We present a method for imaging small scatterers in a homogeneous medium from polarization measurements of the electric field at an array. The electric field comes from illuminating the scatterers with a point source with known location and polarization. We view this problem as a generalized phase retrieval problem with data being the coherency matrix or Stokes parameters of the electric field at the array. We introduce a simple preprocessing of the coherency matrix data that partially recovers the ideal data where all the components of the electric field are known for different source dipole moments. We prove that the images obtained using an electromagnetic version of Kirchhoff migration applied to the partial data are, for high frequencies, asymptotically identical to the images obtained from ideal data. We analyze the image resolution and show that polarizability tensor components in an appropriate basis can be recovered from the Kirchhoff images, which are tensor fields. A time domain interpretation of this imaging problem is provided and numerical experiments are used to illustrate the theory.
We analyze the resolution of imaging functionals that migrate the cross-correlation matrices of passive sensor arrays. These matrices are obtained by cross-correlating signals emitted by ambient noise sources and recorded by the passive sensor array. They contain information about reflectors in the surrounding medium. Therefore, travel time or Kirchhoff migration of the cross-correlations can, under favorable circumstances, produce images of such reflectors. However, migration should be carried out appropriately depending on the type of illumination provided by the ambient noise sources. We present here a detailed resolution analysis of these functionals in a homogeneous medium. Resolution depends on the sensor array diameter, the distance from the array to the reflector and the central frequency, as is the case in active array imaging. When imaging with passive sensor arrays and ambient noise, resolution also depends on the space and time coherence of the noise sources because it determines an effective noise bandwidth. It is well known that the travel time or even the full Green's function between two passive sensors can be estimated from the cross correlation of recorded signal amplitudes generated by ambient noise sources. It is also known that the direction of the energy flux from the noise sources affects the estimation of the travel time. Using the stationary phase method, we show here that the travel time can be effectively estimated when the ray joining the two sensors continues into the noise source region. We extend this analysis to passive sensor imaging of reflectors with different ambient noise source configurations by suitably migrating the cross correlations. If in addition there is multiple scattering in the medium, then reflectors can be imaged with passive sensor networks or arrays by migrating suitable fourth-order cross correlations. Fourth-order cross correlations can also be used with auxiliary passive sensors in order to enhance travel time estimation in a scattering medium. Scatterers in a homogeneous medium are imaged by probing the medium with two point sources of waves modulated by correlated signals and by measuring only intensities at one single receiver. For appropriately chosen source pairs, we show that full waveform array measurements can be recovered from such intensity measurements by solving a linear least squares problem. The least squares solution can be used to image with Kirchhoff migration, even if the solution is determined only up to a known one-dimensional nullspace. The same imaging strategy can be used when the medium is probed with point sources driven by correlated Gaussian processes and autocorrelations are measured at a single location. Since autocorrelations are robust to noise, this can be used for imaging when the probing wave is drowned in background noise. In this paper we provide for a first time, to our knowledge, a mathematical model for imaging an anisotropic, orthotropic medium with polarization-sensitive optical coherence tomography. The imaging problem is formulated as an inverse scattering problem in three dimensions for reconstructing the electrical susceptibility of the medium using Maxwell’s equations. Our reconstruction method is based on the second-order Born-approximation of the electric field.  We propose a direct imaging method based on the reverse time migration method for finding extended obstacles with phaseless total field data in the half space. We prove that the imaging resolution of the method is essentially the same as the imaging results using the scattering data with full phase information when the obstacle is far away from the surface of the half-space where the measurement is taken. Numerical experiments are included to illustrate the powerful imaging quality. In this paper we consider narrow band, active array imaging of weak localized scatterers when only the intensities are recorded at an array with @math transducers. We assume that the medium is homogeneous and, hence, wave propagation is fully coherent. This work is an extension of our previous paper [A. Novikov, M. Moscoso, and G. Papanicolaou, SIAM J. Imaging Sci., 8 (2015), pp. 1547--1573], where we showed that using linear combinations of intensity-only measurements, obtained from @math illuminations, imaging of localized scatterers can be carried out efficiently using imaging methods based on the singular value decomposition of the time-reversal matrix. Here we show that the same strategy can be accomplished with only @math illuminations, therefore enormously reducing 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 diffraction. The key point of this... In this paper we propose a direct imaging method based on reverse time migration for reconstructing extended obstacles by phaseless electromagnetic scattering data. We prove that the imaging resolution of the method is essentially the same as the imaging results using the scattering data with the full phase information. This implies the imaginary part of the cross-correlation imaging functional always peaks on the boundary of the obstacle. Numerical experiments are included to illustrate the powerful imaging quality.  An approach has been developed that involves the fit of a combination of three simple scattering mechanisms to polarimetric SAR observations. The mechanisms are canopy scatter from a cloud of randomly oriented dipoles, evenor double-bounce scatter from a pair of orthogonal surfaces with different dielectric constants and Bragg scatter from a moderately rough surface. This composite scattering model is used to describe the polarimetric backscatter from naturally occurring scatterers. The model is shown to describe the behavior of polarimetric backscatter from tropical rain forests quite well by applying it to data from NASA Jet Propulsion Laboratory's (JPLs) airborne polarimetric synthetic aperture radar (AIRSAR) system. The model fit allows clear discrimination between flooded and nonflooded forest and between forested and deforested areas, for example. The model is also shown to be usable as a predictive tool to estimate the effects of forest inundation and disturbance on the fully polarimetric radar signature. An advantage of this model fit approach is that the scattering contributions from the three basic scattering mechanisms can be estimated for clusters of pixels in polarimetric SAR images. Furthermore, it is shown that the contributions of the three scattering mechanisms to the HH, HV, and VV backscatter can be calculated from the model fit. Finally, this model fit approach is justified as a simplification of more complicated scattering models, which require many inputs to solve the forward scattering problem. We consider imaging with passive arrays of sensors using as illumination ambient noise sources. The first step for imaging under such circumstances is the computation of the cross correlations of the recorded signals, which have attracted a lot of attention recently because of their numerous applications in seismic imaging, volcano monitoring, and petroleum prospecting. Here, we use these cross correlations for imaging reflectors with travel-time migration. While the resolution of the image obtained this way has been studied in detail, an analysis of the signal-to-noise ratio (SNR) is presented in this paper along with numerical simulations that support the theoretical results. It is shown that the SNR of the image inherits the SNR of the computed cross correlations and therefore is proportional to the square root of the bandwidth of the noise sources times the recording time. Moreover, the SNR of the image is proportional to the array size. This means that the image can be stabilized by increasing the si...  We propose a reliable direct imaging method based on the reverse time migration for finding extended obstacles with phaseless total field data. We prove that the imaging resolution of the method is essentially the same as the imaging results using the scattering data with full phase information when the measurement is far away from the obstacle. The imaginary part of the cross-correlation imaging functional always peaks on the boundary of the obstacle. Numerical experiments are included to illustrate the powerful imaging quality 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.