A compressed sensing scheme for near-field imaging of corrugations of relative sparse Fourier components is proposed. The scheme employs random sparse measurement of near field to recover the angular spectrum of the scattered field. Surprisingly, it can be shown heuristically and numerically that under the Rayleigh hypothesis the angular spectrum is compressible and amenable to compressed sensing techniques. Iteration schemes are developed for recovering the surface profile from the angular spectrum. The proposed nonlinear least squares in the Fourier basis produces accurate reconstructions even when the Rayleigh hypothesis is known to be false.
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