Robert A. Gonsalves

Phase Retrieval And Diversity In Adaptive Optics

Wavefront sensing by phase retrieval implies extraction of the Fourier transform of a complex signal based on observation of the modulus of the signal. Only the image intensity from a systems focal plane array is required to estimate the phase aberrations. These estimates are used to derive control signals to align (or to maintain alignment of) the optical system. The concept can be used in both a predetection and postdetection mode. In the former, the control system labors to keep the optics in a diffraction-limited mode all the time. In the latter, the control system induces a phase or wavelength diversity that allows successive images to be restored to nearly diffraction-limited quality by postprocessing of the image. This second mode is particularly interesting because it will reduce the design effort for both the optical system and the control system. How the phase or wavelength diversity is achieved is not clear at this time. If the method has utility, it provides an interesting challenge to designers of optical devices. In this paper we describe the mathematics of the technique and show some computer simulations which involve both point sources and extended objects.

Phase retrieval from modulus data

Phase retrieval implies extraction of the phase of a complex signal f from its modulus |f|. We give examples where an additional constraint is imposed: knowledge of the modulus of F, the Fourier transform of f. The retrieval is accomplished by computer processing of samples of |f| and |F|. The problems of noisy data and nonuniqueness are addressed.

Fluorescent computer tomography: a model for correction of X-ray absorption

The distributions of individual trace elements within a sample can be found using fluorescent computed tomography (FCT). The absorption of incident and fluorescent X-rays results in degraded reconstructions of the distributions. The proposed method uses the absorption density, measured using conventional absorption tomography, to remove absorption effects. A model for FCT with absorption is developed and simulated. The resulting corrected reconstructions are compared to the reconstructions degraded by attenuation efforts. A comparison is made with other methods which utilize knowledge of the sample composition and a standard source to estimate the absorption coefficients used to correct for attenuation effects. >

Small-phase solution to the phase-retrieval problem

A solution to the phase-retrieval problem when the unknown phase is small is presented. The solution specifies the even and odd parts of the unknown phase in two separate equations. The odd part requires a single intensity measurement, and the even part requires two measurements. Phase diversity is used for the second measurement, and computer simulations are given.

Nonisoplanatic imaging by phase diversity

Phase diversity allows one to use multiple images with known phase changes such as defocus to learn the optical characteristics of an imaging medium and to estimate the unknown object. It is shown that the method can be extended to the case in which the optical system is nonisoplanatic; thus it can recover both the extended object and the angle-dependent, aberrating phase of the medium. The technique is a timely extension of the conventional phase-diversity concept and could be used to solve the isoplanatic patch problem of adaptive optics, to determine a spatially varying point-spread function in image restoration, and to image through a single-fiber optic.

Maximum-Likelihood Receiver for Digital Data Transmission

The maximum-likelihood (ML) receiver for binary data, additive Gaussian noise, and nearest-neighbor intersymbol interference (ISI) is shown to be a matched filter followed by a feedback loop and a tapped delay line. The loop and the tap connections contain nonlinear amplifiers, each saturating at the level of the ISI. This receiver minimizes the persymbol probability of error P e ; upper and lower bounds on P e are obtained. The mathematical receiver specifications for multilevel data and for extensive ISI are given; an approximate structure for the latter receiver is presented as an intuitive extension of the binary, limited ISI structure.

Compensation of scintillation with a phase-only adaptive optic

To compensate for scintillation we allow the real, nonnegative field in an aperture to propagate a distance z toward a detector, where the field will be complex. There, we modify its phase to emulate that of a clear aperture. The modified field that is propagated to the detector is nearly diffraction limited. No light is lost, and the Strehl ratio is improved substantially. We show how to modify the phase, and we present a computer simulation.

Entropy-Based Algorithm For Reducing Artifacts In Image Restoration

We introduce a simple algorithm to reduce artifacts that often appear in image restoration techniques such as Wiener filtering. The algorithm starts with the inverse filter solution and iteratively calculates the correction term. At each iteration we use an entropy gradient and an analytically calculated step size. The algorithm uses two Fourier transforms per iteration. We show both 1 -D and 2-D examples to illustrate the algorithm.

Cramér-Rao bounds on mensuration errors

The Cramér-Rao (CR) bound is an inequality that sets a lower bound on the variance of an estimator. The paper presents an application of this bound to the problem of mensuration. Explicit results are given for the width of a pulse, incoherently imaged by an optical system. The results are given in terms of the SNR of the observed signal and the autocorrelation function of the system line spread function. The effects of another unknown parameter (pulse amplitude), an application to sampled data imagery, and the calculation of confidence limits are presented.

Spot Shaping and Incoherent Optical Smoothing for Raster Scanned Imagery

Raster in scanned imagery can be suppressed to improve visual observation of image detail. In the work reported here this is accomplished by optimal choice of the scanning spot and/or by optimal smoothing. Both techniques are based on selection of a Wiener filter. The optimal spot is shown to extend only over two scan lines, for scenes with 1 /f2 power spectra. The filter is designed to be used in an incoherent imaging system. It is constrained to a phase-only approximation of the optimum complex filter to avoid reducing light level in the optical system. Experimental results demonstrate the effectiveness of the filter design.