George Papanicolaou

Miguel Moscoso, Alexei Novikov, George Papanicolaou et al.

We present an ordinary differential equations approach to the analysis of algorithms for constructing $l_1$ minimizing solutions to underdetermined linear systems of full rank. It involves a relaxed minimization problem whose minimum is independent of the relaxation parameter. An advantage of using the ordinary differential equations is that energy methods can be used to prove convergence. The connection to the discrete algorithms is provided by the Crandall-Liggett theory of monotone nonlinear semigroups. We illustrate the effectiveness of the discrete optimization algorithm in some sparse array imaging problems.

Miguel Moscoso, Alexei Novikov, George Papanicolaou et al.

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.

Liliana Borcea, Thomas Callaghan, George Papanicolaou

We study synthetic aperture radar (SAR) imaging and motion estimation of complex scenes consisting of stationary and moving targets. We use the classic SAR setup with a single antenna emitting signals and receiving the echoes from the scene. The known motion estimation methods for such setups work only in simple cases, with one or a few targets in the same motion. We propose to extend the applicability of these methods to complex scenes, by complementing them with a data pre-processing step intended to separate the echoes from the stationary targets and the moving ones. We present two approaches. The first is an iteration designed to subtract the echoes from the stationary targets one by one. It estimates the location of each stationary target from a preliminary image, and then uses it to define a filter that removes its echo from the data. The second approach is based on the robust principle component analysis (PCA) method. The key observation is that with appropriate pre-processing and windowing, the discrete samples of the stationary target echoes form a low rank matrix, whereas the samples of a few moving target echoes form a high rank sparse matrix. The robust PCA method is designed to separate the low rank from the sparse part, and thus can be used for the SAR data separation. We present a brief analysis of the two methods and explain how they can be combined to improve the data separation for extended and complex imaging scenes. We also assess the performance of the methods with extensive numerical simulations.

Miguel Moscoso, Alexei Novikov, George Papanicolaou et al.

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.

Liliana Borcea, Thomas Callaghan, George Papanicolaou

We consider the problem of synthetic aperture radar (SAR) imaging and motion estimation of complex scenes. By complex we mean scenes with multiple targets, stationary and in motion. We use the usual setup with one moving antenna emitting and receiving signals. We address two challenges: (1) the detection of moving targets in the complex scene and (2) the separation of the echoes from the stationary targets and those from the moving targets. Such separation allows high resolution imaging of the stationary scene and motion estimation with the echoes from the moving targets alone. We show that the robust principal component analysis (PCA) method which decomposes a matrix in two parts, one low rank and one sparse, can be used for motion detection and data separation. The matrix that is decomposed is the pulse and range compressed SAR data indexed by two discrete time variables: the slow time, which parametrizes the location of the antenna, and the fast time, which parametrizes the echoes received between successive emissions from the antenna. We present an analysis of the rank of the data matrix to motivate the use of the robust PCA method. We also show with numerical simulations that successful data separation with robust PCA requires proper data windowing. Results of motion estimation and imaging with the separated data are presented, as well.

Anwei Chai, Miguel Moscoso, George Papanicolaou

We present a comprehensive study of the resolution and stability properties of sparse promoting optimization theories applied to narrow band array imaging of localized scatterers. We consider homogeneous and heterogeneous media, and multiple and single scattering situations. When the media is homogeneous with strong multiple scattering between scatterers, we give a non-iterative formulation to find the locations and reflectivities of the scatterers from a nonlinear inverse problem in two steps, using either single or multiple illuminations. We further introduce an approach that uses the top singular vectors of the response matrix as optimal illuminations, which improves the robustness of sparse promoting optimization with respect to additive noise. When multiple scattering is negligible, the optimization problem becomes linear and can be reduced to a hybrid-$\ell_1$ method when optimal illuminations are used. When the media is random, and the interaction with the unknown inhomogeneities can be primarily modeled by wavefront distortions, we address the statistical stability of these methods. We analyze the fluctuations of the images obtained with the hybrid-$\ell_1$ method, and we show that it is stable with respect to different realizations of the random medium provided the imaging array is large enough. We compare the performance of the hybrid-$\ell_1$ method in random media to the widely used Kirchhoff migration and the multiple signal classification methods.

Josselin Garnier, George Papanicolaou, Tzu-Wei Yang

We consider an one-dimensional conservation law with random space-time forcing and calculate using large deviations the exponentially small probabilities of anomalous shock profile displacements. Under suitable hypotheses on the spatial support and structure of random forces, we analyze the scaling behavior of the rate function, which is the exponential decay rate of the displacement probabilities. For small displacements we show that the rate function is bounded above and below by the square of the displacement divided by time. For large displacements the corresponding bounds for the rate function are proportional to the displacement. We calculate numerically the rate function under different conditions and show that the theoretical analysis of scaling behavior is confirmed. We also apply a large-deviation-based importance sampling Monte Carlo strategy to estimate the displacement probabilities. We use a biased distribution centered on the forcing that gives the most probable transition path for the anomalous shock profile, which is the minimizer of the rate function. The numerical simulations indicate that this strategy is much more effective and robust than basic Monte Carlo.

George Papanicolaou, Nicholas West, Tzu-Wei Yang

We consider a reduced order model of an air-breathing hypersonic engine with a time-dependent stochastic inflow that may cause the failure of the engine. The probability of failure is analyzed by the Freidlin-Wentzell theory, the large deviation principle for finite dimensional stochastic differential equations. We compute the asymptotic failure probability by numerically solving the constrained optimization related to the large deviation problem. A large-deviation-based importance sampling suggested by the most probable inflow perturbation is also implemented to compute the probability of failure of the engine. The numerical simulations show that the importance sampling method is much more efficient than the basic Monte Carlo method.

Liliana Borcea, George Papanicolaou, Chrysoula Tsogka

We study detection and imaging of small reflectors in heavy clutter, using an array of transducers that emits and receives sound waves. Heavy clutter means that multiple scattering of the waves in the heterogeneous host medium is strong and overwhelms the arrivals from the small reflectors. Building on the adaptive time-frequency filter of [1], we propose a robust method for detecting the direction of arrival of the direct echoes from the small reflectors, and suppressing the unwanted clutter backscatter. This improves the resolution of imaging. We illustrate the performance of the method with realistic numerical simulations in a non-destructive testing setup.

Miguel Moscoso, Alexei Novikov, George Papanicolaou et al.

We propose an approach for imaging in scattering media when large and diverse data sets are available. It has two steps. Using a dictionary learning algorithm the first step estimates the true Green's function vectors as columns in an unordered sensing matrix. The array data comes from many sparse sets of sources whose location and strength are not known to us. In the second step, the columns of the estimated sensing matrix are ordered for imaging using Multi-Dimensional Scaling with connectivity information derived from cross-correlations of its columns, as in time reversal. For these two steps to work together we need data from large arrays of receivers so the columns of the sensing matrix are incoherent for the first step, as well as from sub-arrays so that they are coherent enough to obtain the connectivity needed in the second step. Through simulation experiments, we show that the proposed approach is able to provide images in complex media whose resolution is that of a homogeneous medium.

Alexander Christie, Matan Leibovich, Miguel Moscoso et al.

We develop an imaging algorithm that exploits strong scattering to achieve super-resolution in changing random media. The method processes large and diverse array datasets using sparse dictionary learning, clustering, and multidimensional scaling. Starting from random initializations, the algorithm reliably extracts the unknown medium properties necessary for accurate imaging using back-propagation, $\ell_2$ or $\ell_1$ methods. Remarkably, scattering enhances resolution beyond homogeneous medium limits. When abundant data are available, the algorithm allows the realization of super-resolution in imaging.

Alexander Christie, Matan Leibovich, Miguel Moscoso et al.

We propose a methodology that exploits large and diverse data sets to accurately estimate the ambient medium's Green's functions in strongly scattering media. Given these estimates, obtained with and without the use of neural networks, excellent imaging results are achieved, with a resolution that is better than that of a homogeneous medium. This phenomenon, also known as super-resolution, occurs because the ambient scattering medium effectively enhances the physical imaging aperture. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.

Miguel Moscoso, Alexei Novikov, George Papanicolaou et al.

We present a novel approach for recovering a sparse signal from cross-correlated data. Cross-correlations naturally arise in many fields of imaging, such as optics, holography and seismic interferometry. Compared to the sparse signal recovery problem that uses linear measurements, the unknown is now a matrix formed by the cross correlation of the unknown signal. Hence, the bottleneck for inversion is the number of unknowns that grows quadratically. The main idea of our proposed approach is to reduce the dimensionality of the problem by recovering only the diagonal of the unknown matrix, whose dimension grows linearly with the size of the problem. The keystone of the methodology is the use of an efficient {\em Noise Collector} that absorbs the data that come from the off-diagonal elements of the unknown matrix and that do not carry extra information about the support of the signal. This results in a linear problem whose cost is similar to the one that uses linear measurements. Our theory shows that the proposed approach provides exact support recovery when the data is not too noisy, and that there are no false positives for any level of noise. Moreover, our theory also demonstrates that when using cross-correlated data, the level of sparsity that can be recovered increases, scaling almost linearly with the number of data. The numerical experiments presented in the paper corroborate these findings.

Miguel Moscoso, Alexei Novikov, George Papanicolaou et al.

The ability to detect sparse signals from noisy high-dimensional data is a top priority in modern science and engineering. 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 is noiseless. Detection of the signal's support from data corrupted by noise is still a challenging problem, especially if the level of noise must be estimated. We propose a new efficient approach that does not require any parameter estimation. We introduce the Noise Collector (NC) 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 the augmented system has zero false discovery rate for any level of noise and with probability that tends to one as the dimension of $ b_0$ increases to infinity. We also 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 to that of the original one. Finally, we demonstrate the effectiveness of the method in applications to passive array imaging.

Miguel Moscoso, Alexei Novikov, George Papanicolaou et al.

We consider the problem of imaging sparse scenes from a few noisy data using an $l_1$-minimization approach. This problem can be cast as a linear system of the form $A \, ρ=b$, where $A$ is an $N\times K$ measurement matrix. We assume that the dimension of the unknown sparse vector $ρ\in {\mathbb{C}}^K$ is much larger than the dimension of the data vector $b \in {\mathbb{C}}^N$, i.e, $K \gg N$. We provide a theoretical framework that allows us to examine under what conditions the $\ell_1$-minimization problem admits a solution that is close to the exact one in the presence of noise. Our analysis shows that $l_1$-minimization is not robust for imaging with noisy data when high resolution is required. To improve the performance of $l_1$-minimization we propose to solve instead the augmented linear system $ [A \, | \, C] ρ=b$, where the $N \times Σ$ 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 $e^N$ 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 $Σ\approx 10 K$.

Miguel Moscoso, Alexei Novikov, George Papanicolaou et al.

We consider imaging the reflectivity of scatterers from intensity-only data recorded by a single moving transducer that both emits and receives signals, forming a synthetic aperture. By exploiting frequency illumination diversity, we obtain multiple intensity measurements at each location, from which we determine field cross-correlations using an appropriate phase controlled illumination strategy and the inner product polarization identity. The field cross-correlations obtained this way do not, however, provide all the missing phase information because they are determined up to a phase that depends on the receiver's location. The main result of this paper is an algorithm with which we recover the field cross-correlations up to a single phase that is common to all the data measured over the synthetic aperture, so all the data are synchronized. Thus, we can image coherently with data over all frequencies and measurement locations as if full phase information was recorded.