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Adaptive Superresolution in Deconvolution of Sparse Peaks

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Document pages: 41 pages

Abstract: The aim of this paper is to investigate superresolution in deconvolutiondriven by sparsity priors. The observed signal is a convolution of an originalsignal with a continuous kernel.With the prior knowledge that the originalsignal can be considered as a sparse combination of Dirac delta peaks, we seekto estimate the positions and amplitudes of these peaks by solving a finitedimensional convex problem on a computational grid. Because, the support of theoriginal signal may or may not be on this grid, by studying the discretedeconvolution of sparse peaks using L1-norm sparsity prior, we confirm recentobservations that canonically the discrete reconstructions will result inmultiple peaks at grid points adjacent to the location of the true peak. Owningto the complexity of this problem, we analyse carefully the de-convolution ofsingle peaks on a grid and gain a strong insight about the dependence of thereconstructed magnitudes on the exact peak location. This in turn allows us toinfer further information on recovering the location of the exact peaks i.e. toperform super-resolution. We analyze in detail the possible cases that canappear and based on our theoretical findings, we propose an self-drivenadaptive grid approach that allows to perform superresolution inone-dimensional and multi-dimensional spaces. With the view that the currentstudy can provide a further step in the development of more robust algorithmsfor the detection of single molecules in fluorescence microscopy oridentification of characteristic frequencies in spectral analysis, wedemonstrate how the proposed approach can recover sparse signals usingsimulated clusters of point sources (peaks) of low-resolution in one andtwo-dimensional spaces.

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