Three new algorithms for deconvolving image blur are presented. All three are based on the computation of the zeros of an image's z-transform and the separation of the zeros into sets belonging to the image and to the point spread function (psf). The zeros lie on a sheet existing in a four-dimensional space. The first two algorithms are applicable when the psf is known a priori. In Algorithm I portions of the 'zero sheet' are matched using a Euclidean measure, then zeros are selected from the remainder and an image is algebraically reconstructed. In Algorithm II the point zeros of 1-dimensional cuts through Fourier space are matched before reconstructing an image estimate via inverse Fourier transformation. Finally, Algorithm III is applicable when an ensemble of differently blurred images are recorded from the same object (e.g. astronomical speckle images); even through the psf is unknown for each member of the ensemble (i.e. deconvolution is to be 'blind'), parts of the zero sheet corresponding to the actual (unblurred) image can be matched over the ensemble and a reconstruction made by inverse Fourier transformation. Encouraging results have been obtained for Algorithms I and III for small positive images contaminated by small amounts of noise; Algorithm II has been successfully applied to larger images. Algorithms I and III have an inherent advantage over conventional Wiener filtering in that the psf does not need to be known precisely to achieve acceptable results.

Two- and higher-dimensional bandlimited functions are almost always non-factorizable, meaning that their zeros form a single analytic curve. In principle one can use this property to separate the product of two bandlimited functions into its respective factors; this is important in Fourier phase retrieval and deconvolution problems. The intersection of this zero structure with the real plane is at points, closed curves or lines stretching to infinity. The location of zero points, curves or lines can only be estimated, in practice, from available noisy data. The estimated locations can be used to write a factorizable approximation to the original function and we explore the consequences of doing this. Of importance is the fact that point zeros can be used to represent a 2D bandlimited function. Hence from intensity data, point zero locations in the intensity can be used directly to estimate the complex spectrum and provide an approximate solution to the phase retrieval problem. Examples will be given and their importance for the general blind deconvolution problem discussed.

In this paper, we present a set theoretic based approach for solving the classical blind deconvolution problem. In this new approach, every piece of available information about both the source function and the system impulse response function is expressed via a constraint set. The link between generalized projection based algorithms and algorithms studied previously by other researchers is established. Motivated by this relation, a new algorithm is proposed which enjoys good convergent properties. Finally, numerical examples are presented to test the proposed algorithm.

The use of support constraints for improving the quality of Fourier spectra, their associated images, and the relationship between the two domains is discussed in this paper. Theoretical relationships are derived which predict the noise reduction in both the image domain and the Fourier domain achieved by single and repeated application of support constraints for the case of wide sense stationary Fourier domain noise. It is shown that the application of support constraints can increase noise inside the support constraint if the application is not done correctly. An iterative algorithm is proposed which enforces support constraints in such a way that noise is never increased inside the support constraint and the algorithm achieves the minimum possible noise in a finite number of steps.

Consider a scenario of imaging through atmospheric turbulence. A digital approach for processing out the turbulence from degraded images is under development. In contrast to past approaches only two short-exposure images are needed as inputs. Here we test the benefits to be gained by inserting object power spectrum information into the algorithm.

We present results obtained from the application of a novel phase retrieval algorithm on experimental scattering data from a microlens array, with the aim of reconstructing the phase profile across the array. The algorithm, originally developed for electron microscopy, requires two intensity measurements at different distances from a weak scatter, where the total transmitted field is composed of the coherent sum of an incident plane wave and the scattered wave. The algorithm is noniterative and does not have the convergence problems associated with iterative algorithms. Our results obtained using the new algorithm on experimental data are comparable to those obtained with the Gerchberg-Saxton algorithm, and the computational cost is much less. The new technique shows great promise for inverse scattering applications, such as optical diffraction tomography and rough surface profiling, where the scattering can be assumed to be weak.

A new algorithm for correcting phase errors in synthetic aperture radar data is described. It employs a gradient search algorithm to minimize the energy of the image outside a support constraint.

Fluctuations, under common conditions regarded as noise or clutter, often are carriers of very useful information. This information may be turned into usable form if the degree of distinctions between fluctuating signals can be determined. To visualize statistical parameters of two-dimensional fields of fluctuations, these parameters are proposed to be converted into videosignals, forming conventional TV image. Recursive technique is applied to such parameter as variance, modified into temporal contrast. Description of visualizer configuration is preceded by a summary of background for speckle-originated fluctuations in turbulent atmosphere when retroreflected from diffuse and mirror surfaces. The possibilities are discussed for controlled speckle displacements using laser beam scanning or phase/frequency swip.

X-ray crystallography is concerned with the determination of molecular structures from measurements of scattered x-rays and, historically, represents the first study of phase problems. The crystallographic problem is distinct from most other image reconstruction problems however, because of the periodic nature of the image. Reconstruction algorithms used in crystallography that make use of various kinds of constraints are described. Uniqueness properties are described and compared with those that apply in general imaging contexts.

The crystallographic X-ray diffraction experiment gives the amplitudes of the Fourier series expansion of the electron density distribution within the crystal. The 'phase problem' in crystallography is the determination of the phase angles of the Fourier coefficients required to calculate the Fourier synthesis and reveal the molecular structure. The magnitude of this task varies enormously as the size of the structures ranges from a few atoms to thousands of atoms, and the number of Fourier coefficients ranges from hundreds to hundreds of thousands. The issue is further complicated for large structures by limited resolution. This problem is solved for 'small' molecules (up to 200 atoms and a few thousand Fourier coefficients) by methods based on probabilistic models which depend on atomic resolution. These methods generally fail for larger structures such as proteins. The phase problem for protein molecules is generally solved either by laborious experimental methods or by exploiting known similarities to solved structures. Various direct methods have been attempted for very large structures over the past 15 years, with gradually improving results -- but so far no complete success. This paper reviews the features of the crystallographic image reconstruction problem which render it recalcitrant, and describes recent encouraging progress in the application of maximum entropy methods to this problem.

The goal in x-ray crystallography is to recover a signal from measurements of the magnitude squared of its Fourier transform. We describe a Bayesian statistical approach using a Markov random field model of the signal and a least squared error reconstruction criteria. The key computation is the computation of the a posteriori mean of the Markov random field given the data. We approximate this mean with the cluster approximation and use a continuation method to solve the resulting fixed-point equation. A small numerical example is presented.

The phase retrieval problem is occurs to time series analysis, astronomical image processing and crystallography. Each field has distinctive methods and terminology which reflect the subject matter problems leading to the phase retrieval problem. The construction of a set of example problems to allow exploration of the algorithmic issues of the Fourier coefficient calculations was started. It became quickly evident that it would be necessary to explore the numerical issues. The example problems were intended to be clean and avoid the issues of statistical error.

The crystallographic image processing techniques of Sayre's equation, molecular averaging, solvent flattening and histogram matching are combined in an integrated procedure for macromolecular phase retrieval. It employs the constraints of the local shape of electron density, equal molecules, solvent flatness and correct electron density distribution. These constraints on electron density image are satisfied simultaneously by solving a system of non- linear equations using fast Fourier transform. The electron density image is further filtered under the constraint of observed diffraction amplitudes. The effect of each constraint on phase retrieval is examined. The constraints are found to work synergistically in phase retrieval. Test results on 2Zn insulin are presented.

The problem of reconstructing a three-dimensional scatterer from measurements of the cylindrical average of the intensity of its Fourier transform is discussed. Reconstruction involves phase retrieval as well as unraveling the effects of cylindrical averaging. The application area considered is x-ray analysis of polymer fibers, and other systems, that consist of collections of rotationally disordered particles. Diffraction from these types of specimens and reconstruction algorithms are described. The particles in these applications often exhibit helical symmetry and the effect of this on reconstruction is described.

Experimental results obtained using holographic laser radar are presented. Holographic laser radar is method for obtaining fine-resolution, 3-D images from laser illuminated objects. A series of complex valued holograms is recorded for a series of laser frequencies. These holograms are assembled into a 3-D data array and Fourier transformation yields a 3-D image. Experimental results obtained using a dye laser are presented.

The ability to accurately measure the phase of the wavefront in an amplitude interferometer is fundamentally limited by the light level. Under high light conditions, the variance of the phase measurement is inversely proportional to the number of photons detected. In this paper, we review the basic theory of phase measurement for an optical heterodyne array imaging system for high light conditions. The theory is then extended to a sheared coherent interferometric photography (SCIP) system. Simulation and laboratory results verifying the theory and extending it to low light levels are then presented.

A new active imaging technique is described which eliminates atmospheric distortion and can give diffraction limited performance without any focusing optics. It has been demonstrated in computer simulations and laboratory experiments.

Sheared Coherent Interferometric Photography (SCIP) is an active imaging technique that allows near-diffraction limited imaging of objects through turbulent media. This paper presents computer simulation and laboratory results that illustrate the viability of the technique.

We have developed a one-dimensional theory and a computer model for synthetically imaging scenes using a novel fringe scanning/Fourier transform technique. Our method probes a scene using two interfering beams of slightly different frequency. These beams form a moving fringe pattern which scans the scene and resonates with any spatial frequency components having the same spatial frequency as the scanning fringe pattern. A simple, non-imaging detector above the scene observes any scattered radiation from the scene falling onto it. If a resonance occurs between the scanning fringe pattern and the scene, then the scattered radiation will be modulated at the difference frequency between the two probing beams. By changing the spatial period of the fringe pattern and then measuring the amplitude and phase of the modulated radiation that is scattered from the scene, the Fourier amplitudes and phases of the different spatial frequency components making up the scene can be measured. A synthetic image of the scene being probed can be generated from this Fourier amplitude and phase data by taking the inverse Fourier transform of this information. This technique could be used to image objects using light, ultrasonic, or other electromagnetic or acoustic waves.

Correlations of coherent backscatter intensities have been used to form Fourier spectra of coherently illuminated objects. Both second and some fourth-order correlations have been studied. We add to this by describing a new method using fourth order field correlations for imaging. If one knows the actual object speckle field, or certain sheared versions of it, then it is possible through fourth-order correlations to obtain the incoherent brightness function of the object. The technique will allow simultaneous averaging-out of both field phase distortions, certain field measurement noises, and laser speckle.

The Hubble Space Telescope image quality is degraded significantly as a result of the spherical aberration in the primary mirror. Although the forthcoming HST servicing mission will deploy corrective optics systems and a second-generation camera with built-in correctors, the current and archival images for the past three years require the use of restoration techniques in order to achieve the full scientific potential of the HST mission. In addition, we expect that the restoration techniques now being developed will continue to be utilized on post- servicing mission imagery in order to remove the remaining diffraction features and optimize dynamic range. A variety of well-known image restoration techniques, such as Wiener filters, Richardson-Lucy, Jansson-van Cittert, CLEAN, and maximum entropy, have been applied to HST imaging with reasonable success. However, all techniques have been hampered by incomplete knowledge of the point spread function and the space variance of the PSF in the Wide Field/Planetary Camera, HST's most frequently used imager. The best restoration results to date have been obtained by utilizing observed PSFs (an optimal exposure which minimizes noise and image defects) on small enough subsets of the total field of view so that the PSF variation can be ignored.

The spherical aberration errors which are present in the Hubble Space Telescope optical system produce a space variant point spread function where only 20% of the photons are concentrated in an area corresponding to the diffraction limit of the telescope. Deconvolution techniques can be used to concentrate the remaining 80% of the photons into the Airy core of the images thus restoring diffraction limited performance. Since the point spread function is space variant the normal Fourier transform convolution techniques are difficult to implement. We have developed an image space based convolution routine on the Connection Machine which can be used with a variety of deconvolution algorithms. The space variant point spread function is approximated by assuming isoplanatic patches in the image which can vary in size from 2 X 2 to 32 X 32 pixels. We present timing examples of two popular deconvolution techniques: the Maximum Entropy Method and the Richardson-Lucy method. Thirty iterations of either algorithm on a 512 X 512 image using 256 different point spread functions is accomplished in 90 seconds.

An image restoration problem will be formulated in the context of nonlinear programming using the conjugate gradient algorithm. The formulation of the objective function used in the conjugate gradient routine is presented. Situations may occur when there is a great deal already known about a certain object of interest which have been optically blurred because of the atmosphere or system imperfections. This paper shows a new and innovative way to incorporate a priori, perfect, partial knowledge of an object into the nonlinear optimization procedure. The topics discussed include the steps which led to the development of this procedure, the incorporation of the a priori knowledge into the nonlinear optimization problem, an analytical, mathematical approach which shows how the improvement should occur, and finally, data from simulated results which demonstrate the improvement using the developed diagnostic metrics.

In the operation of both multiaperture telescopes and segmented mirror telescopes, a primary issue is the phasing of the apertures or mirrors. This is a brief report of the first proof-of- concept experiment of phase diversity using the nonlinear optimization estimation technique for multiaperture systems. A two-telescope system was optically simulated in which piston could be estimated for both a point source and expended objects to .1 (lambda) rms.

A technique for storing high spatial frequency information in low frequency zero regions of an image spectrum has been implemented and evaluated. This technique can be used for sparse arrays of small telescopes acting to synthesize a larger pupil. The number of detectors required in the two-dimensional image focal plane is reduced by a factor of four.

In the past 10 years astronomical image reconstruction based on a variety of speckle techniques have become popular means of enhancing and improving the resolution of turbulence degraded images. These techniques are based on the Fourier processing of a large number of turbulence degraded 'snapshots' or frames of the intensity in the system's image plane. The number of snapshots needed is largely a function of the signal-to-noise ratio (SNR) of the Fourier components of a single snapshot. This paper describes and presents results from a detailed SNR analysis of estimating the modulus of an object's Fourier spectrum from turbulence distorted images. Unlike previous analyses, the work presented in this paper takes into account the proper temporal correlation properties of the atmosphere, the spatial frequency being estimated, as well as the inter-frame correlations that degrade the SNR improvement factor from the ideal case of (root)m.

The recovery of a stellar object's Fourier phase from the bispectrum of the atmospherically- degraded images has been investigated with respect to object complexity, a wide range of light levels, severity of the atmospheric effects and the number of bispectral subplanes used. The incremental improvement in the quality of the recovered phase as the number of near-axis subplanes increases is greatest for complex sources and when photon-noise dominates. At high signal-to-noise ratios it was found that the extended Knox-Thompson transfer function has a higher phase variance than the bispectral transfer function. Analysis has confirmed this difference in behavior.

Using simulation we investigate the difference in performance between low order adaptive optics and computer post-processing techniques. It is shown that even a simple tip-tilt defocus correction implemented with a rigid mirror can improve the quality of speckle images and thus improve the bispectral reconstructions at low photon rates. As an alternative to this low cost adaptive optics system we consider the use of ensemble deconvolution where a Shack- Hartmann sensor is used to augment the speckle images. Results are presented to show that this is a viable alternative to adaptive optics when a simple system for compensating atmospheric turbulence is desired. In particular we investigate the optimal size of Shack- Hartmann array when there is significant CCD readout noise.

Phase-diverse speckle imaging is a novel post-detection correction technique for use in imaging through atmospheric turbulence. When the wavelength dependence is introduced, the data-collection model and problem statement are seen to accommodate a variety of existing and proposed data-collection schemes, including speckle imaging, multi-frame blind deconvolution, deconvolution with wavefront sensing, and hybrid adaptive optics.

In a closed loop adaptive optics system the wavefront sensor is continuously measuring the residual wavefront error. Thus, wavefront information is continuously available to use for image reconstruction. It is shown here that this wavefront information can be used to improve the effective optical transfer function of compensated imaging systems when this information is used in a post processing scheme. This technique is closely related to self referenced speckle holography, and is referred to here as compensated self referenced speckle holography. This method requires good spatial sampling of the residual phase error and degrades gracefully as the wavefront sensor signal-to-noise ratio decreases. The technique can be used to reduce the required closed loop bandwidth of an imaging system, allowing longer integration times in the wavefront sensor, and allowing dimmer objects to be imaged without the use of an artificial guide star.

Deconvolution from wavefront sensing (or self-referenced speckle holography) has previously been proposed as a post-detection processing technique for correcting turbulence-induced wavefront phase-errors in incoherent imaging systems. In this paper, a new methodology is considered for processing the image and wavefront-sensor data in which the method of maximum-likelihood estimation is used to simultaneously estimate the object intensity and phase errors directly from the detected images and wavefront-sensor data. This technique is demonstrated to work well in a situation for which the wavefront sensor's lenslet diameters are such that their images are not simply spots of light translated according to the local slope of the phase errors, but are instead an array of small, interfering speckle patterns.

Astronomical speckle holographic methods are shown to calibrate the image blurring effects of the large fraction of the energy in the side-lobes of the point spread function of a dilute aperture imaging system. This self calibration method works for imagery which contains a local point-like reference within the partially isoplanatic field of view. The reference may be a physical object within the (partially isoplanatic) field of view or it may be reconstructed by iterative deconvolution. Data reductions with an iterative deconvolution algorithm show even more striking performance than speckle holography. Atmospheric modeling was used to simulate multiple observations of the same target object with a 5 m dilute aperture pupil with different point spread functions. The iterative deconvolution algorithm recovers Fourier interpolated results for the equivalent 25 m filled aperture without requiring independent observations of a point-like reference source.

Astronomers have long applied post detection processing to achieve diffraction limited images through atmospheric turbulence. We investigate the extension of these methods to the daylight astronomy problem, and to the problem of satellite imaging. After establishing radiometry and detection limits, we analyze white light speckle imaging in the stringent satellite imaging case. We then analyze the speckle holography (wavefront sensor deconvolution) imaging technique and compare these two methods. We also simulate images to compare performances, and present astronomical data obtained during daylight conditions for both methods.

Wave-Front Sensor Deconvolution is a computer post processing technique that uses wave- front sensor information to correct for the effect of atmospheric turbulence. An approximate analytic expression for the signal to noise ratio (SNR) of the technique has been derived for the full order case of one or more Hartmann sub-apertures per r0. This expression is not valid for cases of fewer than one sub-aperture per r0. In these cases wave-front simulation can be used to numerically evaluate an SNR.

We propose an optimized design for the redeployment of the Multiple Mirror Telescope (MMT) primary optics as an Adaptive Steerable Imaging Array (ASIA) providing direct focal plane image formation. The design incorporates concepts of active telescope alignment and adaptive optics proven in the existing MMT. Existing 25 m radio telescope mount technology could easily be adapted to provide support of the optics with tracking stability, against wind loading, sufficient to allow optical rigidity to be obtained using off-axis stellar references, a possibility only feasible with comounted MMT-like interferometer designs which mitigate the field-of-view limitations of long optical delay-lines. As a result of the Steward Observatory borosilicate honeycomb mirror development program a seventh mirror may also be available, giving the light gathering capability of a 4.8 m aperture but with the resolution of a 25 m aperture, for a facility dedicated to the advancement of interferometric imaging technology and high angular resolution astronomy. By utilizing existing technology and providing weather proofing of the optics on its mount, an expensive telescope enclosure, as well as costly, complex laser metrology and optical delay lines, is avoided in a cost effective installation.

This paper presents a new method for calculating the field angle dependent average OTF of an adaptive optic system and compares this method to calculations based on geometric optics. Geometric optics calculations are shown to be inaccurate due to the diffraction effects created by far-field turbulence and the approximations made in the atmospheric parameters. Our analysis includes diffraction effects and properly accounts for the effect of the atmospheric turbulence scale sizes. We show that for any atmospheric C²_n profile, the actual OTF is always better than the OTF calculated using geometric optics. The magnitude of the difference between the calculation methods is shown to be dependent on the amount of far- field turbulence and the values of the outer scale dimension.

We study the use of a cross-correlation algorithm as a sub-aperture slope estimator in a low light-level adaptive optics scheme for a two-telescope interferometer. Using a computer simulation of a Hartmann-Shack wavefront sensor to compare the algorithm against a conventional intensity-centroid estimator, we find that the cross-correlation algorithm marginally reduces the mean-square slope-measurement error under shot-noise limited conditions and significantly reduces the error in the presence of additive noise. We compare the performance of the two estimators in a two-telescope interferometer scheme by introducing wavefront sensor noise with appropriate variance in a full adaptive optics simulation, calculating measures-of-merit appropriate to interferometric imaging such as fringe visibility and coherent energy.

We present the results of image deconvolution using different techniques and with images obtained during a pupil masking experiment. We discuss the importance and limitation of image deconvolution techniques when dealing with interferometric imaging. Numerical simulations and real data are presented and discussed.

Features smaller than the coherent point spread function of an optical microscope can be detected by looking at the phase of an image rather than simply looking at the reflected intensity. To get an accurate determination of feature size, object reconstruction techniques need to be applied to remove the effects of the optical system. This paper discusses the use of simulated annealing to determine the width and average height of an object feature as well as the defocus and spherical aberration present in the imaging system. Results show that errors of less than 5% are obtained for features as small as 1/6th of the optical resolution of the system and less than 2% for features as small as 1/2 of the resolution. When noise is included in the modeling, errors of 5% are obtained for S/N of 10 and less than 2% for S/N of 100.