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We consider the problem of imaging sparse scenes from a few noisy data using an L1-minimization approach. This problem can be cast as a linear system of the form Ap = b, where A is an N x K measurement matrix. We assume that the dimension of the unknown sparse vector p E Ck is much larger than the dimension of the data vector b E Cn, i.e. K >>N. We provide a theoretical framework that allows us to examine under what conditions the L1-minimization problem admits a solution that is close to the exact one in the presence of noise. Our analysis shows that L1-minimization is not robust for imaging with noisy data when high resolution is required. To improve the performance of L1-minimization we propose to solve instead the augmented linear system [A|C]p = b, where the N = Σ matrix C is a noise collector. It is constructed so as its column vectors provide a frame on which the noise of the data, a vector of dimension N, can be well approximated. Theoretically, the dimension Σ of the noise collector should be eN which would make its use not practical. However, our numerical results illustrate that robust results in the presence of noise can be obtained with a large enough number of columns Σ~10K. Part of this material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, RI, during the Fall 2017 semester. The work of M Moscoso was partially supported by Spanish MICINN grant FIS2016-77892-R. The work of A Novikov was partially supported by NSF grants DMS-1515187, DMS-1813943. The work of C Tsogka was partially supported by AFOSR FA9550-17-1-0238.

In this paper, we study the MUltiple SIgnal Classification (MUSIC) algorithm often used to image small targets when multiple measurement vectors are available. We show that this algorithm may be used when the imaging problem can be cast as a linear system that admits a special factorization. We discuss several active array imaging configurations where this factorization is exact, as well as other configurations where the factorization only holds approximately and, hence, the results provided by MUSIC deteriorate. We give special attention to the most general setting where an active array with an arbitrary number of transmitters and receivers uses signals of multiple frequencies to image the targets. This setting provides all the possible diversity of information that can be obtained from the illuminations. We give a theorem that shows that MUSIC is robust with respect to additive noise provided that the targets are well separated. The theorem also shows the relevance of using appropriate sets of controlled parameters, such as excitations, to form the images with MUSIC robustly. We present numerical experiments that support our theoretical results. Part of this material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, RI, during the Fall 2017 semester. The work of M Moscoso was partially supported by Spanish grant FIS2016- 77892-R. The work of A Novikov was partially supported by NSF grant DMS-1813943. The work of C Tsogka was partially supported by AFOSR FA9550-17-1-0238.

We propose an illumination strategy for interferometric imaging that allows for robust depth recovery from intensity-only measurements. For an array with colocated sources and receivers, we show that all the possible interferometric data for multiple sources, receivers, and frequencies can be recovered from intensity-only measurements provided that we have sufficient source location and frequency illumination diversity. There is no need for phase reconstruction in this approach. Using interferometric imaging methods we show that in homogeneous media there is no loss of resolution when imaging with intensities only. If in these imaging methods we reduce incoherence by restricting the multifrequency interferometric data to nearby array elements and nearby frequencies we obtain robust images in weakly inhomogeneous background media with a somewhat reduced resolution. The work of the authors was partially supported by AFOSR FA9550-14-1-0275. The work of the first author was supported by the Spanish MICINN grants FIS2013-41802-R and FIS2016-77892-R. The work of the second author was partially supported by NSF DMS-1515187.

In this paper we consider narrow band, active array imaging of weak localized scatterers when only the intensities are recorded at an array with N transducers. We consider that the medium is homogeneous and, hence, wave propagation is fully coherent. This work is an extension of our previous paper [21] where we showed that using linear combinations of intensity-only measurements, obtained from N2 illuminations, imaging of localized scatterers can be carried out e_ciently using imaging methods based on the singular value decomposition of the time-reversal matrix. Here we show the same strategy can be accomplished with only 3N¬¬-2 illuminations, herefore reducing enormously the data acquisition process. Furthermore, we show that in the paraxial regime one can form the images by using six illuminations only. In particular, this paraxial regime includes Fresnel and Fraunhofer di_raction. The key point of this work is that if one controls the illuminations, imaging with intensity-only can be easily reduced to a imaging with phases and, therefore, one can apply standard imaging techniques. Detailed numerical simulations illustrate the performance of the proposed imaging strategy with and without data noise.

We propose a new strategy for narrow band, active array imaging of weak localized scatterers when only the intensities are recorded and measured at the array. We consider a homogeneous medium so that wave propagation is fully coherent. We show that imaging with intensity-only measurements can be carried out using the time reversal operator of the imaging system, which can be obtained from intensity measurements using an appropriate illumination strategy and the polarization identity. Once the time reversal operator has been obtained, we show that the images can be formed using its singular value decomposition (SVD). We use two SVD-based methods to image the scatterers. The proposed approach is simple and efficient. It does not need prior information about the sought image, and it guarantees exact recovery in the noise-free case. Furthermore, it is robust with respect to additive noise. Detailed numerical simulations illustrate the performance of the proposed imaging strategy when only the intensities are captured. The work of A. Novikov was supported by NSF grant DMS-0908011. The work of M. Moscoso was supported by Spanish MICINN Grant FIS2013-41802-R. The work of G. Papanicolaou was supported by AFOSR grant FA9550-11-1-0266

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. 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