4 research outputs found
Compressive Inverse Scattering II. SISO Measurements with Born scatterers
Inverse scattering methods capable of compressive imaging are proposed and
analyzed. The methods employ randomly and repeatedly (multiple-shot) the
single-input-single-output (SISO) measurements in which the probe frequencies,
the incident and the sampling directions are related in a precise way and are
capable of recovering exactly scatterers of sufficiently low sparsity.
For point targets, various sampling techniques are proposed to transform the
scattering matrix into the random Fourier matrix. The results for point targets
are then extended to the case of localized extended targets by interpolating
from grid points. In particular, an explicit error bound is derived for the
piece-wise constant interpolation which is shown to be a practical way of
discretizing localized extended targets and enabling the compressed sensing
techniques.
For distributed extended targets, the Littlewood-Paley basis is used in
analysis. A specially designed sampling scheme then transforms the scattering
matrix into a block-diagonal matrix with each block being the random Fourier
matrix corresponding to one of the multiple dyadic scales of the extended
target. In other words by the Littlewood-Paley basis and the proposed sampling
scheme the different dyadic scales of the target are decoupled and therefore
can be reconstructed scale-by-scale by the proposed method. Moreover, with
probes of any single frequency \om the coefficients in the Littlewood-Paley
expansion for scales up to \om/(2\pi) can be exactly recovered.Comment: Add a new section (Section 3) on localized extended target
Compressive Inverse Scattering I. High Frequency SIMO Measurements
Inverse scattering from discrete targets with the
single-input-multiple-output (SIMO), multiple-input-single-output (MISO) or
multiple-input-multiple-output (MIMO) measurements is analyzed by compressed
sensing theory with and without the Born approximation. High frequency analysis
of (probabilistic) recoverability by the -based
minimization/regularization principles is presented. In the absence of noise,
it is shown that the -based solution can recover exactly the target of
sparsity up to the dimension of the data either with the MIMO measurement for
the Born scattering or with the SIMO/MISO measurement for the exact scattering.
The stability with respect to noisy data is proved for weak or widely separated
scatterers. Reciprocity between the SIMO and MISO measurements is analyzed.
Finally a coherence bound (and the resulting recoverability) is proved for
diffraction tomography with high-frequency, few-view and limited-angle
SIMO/MISO measurements.Comment: A new section on diffraction tomography added; typos fixed; new
figures adde