George Papanicolaou

Stanford University | SU · Department of Mathematics

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 p...

We propose a method for imaging in scattering media when large and diverse datasets 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 kno...

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...

We consider imaging of an airborne target and study the stability of correlation based imaging methods to random fluctuations in the target's motion. The imaging system consists of a ground based emitter, and several passive receivers. By migrating the cross correlations of the received signals the two-point interference matrix is obtained. An imag...

Phase retrieval in its most general form is the problem of reconstructing a complex valued function from phaseless information of some transform of that function. This problem arises in various fields such as X-ray crystallography, electron microscopy, coherent diffractive imaging, astronomy, speech recognition, and quantum mechanics. The mathemati...

We present an algorithm for coherent diffractive imaging with phaseless measurements. It treats the forward model as a combination of coherent and incoherent waves. The algorithm reconstructs absorption and phase contrast that quantifies the attenuation and the refraction of the waves propagating through an object. It requires coherent or partially...

We consider imaging of fast moving small objects in space, such as low earth orbit satellites, which are also rotating around a fixed axis. The imaging system consists of ground based, asynchronous sources of radiation and several passive receivers above the dense atmosphere. We use the cross-correlation of the received signals to reduce distortion...

We present a novel approach for recovering a sparse signal from quadratic measurements corresponding to a rank-one tensorization of the data vector. Such quadratic measurements, referred to as interferometric or cross-correlated data, naturally arise in many fields such as remote sensing, spectroscopy, holography and seismology. Compared to the spa...

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. Th...

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 u...

Significance The ability to detect sparse signals from noisy, high-dimensional data is a top priority in modern science and engineering. For optimal results, current approaches need to tune parameters that depend on the level of noise, which is often difficult to estimate. We develop a parameter-free, computationally efficient, ℓ 1 -norm minimizati...

We consider imaging of fast moving small objects in space, such as low earth orbit satellites. The imaging system consists of ground based, asynchronous sources of radiation and several passive receivers above the dense atmosphere. We use the cross correlation of the received signals to reduce distortions from ambient medium fluctuations. Imaging w...

Principal component analysis (PCA) is a useful tool when trying to construct factor models from historical asset returns. For the implied volatilities of U.S. equities there is a PCA-based model with a principal eigenportfolio whose return time series lies close to that of an overarching market factor. The authors show that this market factor is th...

We consider a synthetic aperture imaging configuration, such as synthetic aperture radar (SAR), where we want to first separate reflections from moving targets from those coming from a stationary background, and then to image separately the moving and the stationary reflectors. For this purpose, we introduce a representation of the data as a third...

We analyze synthetic aperture radar (SAR) imaging of complex ground scenes that contain both stationary and moving targets. In the usual SAR acquisition scheme, we consider ways to preprocess the data so as to separate the contributions of the moving targets from those due to stationary background reflectors. Both components of the data, that is, r...

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 $A \, \rho =b$, where $A$ is an $N\times K$ measurement matrix. We assume that the dimension of the unknown sparse vector $\rho \in {\mathbb{C}}^K$ is much larger than the dimension of...

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 Aρ = b can be found efficiently with an l1-norm minimization approach if the data is noiseless. Detection of the signal's support from data corrupted by noise is still a challenging proble...

We analyze synthetic aperture radar (SAR) imaging of complex ground scenes that contain both stationary and moving targets. In the usual SAR acquisition scheme, we consider ways to preprocess the data so as to separate the contributions of the moving targets from those due to stationary background reflectors. Both components of the data, that is, r...

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 a...

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 a...

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 wh...

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 $\ell_1$-minimization based algorithm may be used to solve the imaging problem efficiently. A suitable algorithm in the case of multiple measurement vectors wou...

This paper deals with the risk associated with the mis-estimation of mean-reversion of residuals in statistical arbitrage. The main idea in statistical arbitrage is to exploit short-term deviations in returns from a long-term equilibrium across several assets. This kind of strategy heavily relies on the assumption of mean-reversion of idiosyncratic...

In this paper we consider the problem of imaging a fast moving small object. The imaging system consists of a powerful emitter and several passive receivers located on the ground. Our aim is to compare the well-known matched-filter imaging method with correlation-based imaging. Imaging with correlations has the advantage of not requiring any knowle...

In this paper we introduce and analyze a passive synthetic aperture radar system motivated by space surveillance radar networks for detecting, tracking, imaging, and identifying small debris (aka target) in low-earth orbit. We propose a system with a powerful transmitter on the ground and one or several flying receiver platforms. Each platform can...

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 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...

In dealing with high-dimensional data sets, factor models are often useful for dimension reduction. The estimation of factor models has been actively studied in various fields. In the first part of this paper, we present a new approach to estimate high-dimensional factor models, using the empirical spectral density of residuals. The spectrum of cov...

We consider the Czir\'ok model for collective motion of locusts along a one-dimensional torus. In the model, each agent's velocity locally interacts with other agents' velocities in the system, and there is also exogenous randomness to each agent's velocity. The interaction tends to create the alignment of collective motion. By analyzing the associ...

We consider the Czir\'ok model for collective motion of locusts along a one-dimensional torus. In the model, each agent's velocity locally interacts with other agents' velocities in the system, and there is also exogenous randomness to each agent's velocity. The interaction tends to create the alignment of collective motion. By analyzing the associ...

We present a wideband fast algorithm capable of accurately computing the full numerical solution of the problem of acoustic scattering of waves by multiple finite-sized bodies such as spherical scatterers in three dimensions. By full solution, we mean that no assumption (e.g. Rayleigh scattering, geometrical optics, weak scattering, Born single sca...

Optimal control models for limit order trading often assume that the underlying asset price is a Brownian motion since they deal with relatively short time scales. The resulting optimal bid and ask limit order prices tend to track the underlying price as one might expect. This is indeed the case with the model of Avellaneda and Stoikov (2008), whic...

Optimal control models for limit order trading often assume that the underlying asset price is a Brownian motion since they deal with relatively short time scales. The resulting optimal bid and ask limit order prices tend to track the underlying price as one might expect. This is indeed the case with the model of Avellaneda and Stoikov (2008), whic...

Waves generated by opportunistic or ambient noise sources and recorded by passive sensor arrays can be used to image the medium through which they travel. Spectacular results have been obtained in seismic interferometry, which open up new perspectives in acoustics, electromagnetics, and optics. The authors present, for the first time in book form,...

We consider the problem of seismic velocity change estimation using ambient noise recordings. Motivated by [23] we study how the velocity change estimation is affected by seasonal fluctuations in the noise sources. More precisely, we consider a numerical model and introduce spatio-temporal seasonal fluctuations in the noise sources. We show that in...

Waves generated by opportunistic or ambient noise sources and recorded by passive sensor arrays can be used to image the medium through which they travel. Spectacular results have been obtained in seismic interferometry, which open up new perspectives in acoustics, electromagnetics, and optics. The authors present, for the first time in book form,...

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...

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...

We consider the problem of seismic velocity change estimation using ambient noise recordings. Motivated by [23] we study how the velocity change estimation is affected by seasonal fluctuations in the noise sources. More precisely, we consider a numerical model and introduce spatio-temporal seasonal fluctuations in the noise sources. We show that in...

This paper deals with the risk of mean-reversions in statistical arbitrage. The basic concept of statistical arbitrage is to exploit short-term deviations in returns from a long-term equilibrium across several assets. This kind of strategy heavily relies on the assumption of mean-reversion of idiosyncratic returns - reverting to a long-term mean af...

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 scat...

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 scat...

We consider passive synthetic aperture imaging where a single moving receiver antenna records signals that are generated by distant unknown noise sources and backscattered by one or several reflectors. The reflectors can be imaged by migrating the autocorrelation functions of the received signals. We compare this passive synthetic aperture imaging...

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, where we showed that using linear combinations of...

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, where we showed that using linear combinations of...

We consider a stochastic, continuous state and time opinion model where each agent's opinion locally interacts with other agents' opinions in the system, and there is also exogenous randomness. The interaction tends to create clusters of common opinion. By using linear stability analysis of the associated nonlinear Fokker-Planck equation that gover...

We introduce a synthetic aperture imaging framework that takes into consideration directional dependence of the reflectivity that is to be imaged, as well as its frequency dependence. We use an $\ell_1$ minimization approach that is coordinated with data segmentation so as to fuse information from multiple sub-apertures and frequency sub-bands. We...

We formulate and analyze a multi-agent model for the evolution of individual and systemic risk in which the local agents interact with each other through a central agent who, in turn, is influenced by the mean field of the local agents. The central agent is stabilized by a bistable potential, the only stabilizing force in the system. The local agen...

We consider correlation-based imaging of a reflector located on one side of a passive array where the medium is homogeneous. On the other side of the array the illumination by remote impulsive sources goes through a strongly scattering medium. It has been shown in [J. Garnier and G. Papanicolaou, Inverse Problems, 28 (2012), 075002] that migrating...

We propose a new strategy for narrow band, active array imaging of localized scat- terers 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 imag...

It was shown in [Garnier et al., SIAM J. Imaging Sciences, 2 (2009), 396-437] that it is possible to image reflectors by backpropagating cross correlations of signals generated by ambient noise sources and recorded at sensor arrays. The resolution of the image depends on the directional diversity of the noise signals relative to the sensor array an...

We consider imaging in a scattering medium where the illumination goes through this medium but there is also an auxiliary, passive receiver array that is near the object to be imaged. Instead of imaging with the source-receiver array on the far side of the object we image with the data of the passive array on the near side of the object. The imagin...

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 o...

We consider here the problem of imaging using passive in-coherent recordings due to ambient noise sources. The first step towards imaging in this configuration is the computa-tion of the cross-correlations of the recorded signals. These cross-correlations are computed between pairs of sensors (receivers) and they contain very important information...

We consider narrow band, active array imaging of localized scatterers in a homogeneous medium with and without additive noise. We consider both single and multiple illuminations and study ℓ1 minimization-based imaging methods. We show that for large arrays, with array diameter comparable to range, and when scatterers are sparse and well separated,...

We study imaging of compactly supported scatterers buried deep in layered structures. The layering is unknown and consists of strongly reflecting interfaces as well as weakly reflecting fine layers, which we model with random processes. We consider wave scattering regimes where the unwanted echoes from the layers overwhelm the signal coming from th...

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...

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...

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 pr...

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 (...

Array imaging in a strongly scattering medium is limited because coherent signals recorded at the array and coming from a reflector to be imaged are weak and dominated by incoherent signals coming from multiple scattering by the medium. If, however, an auxiliary passive array can be placed between the reflector to be imaged and the scattering mediu...

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 expone...

We consider a system of diffusion processes that interact through their empirical mean and have a stabilizing force acting on each of them, corresponding to a bistable potential. There are three parameters that characterize the system: the strength of the intrinsic stabilization, the strength of the external random perturbations, and the degree of...

We introduce from first principles a synthetic aperture radar (SAR) imaging and target motion estimation method that is combined with compensation for radar platform trajectory perturbations. The main steps of the method are (a) segmentation of the data into properly calibrated small apertures, (b) preliminary motion estimation from the data using...

An algorithm for solving continuous-time stochastic optimal control problems is presented. The numerical scheme is based on the stochastic maximum principle (SMP) as an alternative to the widely studied dynamic programming principle (DDP). By using the SMP, (Peng, 1990) obtained a system of coupled forward-backward stochastic differential equations...

We investigate the feasibility of using correlation-based methods for estimating the spatial location of distributed receiving nodes in an indoor environment. Our algorithms do not assume any knowledge regarding the transmitter locations or the transmitted sig nal, but do assume that there are ambient signal sources whose location and properties ar...

Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under stochastic volatility in equity, interest-rate, and credit markets. They present and analyze multiscale stochastic volatility models and asymptotic approxima...

Travel time estimation and reflector imaging can be carried out using the cross correlations of signals generated by ambient noise sources and recorded at sensor arrays. We study here the mean and variance of the estimated quantities both with respect to the distribution of the noise sources and with respect to the distribution of the randomly scat...

We analyze the resolution and statistical fluctuations of images when the ambient medium is random and scattering can be modeled primarily by wavefront distortion. We compare the coherent interferometric imaging method to the widely used Kirchhoff migration and show how the latter loses statistical stability at an exponential rate with the distance...

The fueling of air-breathing supersonic combustion engines can cause a shock to develop in the isolator due to thermal choking; if this shock reaches the entrance of the isolator, the engine will stall. The quasi-one-dimensional time-dependent compressible Euler equations with heat release are used to model the flow in engine; both the heat release...

We shall give a fairly selfcontained account of some results on waves in random media and related problems that we have considered in the past few years [l]–[6]. These results rcly upon properties of solutions of differential equations with random coefficients, i.e., stochastic equations. We restrict attention to one-dimensional problems so that we...

Coherent interferometric imaging is based on the backpropagation of local spacetime cross-correlations of array data and was introduced in order to improve images when the medium between the array and the object to be imaged is inhomogeneous and unknown (Borcea et al 2005 Inverse Problems 21 1419). Although this method has been shown to be effectiv...

Echoes from small reflectors buried in heavy clutter are weak and difficult to distinguish from the medium backscatter. Detection and imaging with sensor arrays in such media requires filtering out the unwanted backscatter and enhancing the echoes from the reflectors that we wish to locate. We consider a filtering and detection approach based on th...

We introduce an adaptive approach for the detection of a reflector in a strongly scattering medium using a time-frequency representation of the array response matrix followed by a Singular Value Decomposition (SVD). We use the Local Cosine Transform (LCT) for the time-frequency representation and introduce a detection criterion that identifies anom...