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

Abstract

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.

Mimetic finite difference methods in image processing

“…This lends itself for discretizing the PDEs to obtain numerical schemes that can be solved on a computer. Because of their favorable stability and efficiency properties, semi-implicit schemes have been the methods of choice for the scale discretization [3,4,15,16,18,19,20,24,27,31,32,37,39,41]. As for the space discretization, the most popular choices are finite difference [15,39,41] and finite element [3,4,16,24,37,39,41] methods (in that order of preference).…”

High performance cluster computing with 3-D nonlinear diffusion filters

“…This has triggered a number of researchers to look for alternative algorithmic realisations of nonlinear diffusion filtering and related variational approaches. These alternatives include three-level methods [8], semi-implicit approaches [4,11] and their multiplicative [36] or additive operator splitting variants [37], multigrid methods [1], finite element techniques [2,16,23], finite and complementary volume methods [11], numerical schemes with wavelets as trial functions [7,8], pseudospectral methods [8], lattice Boltzmann techniques [15], and stochastic simulations [24]. Approximations in graphics hardware have been considered in [26], and realisations on analog hardware are discussed in [9,21].…”

Stability and consistency of the semi-implicit co-volume scheme for regularized mean curvature flow equation in level set formulation

“…∇u |∇u| = 0, (1.1) as well as its nontrivial generalizations, is used in the applications as the motion of interfaces (free boundaries) in thermomechanics (solidification, crystal growth) and computational fluid dynamics (free surface flows, multi-phase flows of immiscible fluids, thin films), the smoothing and segmentation of images and the surface reconstructions in the image processing, computer vision and computer graphics (see e.g. [32,29,2,1,6,19,31,15,17]), and in many further situations related to the motion of implicit curves or surfaces. On the other hand, the convergence of numerical schemes to unique viscosity solution [9,14,7] of equation (1.1) is often an open problem, it is an exception to find an analysis of convergence of the methods used for solving the curvature driven flows in the level set formulation.…”

“…Our semi-implicit scheme leads to solution of linear systems in every discrete time step (for other semi-implicit approaches to solving nonlinear diffusion see e.g. [18,22,16,17]), so it is much more efficient than a fully implicit nonlinear scheme [33], and it is unconditionally stable without any restriction to time step in spite of many other explicit schemes [30,32,29,31]. Consistency and stability are two properties, in the theory of Barles and Souganidis [3], which are used to show convergence of a numerical scheme to solution of fully nonlinear second order partial differential equations and we discuss them in this paper.…”

“…Integrating equation (1.1) in the co-volume gives the weak (integral) formulation of the problem from which the computational scheme naturally follows. One of our main motivations to solve the curvature driven level set equation and its generalizations comes from image processing applications [15,16,17,23,24,8]. The co-volume scheme has been applied to smoothing and segmentation of 2D and 3D medical images in [24,8] and is based on the original semi-implicit method studied in [16].…”