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We present a holographic imaging approach for the case in which a single source-detector pair is used to scan a sample. The source-detector pair collects intensity-only data at different frequencies and positions. By using an appropriate illumination strategy, we recover field cross correlations over different frequencies for each scan location. The problem is that these field cross correlations are asynchronized, so they have to be aligned first in order to image coherently. This is the main result of the paper: a simple algorithm to synchronize field cross correlations at different locations. Thus, one can recover full field data up to a global phase that is common to all scan locations. The recovered data are, then, coherent over space and frequency so they can be used to form high-resolution three-dimensional images. Imaging with intensity-only data is therefore as good as coherent imaging with full data. In addition, we use an ${\mathrm{\ell <\#comment/>}}_1$-norm minimization algorithm that promotes the low dimensional structure of the images, allowing for deep high-resolution imaging. 

The ability to detect sparse signals from noisy, high-dimensional data is a top priority in modern science and engineering. It is well known that a sparse solution of the linear system A ρ = b 0 can be found efficiently with an ℓ 1 -norm minimization approach if the data are noiseless. However, detection of the signal from data corrupted by noise is still a challenging problem as the solution depends, in general, on a regularization parameter with optimal value that is not easy to choose. We propose an efficient approach that does not require any parameter estimation. We introduce a no-phantom weight τ and the Noise Collector matrix C and solve an augmented system A ρ + C η = b 0 + e , where e is the noise. We show that the ℓ 1 -norm minimal solution of this system has zero false discovery rate for any level of noise, with probability that tends to one as the dimension of b 0 increases to infinity. We obtain exact support recovery if the noise is not too large and develop a fast Noise Collector algorithm, which makes the computational cost of solving the augmented system comparable with that of the original one. We demonstrate the effectiveness of the method in applications to passive array imaging. 

In this paper, we consider imaging problems that can be cast in the form of an underdetermined linear system of equations. When a single measurement vector is available, a sparsity promoting ℓ1-minimization-based algorithm may be used to solve the imaging problem efficiently. A suitable algorithm in the case of multiple measurement vectors would be the MUltiple SIgnal Classification (MUSIC) which is a subspace projection method. We provide in this work a theoretical framework in an abstract linear algebra setting that allows us to examine under what conditions the ℓ1-minimization problem and the MUSIC method admit an exact solution. We also examine the performance of these two approaches when the data are noisy. Several imaging configurations that fall under the assumptions of the theory are discussed such as active imaging with single- or multiple-frequency data. We also show that the phase-retrieval problem can be re-cast under the same linear system formalism using the polarization identity and relying on diversity of illuminations. The relevance of our theoretical analysis in imaging is illustrated with numerical simulations and robustness to noise is examined by allowing the background medium to be weakly inhomogeneous.