Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimal-sparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.

A robust data interpolation method using curvelets frames is presented. The advantage of this method is that curvelets arguably provide an optimal sparse representation for solutions of wave equations with smooth coefficients. As such curvelets frames circumvent – besides the assumption of caustic-free data – the necessity to make parametric assumptions (e.g. through linear/parabolic Radon or demigration) regarding the shape of events in seismic data. A brief sketch of the theory is provided as well as a number of examples on synthetic and real data. Linear data interpolation Data continuation has been an important topic in seismic processing, imaging and inversion. For instance, our ability to accurately and artifact-free image or predict multiples, as part of surface related multiple elimination, depends for a large part on the availability of densely and equidistantly sampled data. Unfortunately, constraints on marine and land acquisitions preclude such a dense sampling and several efforts have been made to solve issues related to aliasing and missing traces. Typically, data continuation approaches involve some sort of a variational problem that aims to jointly minimize the L 2-mismatch, between measured and interpolated data, and an independent penalty functional (regularization term) on the interpolated data, which we will call the model. For quadratic penalty functionals, the solution for the variational problem [see e.g. 3] can be written explicitly as ˆm = arg min