Laurent Demaret

Image compression by linear splines over adaptive triangulations

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, among all linear splines over D(Y). The significant pixels in Y are selected by an adaptive thinning algorithm, which recursively removes less significant pixels in a greedy way, using a sophisticated criterion for measuring the significance of a pixel. The proposed compression method combines the approximation scheme with a customized scattered data coding scheme. We compare our compression method with JPEG2000 on two geometric images and on three popular test cases of real images.

Jump-Sparse and Sparse Recovery Using Potts Functionals

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 which is based on dynamic programming and the alternating direction method of multipliers (ADMM). A series of experiments shows that the proposed method yields very satisfactory jump-sparse and sparse reconstructions, respectively. We highlight the capability of the method by comparing it with classical and recent approaches such as TV minimization (jump-sparse signals), orthogonal matching pursuit, iterative hard thresholding, and iteratively reweighted ℓ1 minimization (sparse signals).

Total Variation Regularization for Manifold-Valued Data

We consider total variation (TV) minimization for manifold-valued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with

Adaptive Thinning for Terrain Modelling and Image Compression

\ell^p

Adaptive image approximation by linear splines over locally optimal delaunay triangulations

-type data terms in the manifold case. These algorithms are based on iterative geodesic averaging which makes them easily applicable to a large class of data manifolds. As an application, we consider denoising images which take their values in a manifold. We apply our algorithms to diffusion tensor images and interferometric SAR images as well as sphere- and cylinder-valued images. For the class of Cartan--Hadamard manifolds (which includes the data space in diffusion tensor imaging) we show the convergence of the proposed TV minimizing algorithms to a global minimizer.

The

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 removal of the next point is to minimise the error between the resulting linear spline at the bivariate data points and the original function values. This leads to a hierarchy of linear splines of coarser and coarser resolutions.

L ^{ 1 }

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 numerical comparisons.

-Potts Functional for Robust Jump-Sparse Reconstruction

We investigate the nonsmooth and nonconvex

Total variation regularization of shape signals

L^1

Mumford---Shah and Potts Regularization for Manifold-Valued Data

-Potts functional in discrete and continuous time. We show