Preface Through many centuries physics has been one of the most fruitful sources of inspiration for mathematics. As a consequence, mathematics has become an economic language providing a few basic principles which allow to explain a large variety of physical phenomena. Many of them are described in terms of partial diierential equations (PDEs). In recent years, however, mathematics also has been stimulated by other novel elds such as image processing. Goals like image segmentation, multiscale image representation, or image restoration cause a lot of challenging mathematical questions. Nevertheless, these problems frequently have been tackled with a pool of heuristical recipes. Since the treatment of digital images requires very much computing power, these methods had to be fairly simple. With the tremendous advances in computer technology in the last decade, it has become possible to apply more sophisticated techniques such as PDE-based methods which have been inspired by physical processes. Among these techniques, parabolic PDEs have found a lot of attention for smoothing and restoration purposes, see e.g. 113]. To restore images these equations frequently arise from gradient descent methods applied to variational problems. Image smoothing by parabolic PDEs is closely related to the scale-space concept where one embeds the original image into a family of subsequently simpler , more global representations of it. This idea plays a fundamental role for extracting semantically important information. The pioneering work of Alvarez, Guichard, Lions and Morel 11] has demonstrated that all scale-spaces fulllling a few fairly natural axioms are governed by parabolic PDEs with the original image as initial condition. Within this framework, two classes can be justiied in a rigorous way as scale-spaces: the linear diiusion equation with constant dif-fusivity and nonlinear so-called morphological PDEs. All these methods satisfy a monotony axiom as smoothing requirement which states that, if one image is brighter than another, then this order is preserved during the entire scale-space evolution. An interesting class of parabolic equations which pursue both scale-space and restoration intentions is given by nonlinear diiusion lters. Methods of this type have been proposed for the rst time by Perona and Malik in 1987 190]. In v vi PREFACE order to smooth the image and to simultaneously enhance semantically important features such as edges, they apply a diiusion process whose diiusivity is steered by local image properties. These lters are diicult to analyse mathematically , as they may act locally like a backward diiusion process. …

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Anisotropic diffusion in image processing

Basic error estimates for elliptic problems

The description of membrane lipid conformation, order and dynamics by 2H-NMR

Maximum principle and uniform convergence for the finite element method

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation

On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation

On a Finite Difference Analogue of an Elliptic Boundary Problem which is Neither Diagonally Dominant Nor of Non‐negative Type

New monotone type approximations for elliptic problems

In this paper we consider a class of difference approximations to the Dirichlet problem for second-order elliptic operators with smooth coefficients. The main result is that if the order of accuracy of the approximate problem is v, and F (the right-hand side) andf (the boundary values) both belong to O' for X < v, then the rate of convergence is O(hx).

/pdf/convergence-estimates-for-essentially-positive-type-discrete-cmigo5kiwr.pdf

Convergence estimates for essentially positive type discrete Dirichlet problems

Tile region R :is a bounded connected open set in the (x, y) plane whose boundary C consists of the two parts C,, and C2. The symbol k is the Laplace operator _~ (O~/Ox 2) + (02/0y2), and 0/0n denotes differentiation with respect to the outward-directed normal oa C~. The functions f, g and H are defined to be sufficierttly smooth functions on R, C, and C2 respectively. The function a is required to satisfy the following conditions on C~ : (a) piecewise continuity with a finite nmnber of discontinuities, (b) piecewise differentiability, (c) at all points of continuity, either a = 0 (the set C~ ')) or a => am > 0, where a,~ is a constaIlt (the set C~2)). We restrict our considerations to the cases in which either the set C2 or C[ ~) conl rains a nonempty open subset of C. In these eases (1.1) has a unique solution i provided the data and boundary are sufficiently smooth. The case in which C2 is i all of C is just the Diriehlet problem. Results for this special case are contained in [8]. We are interested in fornmlating a finite-difference analogue of (1.1) which has the following properties: (a) The boundary approximations involve at most three interior points (and one boundary point) (b) The matrix of the system is

B. E. Hubbard

Papers

On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation

On a Finite Difference Analogue of an Elliptic Boundary Problem which is Neither Diagonally Dominant Nor of Non‐negative Type

New monotone type approximations for elliptic problems

Convergence estimates for essentially positive type discrete Dirichlet problems

Approximation of Solutions of Mixed Boundary Value Problems for Poisson's Equation by Finite Differences