Richard G. Paxman

Evaluation of phase-diversity techniques for solar-image restoration

Phase-diversity techniques provide a novel observational method for overcomming the effects of turbulence and instrument-induced aberrations in ground-based astronomy. Two implementations of phase-diversity techniques that differ with regard to noise model, estimator, optimization algorithm, method of regularization, and treatment of edge effects are described. Reconstructions of solar granulation derived by applying these two implementations to common data sets are shown to yield nearly identical images. For both implementations, reconstructions from phase-diverse speckle data (involving multiple realizations of turbulence) are shown to be superior to those derived from conventional phase-diversity data (involving a single realization). Phase-diverse speckle reconstructions are shown to achieve near diffraction-limited resolution and are validated by internal and external consistency tests, including a comparison with a reconstruction using a well-accepted speckle-imaging method.

Reconstruction of objects from coded images by simulated annealing

A new Monte Carlo reconstruction procedure is presented for retrieval of objects from their coded images. The reconstruction process is modeled as an optimization problem whose cost function is related to how well the coded image constraints are satisfied. Reduction of the cost function is achieved by an annealing process analogous to the cooling of a melt to produce an ordered crystal. The method is demonstrated by reconstructing two two-dimensional objects from their one-dimensional coded images.

Image reconstruction from coded data: I. Reconstruction algorithms and experimental results.

Two algorithms have been developed for reconstructing objects from their coded images and a priori knowledge of the object class. Reconstructions from both algorithms are presented, but the results appear to be largely independent of the algorithm used. One of the algorithms, a Monte Carlo approach, is used to investigate the quality of the reconstruction of two- and three-dimensional objects from simulated coded-image data with respect to viewing geometry and multiplexing (mixing) of the data. The cases examined include reconstructions from data with and without signal-dependent photon noise. It is found that reconstructing from multiplexed data is not so serious a problem as reconstructing from data obtained with a limited viewing angle. Also, when photon noise is included in the data, reconstructions obtained from multiplexed data are better than those obtained from unmultiplexed data because of the higher photon count made available by multiplexing. It appears that the fidelity of a reconstruction depends much more strongly on the design of the data-taking system (the coded apertures) than on the reconstruction algorithm.

Phase-diverse speckle reconstruction of solar data

Phase-diverse speckle imaging is a novel imaging modality that makes use of both speckle-imaging and phase-diversity concepts. A phase- diverse speckle data set consists of one conventional image and at least one additional image with known phase diversity for each of multiple atmospheric phase realizations. We demonstrate the use of a phase-diverse speckle data set collected at the Swedish Vacuum Solar Telescope on La Palma to overcome the effects of atmospheric turbulence and to restore a fine-resolution image of solar granulation. We present preliminary results of simultaneously reconstructing an object and a sequence of atmospheric phase aberrations from these data using a maximum-likelihood parameter- estimation framework. The consistency of the reconstructions is demonstrated using subsets of the sequence of images pairs. The use of different phase-aberration parameterization schemes and their affect on parameter estimates are discussed. Insight into the desired number of atmospheric realizations is provided.

Phase-diversity correction of turbulence-induced space-variant blur

Space-variant blur is encountered when objects extend beyond the isoplanatic patch associated with the intervening atmospheric turbulence. The method of phase diversity, used to estimate jointly the object and the aberrations, is generalized to accommodate turbulence-induced space-variant blur. This generalization utilizes a parametric model for the blur function that is constructed with multiple phase screens. Simulation results are presented that demonstrate the recovery of near-diffraction-limited imagery from phase-diversity imagery that has been degraded with rather severe anisoplanatism.

Maximum a posteriori estimation of fixed aberrations, dynamic aberrations, and the object from phase-diverse speckle data

In phase-diverse speckle imaging one collects a time series of phase-diversity image sets that are used to jointly estimate the object and each of the phase-aberration functions. Current approaches model the total phase aberration in some deterministic parametric fashion. For many imaging schemes, however, additional information can be exploited. Specifically, the total aberration function consists of the fixed aberrations combined with dynamic (time-varying), turbulence-induced aberrations, about whose stochastic behavior we often have some knowledge. One important example is that in which the wave-front phase error corresponds to Kolmogorov turbulence. In this context using the extra statistical information available may be a powerful aid in the joint aberration/object estimation. In addition, such a framework provides an attractive method for calibrating fixed aberrations in an imaging system. The discipline of Bayesian statistical inference provides a natural framework for using the stochastic information regarding the wave fronts. Here one imposes an a priori probability distribution on the turbulence-induced wave fronts. We present the general Bayesian approach for the joint-estimation problem of fixed aberrations, dynamic aberrations, and the object from phase-diverse speckle data that leads to a maximum a posteriori estimator. We also present results based on simulated data, which show that the Bayesian approach provides an increase in accuracy and robustness for this joint estimation.

Phase retrieval from experimental far-field speckle data.

Phase retrieval from experimental (laboratory) data has been successfully demonstrated. A diffuse object was coherently illuminated and Fourier intensity data were collected by a charge-coupled device detector and a video digitizer. By using the data and an a priori triangular image support constraint, an iterative Fourier-transform algorithm was used to estimate the phase of the Fourier transform of the object. The reconstructed image compares favorably with a conventional image with the same spatial-frequency bandwidth.

Comparison of phase diversity and curvature wavefront sensing

We compare phase diversity and curvature wavefront sensing. Besides having completely different reconstruction algorithms, the two methods measure data in different domains: phase diversity very near to the focal plane, and curvature wavefront sensing far from the focal plane in quasi-pupil planes, which enable real-time computation of the wavefront using analog techniques. By using information- theoretic lower bounds, we show that the price of measuring far from the focal lane is an increased error in estimating the phase. Curvature wavefront sensing is currently operating in the field, but phase diversity should produce superior estimates as real-time computing develops.

Image reconstruction from coded data: II. Code design

A strategy is given for the design of coded apertures with respect to a given class of objects that are to be imaged. Previous knowledge of the first- and second-order statistics for the object class is assumed. The object class is characterized by its Karhunen-Loève eigenvectors and eigenvalues, whereas the imaging system is characterized by its singular-value decomposition. We introduce the concept of alignment in which the aperture parameters are adjusted until the system is tuned to measure the given object class well. A mean-square-error figure of merit that indicates degree of alignment is given, and alignment is performed by standard optimization techniques. We illustrate this technique with a simple proof-of-principle experiment. These concepts are general and may be applied to any linear imaging system.

Closed-loop wavefront sensing for a sparse-aperture multitelescope array using broadband phase diversity

The Lockheed Martin phased-array telescope developed at the Palo Alto Research Laboratory is a 0.75-m imaging system consisting of 9 separate 90-mm telescopes. One of the technology drivers behind this design is the ability to maintain the phasing of the individual telescopes to sub- wavelength tolerances. We demonstrate here the use of the focal-plane method of phase diversity for maintaining the phased-array alignment. The telescope is designed to operate with white light, so the phase diversity concept is extended to accommodate a broad optical bandwidth. A simulation of white-light phase-diverse wavefront sensing is presented as a demonstration of the robustness of the method with respect to sparse pupil and wavelength sampling. The simulation is validated with laboratory experiments using a point source. Finally, a closed-loop experiment is conducted that demonstrates the ability of phase diversity to sense piston error and maintain the alignment of the phased-array system.