Highly nonlinear approximations for sparse signal representation

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The problem of best approximating a target signal by superposition of elementary signals, called atoms, which are drawn from a large and redundant set called dictionary is, in general, a Nonpolynomial (NP)-hard problem. The mathematical methods for signal representation within this framework are referred to as `highly nonlinear approximations'.Such types of signal representations have been proved to be very powerful even when addressed by heuristic algorithms in the line of pursuit approaches, which do not seek for the optimal solution with regard to sparseness. The basic matching pursuit approach, for instance, is used for images, audio and video coding, as well as for biomedical applications. We have recently proposed a number of greedy strategies which significantly improve upon the original matching pursuit approach. Our strategies, effectively implemented by adaptive biorthogonalisation techniques, evolve by stepwise selection of atoms through the following operations:i)Forward steps to increase the number of atoms in the signal decomposition by selecting them one by one from the dictionary.ii)Backward steps for deleting atoms one by one from the signal decomposition and adapting the remaining representation.iii)Swapping operations to interchange atoms in the decomposition with atoms from the dictionary (For more information see http://www.ncrg.aston.ac.uk/Projects/BiOrthog)The success of the techniques evolving by stepwise selection of single atoms motivates the endeavour of designing well founded advanced greedy methodologies evolving by simultaneous selection of multiple atoms. One can certainly envisage the enormous gain in the sparseness of a representation that such technique should be able to produce. The consequent increment in complexity will be compensated by designing algorithms specially devised for parallel implementation. Furthermore, from the information collected by such new techniques it is expected to gather an insight leading to new heuristic methodologies of low complexity.