  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 $\ell_1$ 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.
