Publisher:
Society for Industrial and Applied Mathematics
Issued date:
2014-06-12
Citation:
SIAM Journal on Imaging Sciences, v. 7, Issue 2, pp. 1358-1387
ISSN:
1936-4954
DOI:
10.1137/130943200
Sponsor:
M. Moscoso was supported by AFOSR grant FA9550-11-1-0266 and by AFOSR NSSEFF fellowship. G. Papanicolaou was supported by AFOSR grant FA9550-11-1-0266
Rights:
Atribución-NoComercial-SinDerivadas 3.0 España
Abstract:
We study active array imaging of small but strong scatterers in homogeneous media when multiple
scattering between them is important. We use the Foldy-Lax equations to model wave propagation
with multiple scattering when the scatterers are small relative to We study active array imaging of small but strong scatterers in homogeneous media when multiple
scattering between them is important. We use the Foldy-Lax equations to model wave propagation
with multiple scattering when the scatterers are small relative to the wavelength. In active array
imaging we seek to locate the positions and reflectivities of the scatterers, that is, to determine
the support of the reflectivity vector and the values of its nonzero elements from echoes recorded
on the array. This is a nonlinear inverse problem because of the multiple scattering. We show
in this paper how to avoid the nonlinearity and form images non-iteratively through a two-step
process which involves 1 norm minimization. However, under certain illuminations imaging may be
affected by screening, where some scatterers are obscured by multiple scattering. This problem can
be mitigated by using multiple and diverse illuminations. In this case, we determine solution vectors
that have a common support. The uniqueness and stability of the support of the reflectivity vector
obtained with single or multiple illuminations are analyzed, showing that the errors are proportional
to the amount of noise in the data with a proportionality factor dependent on the sparsity of the
solution and the mutual coherence of the sensing matrix, which is determined by the geometry of the
imaging array. Finally, to filter out noise and improve the resolution of the images, we propose an
approach that combines optimal illuminations using the singular value decomposition of the response
matrix together with sparsity promoting optimization jointly for all illuminations. This work is an
extension of our previous paper [5] on imaging using optimization techniques where we now account
for multiple scattering effects.[+][-]