Mathematics Colloquia and Seminars

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In the process of discretizing continuum imaging problems,a gridding error,roughly proportional to the grid spacing, arises. When the grid spacing is above the Rayleigh length, the gridding error can be as large as the data themselves, creating an unfavorable signal to noise ratio. To reduce the gridding error, it’s natural to refine the grid. However, the sensing matrices become underdetermined and highly coherent when the grid spacing is reduced below the Rayleigh length. In this case, existing compressive sensing(CS) algorithms fail due to the absence of incoherence. In order to fill the gap, we propose the techniques of band exclusion(BE) and local optimization(LO) to deal with coherent sensing matrices on fine grid. These techniques are embedded in the existing CS algorithms, such as Orthogonal Matching Pursuit(OMP) and Basis Pursuit(BP), and result in the modified algorithms, such as BLO-based OMP and BLO-based BP respectively. We have proved that under certain conditions, BLO-based OMP is capable of reconstructing sparse, widely separated objects within one Rayleigh length in bottleneck distance independent of the grid spacing.Detailed numerical comparisons with other algorithms designed for the same purpose, such as Spectral Iterative Hard Thresholding(SIHT) and the analysis-based BP, demonstrate the superiority of BLO-based OMP and BLO-based BP.

Pizzas and sodas will be provided.