Compressive imaging of subwavelength structures: periodic rough surfaces

Abstract

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.

ℓ 1 − αℓ 2 minimization methods for signal and image reconstruction with impulsive noise removal

“…For example, for A ∈ R 32×640 , μ(A) ≈ 0.97 when F = 5, and μ(A) easily exceeds 0.99 when F = 10. The over-sampled DCT matrices are derived from the problem of spectral estimation [49][50][51] in signal processing, and radar imagining [49,52] and surface scattering [53] in image processing.…”

Sparse representation of astronomical images

“…The B-spline basis for the cardinal spline space corresponding to m = 2 is constructed by considering l = 1 in (10) and translating the prototype every knot. Dictionaries for the identical space of functions of broader support arise by setting l ∈ N in order to fix the desired support.…”

“…Obviously for a finite dimension Euclidean space one can construct arbitrary dictionaries. In particular, redundant B-spline based dictionaries with prototypes of different support and shapes arising from the functions (10) and (11) and their corresponding derivatives.…”

“…For a significant reduction in the data dimensionality, from say N to K < N points, the image is said to be K-sparse in the transformed domain. In addition to the many applications that benefit from sparse representation of information [1][2][3][4][5], the emerging theory of sampling, called compressive sensing/sampling, asserts that sparsity of a representation may also lead to more economical data collection [6][7][8][9][10]. The relevance of compressive sensing within the context of astronomical data is discussed in [11,12], where algorithms for signal recovery are advanced and illustrated by these types of data.…”

The Dantzig selector: Recovery of Signal via $\ell_1-α\ell_2$ Minimization

“…For example, for A ∈ R 32×640 , µ(A) ≈ 0.97 when F = 5, and µ(A) easily exceeds 0.99 when F = 10. The over-sampled DCT matrices are derived from the problem of spectral estimation [19,29] in signal processing, and radar imagining [19,21] and surface scattering [20] in image processing.…”