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We report a multiframe blind deconvolution algorithm that we have developed for imaging through the atmosphere. The algorithm has been parallelized to a significant degree for execution on high-performance computers, with an emphasis on distributed-memory systems so that it can be hosted on commodity clusters. As a result, image restorations can be obtained in seconds to minutes. We have compared and quantified the quality of its image restorations relative to the associated Cramér-Rao lower bounds (when they can be calculated). We describe the algorithm and its parallelization in detail, demonstrate the scalability of its parallelization across distributed-memory computer nodes, discuss the results of comparing sample variances of its output to the associated Cramér-Rao lower bounds, and present image restorations obtained by using data collected with ground-based telescopes.

Unbiased Cramér-Rao lower bound (CRB) theory can be used to calculate lower bounds to the variances of unbiased estimates of a set of parameters given only the probability density function of a random vector conditioned on the true parameter values. However, when the estimated parameter values are required to satisfy inequality constraints such as positivity, the resulting estimator is typically biased. To calculate CRBs for biased estimates of the parameter values, an expression for the bias gradient matrix must also be known. Unfortunately, this expression often does not exist. Because expressions for biased CRBs are preferable to sample variance calculations, alternative methods for deriving biased CRB expressions associated with inequality constraints are needed. We present an alternative approach that is based upon creating the probability density function associated with a given biased estimate of these parameters using the available knowledge of the estimator properties. We apply this approach to the calculation of biased CRBs for estimators that use a positivity constraint with and without a support constraint for a specific measurement model and discuss the benefits and limitations of this approach.

A hybrid technique to simulate the imaging of space-based objects through cirrus clouds is presented. The method makes use of standard Huygens–Fresnel propagation beyond the cloud boundary and a novel vector trace approach within the cloud. At the top of the cloud, the wave front is divided into an array of input gradient vectors, which are in turn transmitted through the cloud model by use of the Coherent Illumination Ray Trace and Imaging Software for Cirrus. At the bottom of the cloud, the output vector distribution is used to reconstruct a wave front that continues propagating to the ground receiver. Images of the object as seen through cirrus clouds with different optical depths are compared with a diffraction-limited image. Turbulence effects from the atmospheric propagation are not included.

Optical diffusion tomography is a technology that is employed to obtain images of the heterogeneous nature of turbid media by using optical radiation. Noise ultimately limits the achievable spatial resolution in these reconstructed images; therefore it is of interest to develop signal-to-noise-ratio expressions that relate spatial resolution in the images to the underlying system and material properties. In this study, Fourier-domain signal-to-noise-ratio expressions are derived for two types of optical diffusion tomography systems: those that use amplitude-modulated illumination sources and those that use continuous-wave illumination sources. The signal-to-noise-ratio expressions are compared for these two types of systems and are validated by laboratory data.

The role that deconvolution plays in the achievable spatial resolution in optical diffusion tomography is examined for the case of imaging an inhomogeneity in an otherwise homogeneous medium. It is shown that, in the measured data, it is the shape of the turbid medium modulation transfer function that determines the maximum spatial resolution. When the turbid medium transfer function is deconvolved from the measured data, it is the signal-to-noise ratio properties of the Fourier-transformed measured data that determine the maximum spatial resolution. It is shown that deconvolution-based methods can improve the spatial resolution in measured data up to a factor of 5 for realistic noise levels. These results are demonstrated with computer-simulated data.

We report on the first satellite feature reconstruction, to our knowledge, by use of range-resolved reflective tomography techniques from nonimaging laser radar data collected on an orbiting satellite. The reconstructed features were two retroreflectors mounted on a satellite. The data were collected with a coherent laser radar system located at the Maui Space Surveillance Site in Maui, Hawaii. We utilized a filtered backpropagation algorithm to reconstruct the images using intensity projections created from the raw data. Key components of the image reconstruction process included the calculation of tomographic angles, the alignment of the intensity projections to an appropriate center of rotation, and compensation for the lack of data for a full 360° set of views.

A new technique for modeling image transfer through cirrus clouds is presented. The technique uses a ray trace to model beam propagation through a three-dimensional volume of polydisperse, hexagonal ice crystals. Beyond the cloud, the technique makes use of standard Huygens–Fresnel propagation methods. At the air–cloud interface, each wave front is resolved into a ray distribution for input to the ray trace software. Similarly, a wave front is reconstructed from the output ray distribution at the cloud–air interface. Simulation output from the ray trace program is presented and the modulation transfer function for stars imaged through cirrus clouds of varying depths is discussed.

We extend the backpropagation algorithm of standard diffraction tomography to backpropagation in turbid media. We analyze the behavior of the backpropagation algorithm both for a single-view geometry, as is common in mammography, and for multiple views. The most general form of the algorithm permits arbitrary placement of sources and detectors in the background medium. In addition, we specialize the algorithm for the case of a planar array of detectors, which permits the backpropagation algorithm to be implemented with fast-Fourier-domain noniterative algebraic methods. In this case the algorithm can be used to reconstruct three-dimensional images in a minute or less, depending on the number of views. We demonstrate the theoretical results with computer simulations.

We extend our previously developed diffraction tomography model of diffuse photon density wave propagation in turbid media to analyze the forward problem associated with detecting and resolving both absorptive and scattering inhomogeneities. Our results assume that detection occurs in a plane but no restrictions are placed on the illumination source geometry. We then specialize these results to plane-wave illumination and derive the turbid media version of the Fourier diffraction theorem. We also develop a shot-noise-limited Fourier-domain signal-to-noise-ratio (SNR) expression to determine how background, system, and inhomogeneity parameters affect one’s ability to detect and resolve inhomogeneities. We show that, in general, scattering inhomogeneities are more easily resolved than absorbing inhomogeneities. We also show that lower temporal modulation frequencies enhance one’s ability to detect and resolve inhomogeneities. These theoretical results are compared with previously published image-domain SNR results, and qualitative agreement is demonstrated.

The use of support constraints for improving the quality of Fourier spectra and their associated images is discussed as well as the relationship between the two domains. Theoretical relationships are derived that 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 that 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.

Ultimately, the band-limited nature of imaging systems restricts image quality in measured data. However, prior knowledge can be employed to improve image quality beyond that available from measured data. Kinds of prior knowledge include knowledge of the support of the object and knowledge that the object has only positive intensities. In previous work it has been shown that prior knowledge increases image quality by two means: superresolution and improvements in the signal-to-noise ratio in the Fourier domain. However, after prior knowledge is enforced, the resulting filter that multiplies the Fourier data may unduly limit resolution in the constrained image. Here maximum achievable resolutions are derived for one- and two-dimensional filters. In addition, it is shown that requiring a signal to be positive results in lowering its maximum achievable resolution by as much as a factor of 2. As a result, algorithms that use positivity to improve the quality of Fourier-domain data may benefit from a final postprocessing step to increase resolution.

Two new weighted-least-squares algorithms for magnitude-spectrum estimation from the bispectrum are presented. The performance of these algorithms is compared both with direct-magnitude-spectrum estimation and with several previously proposed algorithms for magnitude-spectrum estimation from the bispectrum. It is shown that the new algorithms result in more accurate magnitude-spectrum estimates than these other approaches.

A new weighted-least-squares algorithm for phase estimation from the bispectrum is presented. The algorithm uses phasors instead of phases to avoid modulo-2π ambiguities present in the bispectrum phases. The performance of this algorithm is compared with the performances of several other previously proposed algorithms from both simulated and field data. It is shown that the new algorithm results in more accurate phase-spectrum estimates than the other approaches.

The Air Force Phillips Laboratory is in the process of demonstrating an advanced space surveillance capability with a heterodyne laser radar (ladar) system. Notable features of this ladar system include its narrow micropulses (<1.5 ns) contained in a pulse-burst wave form that allows high-resolution range data to be obtained and its high power (30 J/pulse burst) that permits reasonable signal returns from satellites. Recently, time-resolved image-domain signal-to-noise ratios have been derived for both the intensity projections calculated from the range-resolved reflective data and image information obtained with linear combinations of the projections. In this paper the signal-to-noise expression for intensity projections is validated by laboratory data.

Speckle imaging techniques have been shown to mitigate atmospheric-resolution limits, allowing near-diffraction-limited images to be reconstructed. Few images of extended objects reconstructed by use of these techniques have been published, and most of these results are for relatively bright objects. We present image reconstructions of an orbiting Molniya 3 spacecraft from data collected by use of a 2.3-m ground-based telescope. The apparent brightness of the satellite was 15th visual magnitude. Power-spectrum and bispectrum speckle imaging techniques are used prior to image reconstruction to ameliorate atmospheric blurring. We discuss how these images, although poorly resolved, can be used to provide information on the satellite’s functional status. It is shown that our previously published optimal algorithms produce a higher-quality image than do conventional speckle imaging methods.

Previous analyses have predicted that improved power-spectrum estimation results from application of speckle-imaging postprocessing to compensated astronomical images. We report the first results, to our knowledge, of compensated-speckle-imaging experiments, conducted at a compensated telescope operated by the U.S. Air Force, that confirm these predictions. The power-spectrum signal-to-noise ratio is used as the metric for evaluating the performance. We report the results of power-spectrum estimation for a single star and three binary stars, and we reconstruct images of the binary stars using the bispectrum method to obtain the Fourier phase of the object. Compensated and uncompensated results are compared. A previously derived expression that expresses the power-spectrum signal-to-noise ratio in terms of the compensated optical transfer function statistics and object parameters is verified by experimental data.

The use of predetection compensation for the effects of atmospheric turbulence combined with postdetection image processing for imaging applications with large telescopes is addressed. Full and partial predetection compensation with adaptive optics is implemented by varying the number of actuators in the deformable mirror. The theoretical expression for the single-frame power spectrum signal-to-noise ratio (SNR) is reevaluated for the compensated case to include the statistics of the compensated optical transfer function. Critical to this analysis is the observation that the compensated optical transfer function does not behave as a circularly complex Gaussian random variable except at high spatial frequencies. Results from a parametric study of performance are presented to demonstrate improvements in power spectrum estimation for both point sources and an extended object and improvements in the Fourier phase spectrum estimation for an extended object. Full compensation is shown to provide a large improvement in the power spectrum SNR over the uncompensated case, while successively smaller amounts of predetection compensation provide smaller improvements, until a low degree of compensation gives results essentially identical to those of the uncompensated case. Three regions of performance were found with respect to the object Fourier phase spectrum estimate obtained from bispectrum postprocessing: (1) the fully compensated case in which bispectrum postprocessing provides no improvement in the phase estimate over that obtained from a fully compensated long-exposure image, (2) a partially compensated regime in which applying bispectrum postprocessing to the compensated images provides a phase spectrum estimation superior to that of the uncompensated bispectrum case, and (3) a poorly compensated regime in which the results are essentially indistinguishable from those of the uncompensated case. Accurate simulations were used to obtain some parameters for the power spectrum SNR analysis and to obtain the Fourier phase spectrum results.

Range-resolved reflective tomography has been shown to provide useful two-dimensional image reconstructions from a collection of one-dimensional laser radar returns. In this paper, we explore quantitatively the usefulness of this technique to provide two-dimensional images of space objects illuminated by a ground-based laser radar. We do this by developing image domain signal-to-noise ratios (SNRs) both for the intensity projections calculated from the range-resolved data and for image information obtained using linear combinations of the projections. This includes range-resolved reflective tomography as a special case. The two noise sources considered are photon and laser speckle noise. It is shown that the SNR of a projection is bounded from above by one due to laser speckle and goes to zero as the signal level goes to zero, while the reconstructed image typically has SNRs well above one. The SNR expressions are validated using computer simulations.

A new approach to three-dimensional tumor localization in turbid media with the use of measurements in a single plane is presented. Optical diffuse photon-density waves are used to probe the turbid medium. Relative amplitudes and phases are measured in the detection plane. Lateral localization is accomplished in the detection plane. With a Fourier optics approach, the scattered wave is reconstructed throughout the volume to provide depth localization. Computer-simulation results that validate this technique are presented. Applications of this technique to multiple tumors and to optical mammography are discussed.

The use of support constraints for noise reduction in images obtained with telescopes that use adaptive optics for atmospheric correction is discussed. Noise covariances are derived for these type of data, including the effects of photon noise and CCD read noise. The effectiveness of support constraints in achieving noise reduction is discussed in terms of these noise properties and in terms of the types of algorithms used to enforce the support constraint. Both a convex-projections and a cost-function minimization algorithm are used to enforce the support constraints, and it is shown with the use of computer simulations and field data that the cost-function algorithm results in artifacts in the reconstructions. The convex-projections algorithms produced mean-square-error decreases in the image domain of approximately 10% for high light levels but essentially no error decreases for low light levels. We emphasize images that are well resolved by the telescope and adaptive-optics system.

We present a model of the forward problem for diffuse photon-density waves in a turbid medium containing a spherical inhomogeneity, using diffraction tomographic (DT) methods. Recently, by use of DT methods, the forward problem was investigated [Opt.??Express1, 6 (1997); www.osa.org] assuming weak perturbations from the background medium but an arbitrary inhomogeneity structure. We apply DT concepts to a forward problem solution that permits strong perturbations but requires a spherical inhomogeneity. We show that this model is consistent with previous DT results and discuss the application of this new model.

We present a model of the forward problem for diffuse photon-density waves in a turbid medium containing a spherical inhomogeneity, using diffraction tomographic (DT) methods. Recently, by use of DT methods, the forward problem was investigated [Opt. Express 1, 6 (1997); www.osa.org] assuming weak perturbations from the background medium but an arbitrary inhomogeneity structure. We apply DT concepts to a forward problem solution that permits strong perturbations but requires a spherical inhomogeneity. We show that this model is consistent with previous DT results and discuss the application of this new model.