Within the area of applied harmonic analysis, various multiscale systems such as wavelets, ridgelets, curvelets, and shearlets have been introduced and successfully applied. The key property of each of those systems are their (optimal) approximation properties in terms of the decay of the L 2 -error of the best N -term approximation for a certain class of functions. In this paper, we introduce the general framework of α-molecules, which encompasses most multiscale systems from applied harmonic analysis, in particular, wavelets, ridgelets, curvelets, and shearlets as well as extensions of such with α being a parameter measuring the degree of anisotropy, as a means to allow a unified treatment of approximation results within this area. Based on an α-scaled index distance, we first prove that two systems of α-molecules are almost orthogonal. This leads to a general methodology to transfer approximation results within this framework, provided that certain consistency and time-frequency localization conditions of the involved systems of α-molecules are satisfied. We finally utilize these results to enable the derivation of optimal sparse approximation results for a specific class of cartoonlike functions by sufficient conditions on the 'control' parameters of a system of α-molecules.

It is well-known that curvelets provide optimal approximations for so-called cartoon images which are defined as piecewise C 2 -functions, separated by a C 2 singularity curve. In this paper, we consider the more general case of piecewise C β -functions, separated by a C β singularity curve for β ∈ (1, 2]. We first prove a benchmark result for the possibly achievable best N -term approximation rate for this more general signal model. Then we introduce what we call α-curvelets, which are systems that interpolate between wavelet systems on the one hand (α = 1) and curvelet systems on the other hand (α = 1 2 ). Our main result states that those frames achieve this optimal rate for α = 1 β , up to log-factors.

The need of reconstructing discrete-valued sparse signals from few measurements, that is solving an undetermined system of linear equations, appears frequently in science and engineering. Whereas classical compressed sensing algorithms do not incorporate the additional knowledge of the discrete nature of the signal, classical lattice decoding approaches such as the sphere decoder do not utilize sparsity constraints.In this work, we present an approach that incorporates a discrete values prior into basis pursuit. In particular, we address unipolar binary and bipolar ternary sparse signals, i.e., sparse signals with entries in {0, 1}, respectively in {−1, 0, 1}. We will show that phase transition takes place earlier than when using the classical basis pursuit approach and that, independently of the sparsity of the signal, at most N/2, respectively 3N/4, measurements are necessary to recover a unipolar binary, and a bipolar ternary signal uniquely, where N is the dimension of the ambient space. We will further discuss robustness of the algorithm and generalizations to signals with entries in larger alphabets.

No abstract

No abstract