No abstract

A recursive approach for shrinking coefficients of an atomic decomposition is proposed. The corresponding algorithm evolves so as to provide at each iteration a) the orthogonal projection of a signal onto a reduced subspace and b) the index of the coefficient to be disregarded in order to construct a coarser approximation minimizing the norm of the residual error.

A prescription for constructing dictionaries for cardinal spline spaces on a compact interval is provided. It is proved that such spaces can be spanned by dictionaries which are built by translating a prototype B-spline function of fixed support into the knots of the required cardinal spline space. This implies that cardinal spline spaces on a compact interval can be spanned by dictionaries of cardinal B-spline functions of broader support that the corresponding basis function.

Wavelet families arise by scaling and translations of a prototype function, called the mother wavelet. The construction of wavelet bases for cardinal spline spaces is generally carried out within the multi-resolution analysis scheme. Thus, the usual way of increasing the dimension of the multi-resolution subspaces is by augmenting the scaling factor. We show here that, when working on a compact interval, the identical effect can be achieved without changing the wavelet scale but reducing the translation parameter. By such a procedure we generate a redundant frame, called a dictionary, spanning the same spaces as a wavelet basis but with wavelets of broader support. We characterise the correlation of the dictionary elements by measuring their 'coherence' and produce examples illustrating the relevance of highly coherent dictionaries to problems of sparse signal representation.

We present spline wavelets of class C n (R) supported by sequences of aperiodic discretizations of R. The construction is based on multiresolution analysis recently elaborated by G. Bernuau. At a given scale, we consider discretizations that are sets of left-hand ends of tiles in a self-similar tiling of the real line with finite local complexity. Corresponding tilings are determined by two-letter Sturmian substitution sequences. We illustrate the construction with examples having quadratic Pisot-Vijayaraghavan units (like τ = (1 + √ 5)/2 or τ 2 = (3 + √ 5)/2) as scaling factor. In particular, we present a comprehensive analysis of the Fibonacci chain and give the analytic form of related scaling functions and wavelets. We also give some hints for the construction of multidimensional spline wavelets based on stone-inflation tilings in arbitrary dimension.