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A finite difference scheme is presented for a density-dependent diffusion equation that arises in the mathematical modelling of bacterial biofilms. The peculiarity of the underlying model is that it shows degeneracy as the dependent variable vanishes, as well as a singularity as the dependent variable approaches its a priori known upper bound. The first… (More)

- Martin Storath, Andreas Weinmann, Laurent Demaret
- IEEE Transactions on Signal Processing
- 2014

We recover jump-sparse and sparse signals from blurred incomplete data corrupted by (possibly non-Gaussian) noise using inverse Potts energy functionals. We obtain analytical results (existence of minimizers, complexity) on inverse Potts functionals and provide relations to sparsity problems. We then propose a new optimization method for these functionals… (More)

- Laurent Demaret, Nira Dyn, Armin Iske
- Signal Processing
- 2006

This paper proposes a new method for image compression. The method is based on the approximation of an image, regarded as a function , by a linear spline over an adapted triangulation, D(Y), which is the Delaunay triangulation of a small set Y of significant pixels. The linear spline minimizes the distance to the image, measured by the mean square error,… (More)

- Felix Friedrich, Laurent Demaret, Hartmut Führ, Katrin Wicker
- SIAM J. Scientific Computing
- 2007

Many algorithms in image processing rely on the computation of sums of pixel values over a large variety of subsets of the image domain. This includes the computation of image moments for pattern recognition purposes, or adaptive smoothing and regression methods, such as wedgelets. In the first part of the paper, we present a general method which allows the… (More)

- L. Demaret
- 2005

We present a heuristic algorithm for the choice of the wedgelet regularization parameter for the purpose of denois-ing in the case where the noise variance σ 2 is not known. Numerical experiments comparing wavelet thresholding with wedgelet denoising, and with the related schemes quadtree approximation and platelet approximation, allow to assess the… (More)

It is by now a well-established fact that the usual two-dimensional tensor product wavelet bases are not optimal for representing images consisting of different regions of smoothly varying greyvalues, separated by smooth boundaries. The chapter starts with a discussion of this phenomenon from a nonlinear approximation point of view, and then proceeds to… (More)

- Laurent Demaret, Nira Dyn, Michael S. Floater, Armin Iske
- Advances in Multiresolution for Geometric…
- 2005

Adaptive thinning algorithms are greedy point removal schemes for bivariate scattered data sets with corresponding function values, where the points are recursively removed according to some data-dependent criterion. Each subset of points, together with its function values, defines a linear spline over its Delaunay triangulation. The basic criterion for the… (More)

- Laurent Demaret
- 2003

This paper proposes a novel concept for digital image compression. The resulting compression scheme relies on adap-tive thinning algorithms, which are recent multiresolution methods from scattered data approximation. Adaptive thinning algorithms are recursive point removal schemes, which are combined with piecewise linear interpolation over decre-mental… (More)

- Laurent Demaret, Armin Iske
- IEEE Signal Processing Letters
- 2006

Locally optimal Delaunay triangulations are constructed to improve previous image approximation schemes. Our construction relies on a local optimization procedure, termed exchange. The efficient implementation of the exchange algorithm is addressed, and its complexity is discussed. The good performance of our improved image approximation is illustrated by… (More)

This paper concerns digital image compression using adaptive thinning algorithms. Adaptive thinning is a recursive point removal scheme which works with decremental Delaunay triangula-tions. When applied to digital images, adaptive thinning returns a scattered set of most significant pixels. This requires efficient and customized methods for coding these… (More)