two adjacent finite volumes is perpendicular to their common boundary enabling us to apply the classical finite volume methods. On the other hand, for such an adjusted grid, the intersection of representative points connection with a finite volume boundary is not a middle point of their common edge and standard methods cannot achieve a good accuracy. In this paper we present a new cell-centered finite volume method to evaluate solution gradients, which results into a solution of a simple linear algebraic system and we prove its unique solvability. Finally we present numerical experiments for the regularized Perona-Malik model in which we applied this new method.

Study of a finite volume scheme for the regularized mean curvature flow level set equation

This paper solves the problem of how to evaluate gradients and their norms on a quadtree grid, which is deformed in such a way that the connections of centers of its adjacent elements are perpendicular to their common boundaries. On the grid, we solve the parabolic PDEs representing the curvature-driven filters based on the mean curvature flow and geodetic mean curvature flow equations in a level set formulation. The numerical solution of these equations is based on the finite volume method, where the finite volumes correspond to elements of the deformed quadtree. The described method utilizes representative points not only for the finite volumes but also for the edges forming the boundaries of grid elements. Using these points we evaluate the gradients locally. Solution values in the edge representative points are updated by balancing the fluxes in such a way that we always need only neighbors of a finite volume sharing a common edge with it, not only a vertex. This fact is important for the efficiency of the algorithm. The edge representative points have been chosen in such a way that they lie on a connection of volume representative points enabling to derive a special formula for the gradient norm. We discuss the ways of approximating the norms of the gradients, and on selected examples we show their properties.

Error estimates of the finite volume scheme for the nonlinear tensor-driven anisotropic diffusion

Many problems described by nonlinear PDEs need good approximations of gradients on finite volumes. Using finite volume methods, this can be difficult task if discretization of a computational domain does not fulfill the classical orthogonality property. Such a situation can occur, e.g., during coarsening in image processing using quadtree grids. We present a construction of an adjusted quadtree grid for which the connection of representative points of two adjacent finite volumes is perpendicular to their common boundary. On the other hand, for such an adjusted grid, the intersection of representative points connection with a finite volume boundary is not a middle point of their common edge. In this paper we present a new method of gradient evaluation for such a situation.