David R. Gerwe

Superresolved image reconstruction of images taken through the turbulent atmosphere

Superresolved image reconstruction is demonstrated by use of multiple images taken through atmospheric turbulence under photon-limited conditions. An iterative reconstruction algorithm applies estimate-maximize techniques to a series of short-exposure images of the desired object scene along with the corresponding image sequence of a guide star. Simulations show that estimates of the Fourier components both below and above the diffraction limit are improved at successive iterations. The estimated images give finer detail of the original object than does the diffraction-limited image. Effects of photon-noise levels on restoration performance are investigated, and a modification to the reconstruction algorithm is derived that accounts for the effects of CCD read noise.

Regularization for non-linear image restoration using a prior on the object power spectrum

Image restoration algorithms compensate for blur induced attenuation of frequency components that correspond to fine scale image features. However, for Fourier spatial frequency components with low signal to noise ratio, noise amplification outweighs the benefit of compensation and regularization methods are required. This paper investigates a generalization of the Wiener filter approach developed as a maximum a priori estimator based on statistical expectations of the object power spectrum. The estimate is also required to agree with physical properties of the system, specifically object positivity and Poisson noise statistics. These additional requirements preclude a closed form expression. Instead, the solution is determined by an iterative approach. Incorporation of the additional constraints results in significant improvement in the mean square error and in visual interpretability. Equally important, it is shown that the performance has weak sensitivity to the weight of the prior over a large range of SNR values, blur strengths, and object morphology, greatly facilitating practical use in an operational environment.

Cramer-Rao lower bound and object reconstruction performance evaluation for intensity interferometry

This paper addresses the fundamental performance limits of object reconstruction methods using intensity interferometry measurements. It shows examples of reconstructed objects obtained with the FIIRE (Forward-model Interferometry Image Reconstruction Estimator) code developed by Boeing for AFRL. It considers various issues when calculating the multidimensional Cramér-Rao lower bound (CRLB) when the Fisher information matrix (FIM) is singular. In particular, when comparing FIIRE performance, characterized as the root mean square difference between the estimated and pristine objects with the CRLB, we found that FIIRE performance improved as the singularity became worse, a result not expected. We found that for invertible FIM, FIIRE yielded lower root mean squared error than the square root of the CRLB (by a factor as large as 100). This may be due to various regularization constraints (positivity, support, sharpness, and smoothness) included in FIIRE, rendering it a biased estimator, as opposed to the unbiased CRLB framework used. Using the sieve technique to mitigate false high frequency content inherent in point-by-point object reconstruction methods, we also show further improved FIIRE performance on some generic objects. It is worth noting that since FIIRE is an iterative algorithm searching to arrive at an object estimate consistent with the collected data and various constraints, an initial object estimate is required. In our case, we used a completely random initial object guess consisting of a 2-D array of uniformly distributed random numbers, sometimes multiplied with a 2-D Gaussian function.

Cramer-Rao bound analysis of target characterization accuracy limits for imaging systems

A methodology for analyzing an imaging sensors ability to assess target properties is developed. By applying Cramer- Rao covariance analysis to a statistical model relating the sensor measurements to the target, a bound on the accuracy with which target properties can be estimated can be calculated. Such calculations are important in understanding how a sensors design effects its performance for a given assessment task, and in performing feasibility studies or trade studied between sensor designs and sensing modalities. A novel numerical model relating a sensors measurements to a targets three-dimensional geometry is developed in order to overcome difficulties in accurately performing the required numerical computations. An example use of the approach is presented in which the influence of viewing perspective on orientation accuracy limits is analyzed. The example is also used to examine the potential for improving the accuracy bound by fusing multi-perspective data.

Comparison of maximum-likelihood image and wavefront reconstruction using conventional image, phase diversity, and lenslet diversity data

An image reconstruction approach is developed that makes joint use of image sequences produced by a conventional imaging channel and a Shack-Hartmann (lenslet) channel. Iterative maximization techniques are used to determine the reconstructed object that is most consistent with both the conventional and Shack-Hartmann raw pixel-level data. The algorithm is analogous to phase diversity, but with the wavefront diversity provided by a lenslet array rather than a simple defocus. The log-likelihood cost function is matched to the Poisson statistics of the signal and Gaussian statistics of the detector noise. Addition of a cost term that encourages the estimated object to agree with a priori knowledge of an ensemble averaged power spectrum regularizes the reconstruction. Techniques for modeling FPA sampling are developed that are convenient for performing both the forward simulation and the gradient calculations needed for the iterative maximization. The model is computationally efficient and accurately addresses all aspects of the Shack-Hartmann sensor, including subaperture cross-talk, FPA aliasing, and geometries in which the number of pixels across a subaperture is not an integer. The performance of this approach is compared with multi-frame blind deconvolution and phase diversity using simulations of image sequences produced by the visible band GEMINI sensor on the AMOS 1.6 meter telescope. It is demonstrated that wavefront information provided by the second channel improves image reconstruction by avoiding the wavefront ambiguities associated with multiframe blind deconvolution and to a lesser degree, phase diversity.

Closed-Loop Control for Adaptive Optics Wavefront Compensation in Highly Scintillated Conditions

Conventional adaptive optics methods for controlling the phase control element based on least-squares reconstructions of the measured residual phase error exhibit poor performance as scintillation becomes strong. This paper compares the performance of various closed-loop control methods for different phase sensor types (self-referencing interferometer, shearing, and Shack-Hartmann interferometers), and for both conventional and segmented piston-type deformable mirrors (DMs). Significant performance improvements are demonstrated using a weighted least-squares reconstructor that adaptively optimizes the weights at each frame based on the intensities associated with each phase difference measurement and their sums around closed loops. Although the reconstructors considered do not explicitly place branch-cuts in the reconstructed residual phase, branch-cut like features can appear in both the single frame reconstructions and the closed-loop actuator commands. It is also found that at higher Rytov numbers, segmented piston-type DMs outperform conventional deformable mirrors. It is believed that conventional DMs suffer a fitting error associated with branch cuts in the actuator commands that the piston-type DMs are immune to. Performance trends corresponding to self-referencing interferometers provides a useful benchmark since, unlike Shack-Hartmann and shearing interferometers, the phase measurements are not corrupted by scintillation effects.

Phase diverse wavefront estimation and image restoration with a magnification change between imaging channels

Ideally phase diversity determines the object and wavefront that are consistent with two images taken identically except that the wavefront of the diversity channel is perturbed by a known additive aberration. In practice other differences may occur such as image rotation, magnification, changes in detector response, and non-common image motion. This paper develops a mathematical forward model for addressing magnification changes and a corresponding maximumlikelihood implementation of phase diversity. Performance using this physically correct forward model is compared with the more simple approach of resampling the data of the diversity channel.

Image restoration of multiple noisy images by use of a priori knowledge of the anisoplanatic point-spread function

We propose and simulate an iterative maximum-likelihood image reconstruction method that uses multiple images taken through the turbulent atmosphere under photon-limited conditions. The method assumes that the point-spread function corresponding to each image has been measured by a guide star or other technique. Although a guide star is still required, wave-front correction is achieved through postprocessing instead of by the mechanical methods currently used with adaptive optics. A restored image of greatly increased quality was found to be produced after only 10 iterations. Additional iterations continued to reduce the mean-square error.

Double-passage imaging through a random phase screen

We evaluate the time-averaged double-passage image of a coherently illuminated object that is obscured by a random phase screen. In particular, we study the effect of changing the location of the random screen on the average intensity spectrum of the image. We consider two cases, that when the random screen is at an arbitrary location between the pupil plane of the imaging system and the object plane and that when it is located right next to the object. In both cases we find that the average intensity spectrum of the image is diffraction limited. The results are proved analytically and are supported by numerical calculations.

Speckle imaging of coherent sources

Two methods for restoration of images of coherently illuminated objects or coherent sources using multiple short-exposure images are proposed. The first method is based on the same concepts as the Knox-Thompson technique used for incoherent imaging. Sampling resolution requirements are calculated as a function of turbulence strength and lens size. The second method uses a probability maximization technique to form an estimate of the object based on the images taken and knowledge of the statistical nature of atmospheric turbulence. Both techniques require phase information of the electric field in the image plane. Methods of obtaining this information are also discussed.