Regularized Estimation of Stokes Images from Polarimetric Measurements

In the remote sensing context the goal of imaging polarimetry is to map the state of polarization of a scene of interest. The polarization state of a scene can be represented by the Stokes parameters. Since the Stokes parameters are not directly measurable one must first make several individual measurements and then the infer the Stokes parameters. We approach the Stokes parameter construction problem using penalized-likelihood estimation. Given the measured linearly polarized images, what is the optimal means by which to deblur and denoise and construct the Stokes parameters? In traditional image restoration one attempts to restore the blurred and noise corrupted data directly. In the case of imaging polarimetry we must answer the question of the optimality of restoring the measured data and then forming the Stokes images or restoring the Stokes images directly. An alternative approach is to estimate the Stokes parameters directly. We define our cost function for reconstruction by a weighted least squares data fit penalty and a regularization penalty. We show that for quadratic regularization the estimators of Stokes and intensity images can be made equal by appropriate choice of regularization parameters. It is empirically shown that, when using edge preserving regularization, estimating Stokes parameters directly leads to somewhat lower error.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85937/1/Fessler231.pd

Polca SARA - Full polarization, direction-dependent calibration and sparse imaging for radio interferometry

New generation of radio interferometers are envisaged to produce high quality, high dynamic range Stokes images of the observed sky from the corresponding under-sampled Fourier domain measurements. In practice, these measurements are contaminated by the instrumental and atmospheric effects that are well represented by Jones matrices, and are most often varying with observation direction and time. These effects, usually unknown, act as a limiting factor in achieving the required imaging performance and thus, their calibration is crucial. To address this issue, we develop a global algorithm, named Polca SARA, aiming to perform full polarization, direction-dependent calibration and sparse imaging by employing a non-convex optimization technique. In contrast with the existing approaches, the proposed method offers global convergence guarantees and flexibility to incorporate sophisticated priors to regularize the imaging as well as the calibration problem. Thus, we adapt a polarimetric imaging specific method, enforcing the physical polarization constraint along with a sparsity prior for the sought images. We perform extensive simulation studies of the proposed algorithm. While indicating the superior performance of polarization constraint based imaging, the obtained results also highlight the importance of calibrating for direction-dependent effects as well as for off-diagonal terms (denoting polarization leakage) in the associated Jones matrices, without inclusion of which the imaging quality deteriorates

Joint Reconstruction of Stokes Images from Polarimetric Measurements

In the field of imaging polarimetry Stokes parameters are sought and must be inferred from noisy and blurred intensity measurements. Using a penalized-likelihood estimation framework we investigate reconstruction quality when estimating intensity images and then transforming to Stokes parameters, and when estimating Stokes parameters directly. We define our cost function for reconstruction by a weighted least-squares data fit term and a regularization penalty. We show that for quadratic regularization the estimators of Stokes and intensity images can be made equal by appropriate choice of regularization parameters. It is empirically shown that, when using edge preserving regularization, estimating the Stokes parameters directly leads to lower RMS error. Also, the addition of a cross channel regularization term further lowers the RMS error for both methods, especially in the case of low SNR.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85916/1/Fessler20.pd

Joint Estimation of Stokes Images and Aberrations from Phase-Diverse Polarimetric Measurements

The technique of phase diversity has been used in traditional incoherent imaging systems to jointly estimate an object and optical system aberrations. This paper extends the technique of phase diversity to polarimetric imaging systems. Specifically, we describe penalized-likelihood methods for jointly estimating Stokes images and optical system aberrations from measurements that contain phase diversity. Jointly estimating Stokes images and optical system aberrations involves a large parameter space. A closed-form expression for the estimate of the Stokes images in terms of the aberration parameters is derived and used in a formulation that reduces the dimensionality of the search space to the number of aberration parameters only. We compare the performance of the joint estimator under both quadratic and edge-preserving regularization; we also compare the performance of the reduced parameter search strategy to the full parameter search strategy under quadratic regularization. The joint estimator with edge-preserving regularization yields higher fidelity polarization estimates than with quadratic regularization. With the reduced parameter search strategy, accurate aberration estimates can be obtained without recourse to regularization “tuning.”Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85915/1/Fessler9.pd

Polarimetric Image Reconstruction Algorithms.

In the field of imaging polarimetry Stokes parameters are sought and must be inferred from noisy and blurred intensity measurements. Using a penalized-likelihood estimation framework we investigate reconstruction quality when estimating intensity images and then transforming to Stokes parameters (traditional estimator), and when estimating Stokes parameters directly (Stokes estimator). We define our cost function for reconstruction by a weighted least squares data fit term and a regularization penalty. It is shown that under quadratic regularization, the traditional and Stokes estimators can be made equal by appropriate choice of regularization parameters. It is empirically shown that, when using edge preserving regularization, estimating the Stokes parameters directly leads to lower textsc{RMS} error in reconstruction. Also, the addition of a cross channel regularization term further lowers the textsc{RMS} error for both methods especially in the case of low textsc{SNR}. The technique of phase diversity has been used in traditional incoherent imaging systems to jointly estimate an object and optical system aberrations. We extend the technique of phase diversity to polarimetric imaging systems. Specifically, we describe penalized-likelihood methods for jointly estimating Stokes images and optical system aberrations from measurements that contain phase diversity. Jointly estimating Stokes images and optical system aberrations involves a large parameter space. A closed-form expression for the estimate of the Stokes images in terms of the aberration parameters is derived and used in a formulation that reduces the dimensionality of the search space to the number of aberration parameters only. We compare the performance of the joint estimator under both quadratic and edge-preserving regularization. The joint estimator with edge-preserving regularization yields higher fidelity polarization estimates than with quadratic regularization. Under quadratic regularization, using the reduced-parameter search strategy, accurate aberration estimates can be obtained without recourse to regularization ``tuning''. Phase-diverse wavefront sensing is emerging as a viable candidate wavefront sensor for adaptive-optics systems. In a quadratically penalized weighted least squares estimation framework a closed form expression for the object being imaged in terms of the aberrations in the system is available.Ph.D.Applied PhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/77925/1/jvalenz_1.pd

Super-Resolution for Imagery from Integrated Microgrid Polarimeters

Imagery from microgrid polarimeters is obtained by using a mosaic of pixel-wise micropolarizers on a focal plane array (FPA). Each distinct polarization image is obtained by subsampling the full FPA image. Thus, the effective pixel pitch for each polarization channel is increased and the sampling frequency is decreased. As a result, aliasing artifacts from such undersampling can corrupt the true polarization content of the scene. Here we present the first multi-channel multi-frame super-resolution (SR) algorithms designed specifically for the problem of image restoration in microgrid polarization imagers. These SR algorithms can be used to address aliasing and other degradations, without sacrificing field of view or compromising optical resolution with an anti-aliasing filter. The new SR methods are designed to exploit correlation between the polarimetric channels. One of the new SR algorithms uses a form of regularized least squares and has an iterative solution. The other is based on the faster adaptive Wiener filter SR method. We demonstrate that the new multi-channel SR algorithms are capable of providing significant enhancement of polarimetric imagery and that they outperform their independent channel counterparts

First M87 Event Horizon Telescope Results. VII. Polarization of the Ring

In 2017 April, the Event Horizon Telescope (EHT) observed the near-horizon region around the supermassive black hole at the core of the M87 galaxy. These 1.3 mm wavelength observations revealed a compact asymmetric ring-like source morphology. This structure originates from synchrotron emission produced by relativistic plasma located in the immediate vicinity of the black hole. Here we present the corresponding linear-polarimetric EHT images of the center of M87. We find that only a part of the ring is significantly polarized. The resolved fractional linear polarization has a maximum located in the southwest part of the ring, where it rises to the level of similar to 15%. The polarization position angles are arranged in a nearly azimuthal pattern. We perform quantitative measurements of relevant polarimetric properties of the compact emission and find evidence for the temporal evolution of the polarized source structure over one week of EHT observations. The details of the polarimetric data reduction and calibration methodology are provided. We carry out the data analysis using multiple independent imaging and modeling techniques, each of which is validated against a suite of synthetic data sets. The gross polarimetric structure and its apparent evolution with time are insensitive to the method used to reconstruct the image. These polarimetric images carry information about the structure of the magnetic fields responsible for the synchrotron emission. Their physical interpretation is discussed in an accompanying publication

Sparse interferometric Stokes imaging under polarization constraint (Polarized SARA)

We develop a novel algorithm for sparse Stokes parameters imaging in radio interferometry under the polarization constraint. The latter is a physical non-linear relation between the Stokes parameters, imposing that the polarization intensity is a lower bound on the total intensity. To solve the joint inverse Stokes imaging problem including this bound, we leverage epigraphical projection techniques in convex optimization and design a primal-dual method offering a highly flexible and parallelizable structure. In addition, we propose to regularize each Stokes parameter map through an average sparsity prior in the context of a reweighted analysis approach (SARA). The resulting approach is dubbed Polarized SARA. We demonstrate on simulated observations of M87 with the Event Horizon Telescope that imposing the polarization constraint leads to superior image quality. The results also confirm that the performance of the average sparsity prior surpasses the alternative state-of-the-art priors for polarimetric imaging