Speckle masking in astronomy: triple correlation theory and applications

Super-resolution imaging through scattering media based on improved triple correlation recursion and deterministic iterative estimation

Iterative phase retrieval algorithms are commonly used in computational techniques and optimization methods to obtain the reconstruction of objects hidden behind opaque scattering media. However, these methods are susceptible to converging to incorrect local minima, and the calculation results tend to be unstable. In this paper, a triple-correlation-based super-resolution imaging (TCSI) framework is proposed to achieve single-shot imaging of unknown objects hidden behind the scattering medium. The amplitude spectrum of the object is obtained by a speckle correlation (SC) method. Iterative relaxation recursion (IRR) sufficiently extracts object information from the triple correlation (TC) of the speckle patterns, serving as the prior initial guess for the iterative estimation algorithm (IE) to obtain a deterministic phase spectrum. Blur correction (BC) is then applied to the diffraction-limited image to achieve super-resolution imaging. Experimental results demonstrate that the flexible framework could effectively overcome the influence of speckle resolution and outperform traditional methods in terms of performance. Our approach provides a basis for non-invasively visualizing various samples behind scattering media.

Ab initio spatial phase retrieval via intensity triple correlations

Second-order intensity correlations from incoherent emitters can reveal the Fourier transform modulus of their spatial distribution, but retrieving the phase to enable completely general Fourier inversion to real space remains challenging. Phase retrieval via the third-order intensity correlations has relied on special emitter configurations which simplified an unaddressed sign problem in the computation. Without a complete treatment of this sign problem, the general case of retrieving the Fourier phase from a truly arbitrary configuration of emitters is not possible. In this paper, a general method for ab initio phase retrieval via the intensity triple correlations is described. Simulations demonstrate accurate phase retrieval for clusters of incoherent emitters which could be applied to imaging stars or fluorescent atoms and molecules. With this work, it is now finally tractable to perform Fourier inversion directly and reconstruct images of arbitrary arrays of independent emitters via far-field intensity correlations alone.

Recursion-driven bispectral imaging for dynamic scattering scenes

Imaging dynamic strongly scattering scenes remains a significant challenge because it is typically believed that moving objects and dynamic media provide huge barriers. Instead, we use the dynamics of objects and media and put forward a recursion-driven bispectral imaging (ReDBI) framework here for the reconstruction of a stationary or moving object hidden behind the dynamic media. ReDBI avoids the errors introduced by speckle modulation and phase-retrieval algorithms in the existing studies. We also quantitatively assess the reconstruction difficulty of character and shape objects with the benchmark of the minimum number of speckle images (MNSI) required to achieve a high-quality reconstruction, which can help to comprehend the media's transfer properties.

Image-plane self-calibration in interferometry

We develop a process of image-plane self-calibration for interferometric imaging data. The process is based on shape-orientation-size (SOS) conservation for the principal triangle in an image generated from the three fringes made from a triad of receiving elements, in situations where interferometric phase errors can be factorized into element-based terms. The basis of the SOS conservation principle is that, for a three-element array, the only possible image corruption due to an element-based phase screen is a tilt of the aperture plane, leading to a shift in the image plane. Thus, an image made from any three-element interferometer represents a true image of the source brightness, modulo an unknown translation. Image-plane self-calibration entails deriving the unknown translations for each triad image via cross-correlation of the observed triad image with a model image of the source brightness. After correcting for these independent shifts, and summing the aligned triad images, a good image of the source brightness is generated from the full array, recovering source structure at diffraction-limited resolution. The process is iterative, using improved source models based on previous iterations. We demonstrate the technique in a high signal-to-noise context, and include a configuration based on radio astronomical facilities, and simple models of double sources. We show that the process converges for the simple models considered, although convergence is slower than for aperture-plane self-calibration for large-N arrays. As currently implemented, the process is most relevant for arrays with a small number of elements. More generally, the technique provides geometric insight into closure phase and the self-calibration process. The technique is generalizable to non-astronomical interferometric imaging applications across the electromagnetic spectrum.

Tuning Axial Resolution Independent of Lateral Resolution in a Computational Imaging System Using Bessel Speckles

Speckle patterns are formed by random interferences of mutually coherent beams. While speckles are often considered as unwanted noise in many areas, they also formed the foundation for the development of numerous speckle-based imaging, holography, and sensing technologies. In the recent years, artificial speckle patterns have been generated with spatially incoherent sources using static and dynamic optical modulators for advanced imaging applications. In this report, a basic study has been carried out with Bessel distribution as the fundamental building block of the speckle pattern (i.e., speckle patterns formed by randomly interfering Bessel beams). In general, Bessel beams have a long focal depth, which in this scenario is counteracted by the increase in randomness enabling tunability of the axial resolution. As a direct imaging method could not be applied when there is more than one Bessel beam, an indirect computational imaging framework has been applied to study the imaging characteristics. This computational imaging process consists of three steps. In the first step, the point spread function (PSF) is calculated, which is the speckle pattern formed by the random interferences of Bessel beams. In the next step, the intensity distribution for an object is obtained by a convolution between the PSF and object function. The object information is reconstructed by processing the PSF and the object intensity distribution using non-linear reconstruction. In the computational imaging framework, the lateral resolution remained a constant, while the axial resolution improved when the randomness in the system was increased. Three-dimensional computational imaging with statistical averaging for different cases of randomness has been synthetically demonstrated for two test objects located at two different distances. The presented study will lead to a new generation of incoherent imaging technologies.

Phase-Sensitive Measurements of Depth-Dependent Signal Transduction in the Inner Plexiform Layer

Non-invasive spatially resolved functional imaging in the human retina has recently attracted considerable attention. Particularly functional imaging of bipolar and ganglion cells could aid in studying neuronal activity in humans, including an investigation of processes of the central nervous system. Recently, we imaged the activity of the inner neuronal layers by measuring nanometer-size changes of the cells within the inner plexiform layer (IPL) using phase-sensitive optical coherence tomography (OCT). In the IPL, there are connections between the neuronal cells that are dedicated to the processing of different aspects of the visual information, such as edges in the image or temporal changes. Still, so far, it was not possible to assign functional changes to single cells or cell classes in living humans, which is essential for studying the vision process. One characteristic of signal processing in the IPL is that different aspects of the visual impression are only processed in specific sub-layers (strata). Here, we present an investigation of these functional signals for three different sub-layers in the IPL with the aim to separate different properties of the visual signal processing. Whereas the inner depth-layer, closest to the ganglion cells, exhibits an increase in the optical path length, the outer depth-layer, closest to the bipolar cell layer, exhibits a decrease in the optical path length. Additionally, we found that the central depth is sensitive to temporal changes, showing a maximum response at a stimulation frequency of around 12.5 Hz. The results demonstrate that the signals from different cell types can be distinguished by phase-sensitive OCT.

Shift-and-add image processing incorporated with the unsharp masking method

Shift-and-add (SAA) is a simple image processing procedure. SAA was devised to reconstruct a diffraction-limited image from atmospherically degraded stellar images. Recently SAA has been applied to biological imaging. There are several variants of SAA. Here proposed is an SAA procedure incorporated with unsharp masking (USM). The SAA procedure proposed here encompasses an extended version of USM. The proposed SAA method retains the simplicity and easiness, and the basic features of SAA. The effectiveness of the proposed method is examined by restoring atmospherically degraded solar images. It is shown that the USM SAA reconstructed image exhibits high contrast and reveals fine structures blurred by atmospheric turbulence. It is also shown that the USM SAA performs better with a data frame selection scheme.

Single-shot multi-view imaging enabled by scattering lens

Imaging three-dimensional (3D) objects has been realized by methods such as binocular stereo vision and multi-view imaging. These methods, however, needs multiple cameras or multiple shots to get elemental images. In this paper, we develop a single-shot multi-view imaging technique by utilizing the natural randomness of scattering media. By exploiting the memory effect and uncorrelated point spread functions (PSF) among scattering media, we demonstrate that both stereo imaging with large disparity and up to seven-view imaging of a 3D object can be reconstructed from only one speckle pattern by deconvolution. The elemental images are consistent with 3D object projection and images taken by multi-shot imaging. Our technique provides a feasible method to capture multi-view imaging with short acquisition time and easy calibration.

Fast diffraction-limited image recovery through turbulence via subsampled bispectrum analysis

Imaging through temporally changing aberrations is a common challenge that can be found in many different fields such as astronomy, long-range surveillance, and deep tissue bioimaging. Based on the notions originally developed in speckle interferometry, time-varying aberrations can be used to our advantage to obtain diffraction-limited resolution images through turbulence via bispectrum analysis. However, due to the heavy computational load brought on by the triple correlation and the phase extraction process, widespread use has been limited. Here, we demonstrate a Fourier domain subsampling scheme that can accelerate the speed of bispectrum analysis by more than 2 orders of magnitude. In contrast to other approaches for parallelization such as those based on Radon transform or image segmentation, our proposed method enables diffraction-limited imaging without suffering from resolution loss or image artifacts.

Phase-sensitive interferometry of decorrelated speckle patterns

Phase-sensitive coherent imaging exploits changes in the phases of backscattered light to observe tiny alterations of scattering structures or variations of the refractive index. But moving scatterers or a fluctuating refractive index decorrelate the phases and speckle patterns in the images. It is generally believed that once the speckle pattern has changed, the phases are scrambled and any meaningful phase difference to the original pattern is removed. As a consequence, diffusion and tissue motion that cannot be resolved, prevent phase-sensitive imaging of biological specimens. Here, we show that a phase comparison between decorrelated speckle patterns is still possible by utilizing a series of images acquired during decorrelation. The resulting evaluation scheme is mathematically equivalent to methods for astronomic imaging through the turbulent sky by speckle interferometry. We thus adopt the idea of speckle interferometry to phase-sensitive imaging in biological tissues and demonstrate its efficacy for simulated data and imaging of photoreceptor activity with phase-sensitive optical coherence tomography. We believe the described methods can be applied to many imaging modalities that use phase values for interferometry.

Proof of concept for adaptive sequential optimization of free-space communication receivers

In a downlink scenario, the performance of laser satellite communications is limited due to atmospheric turbulence, which causes fluctuations in the intensity and the phase of the received signal, leading to an increase in bit error probability. In principle, a single-aperture phase-compensated receiver, based on adaptive optics, can overcome atmospheric limitations by adaptive tracking and correction of atmospherically induced aberrations. However, under strong turbulence situations, the effectiveness of traditional adaptive optics systems is severely compromised. We have developed an alternative intensity-based technique that corrects the wavefront by iteratively updating the phases of individual focal-plane speckles, which maximizes the power coupled into a single-mode fiber. Here, we present the proof of concept for this method. We show how this technique offers around 4 dB power gain with fewer than 60 power measurements under strong turbulence conditions. It delivers a good performance in different turbulent regimes, and it shows robustness against severe deterioration of the signal-to-noise ratio.

Partially coherent microscope in phase space

Explicit expressions are presented for different phase-space representations (mutual intensity, Wigner distribution function, and ambiguity function) of the partially coherent image wave field in a microscope system. These are separated into system- and object-dependent parts. The partially coherent image in phase space can be described in terms of different 6D system-dependent kernels, all Fourier transforms of the system mutual spectrum, the region of overlap of two displaced objective pupils and the effective source. The image intensity can be expressed in terms of a 4D kernel, the convolution in spatial frequency of the source, and the Wigner distribution function of the objective pupil, given by a marginal of, or a section through, the respective phase-space kernels.

Imaging through a thin scattering layer and jointly retrieving the point-spread-function using phase-diversity

Recently introduced angular-memory-effect based techniques enable non-invasive imaging of objects hidden behind thin scattering layers. However, both the speckle-correlation and the bispectrum analysis are based on the statistical average of large amounts of speckle grains, which determines that they can hardly access the important information of the point-spread-function (PSF) of a highly scattering imaging system. Here, inspired by notions used in astronomy, we present a phase-diversity speckle imaging scheme, based on recording a sequence of intensity speckle patterns at various imaging planes, and experimentally demonstrate that in addition to being able to retrieve the image of hidden objects, we can also simultaneously estimate the pupil function and the PSF of a highly scattering imaging system without any guide-star nor reference.

Randomized apertures: high resolution imaging in far field

We explore opportunities afforded by an extremely large telescope design comprised of ill-figured randomly varying subapertures. The veracity of this approach is demonstrated with a laboratory scaled system whereby we reconstruct a white light binary point source separated by 2.5 times the diffraction limit. With an inherently unknown varying random point spread function, the measured speckle images require a restoration framework that combine support vector machine based lucky imaging and non-negative matrix factorization based multiframe blind deconvolution. To further validate the approach, we model the experimental system to explore sub-diffraction-limited performance, and an object comprised of multiple point sources.

Hybrid correlation holography with a single pixel detector

Correlation holography reconstructs three-dimensional (3D) objects as a distribution of two-point correlations of the random field detected by two-dimensional detector arrays. Here, we describe a hybrid method, a combination of optical and computational channels, to reconstruct the objects from only a single pixel detector. An experimental arrangement is proposed as a first step to realizing the technique; we have simulated the experimental model for both scalar and vectorial objects. The proposed technique provides depth focusing and 3D reconstruction with digital suppression of unwanted frequency spectrum.

Principles of image reconstruction in optical interferometry: tutorial

This paper provides a general introduction to the problem of image reconstruction from interferometric data. A simple model of the interferometric observables is given, and the issues arising from sparse Fourier data are discussed. The effects of various regularizations are described. In the proposed general framework, most existing algorithms can be understood. For an astronomer, such an understanding is crucial not only for selecting and using an algorithm but also to ensure correct interpretation of the resulting image.

Single-shot diffraction-limited imaging through scattering layers via bispectrum analysis

Recently introduced speckle correlations-based techniques enable noninvasive imaging of objects hidden behind scattering layers. In these techniques, the hidden object Fourier amplitude is retrieved from the scattered light autocorrelation, and the lost Fourier phase is recovered via iterative phase-retrieval algorithms, which suffer from convergence to wrong local minimums solutions and cannot solve ambiguities in object orientation. Here, inspired by notions used in astronomy, we experimentally demonstrate that in addition to Fourier amplitude, the object-phase information is naturally and inherently encoded in the scattered light bispectrum (the Fourier transform of triple correlation) and can also be extracted from a single high-resolution speckle pattern, based on which we present a single-shot imaging scheme to deterministically and unambiguously retrieve diffraction-limited images of objects hidden behind scattering layers.

High speed color imaging through scattering media with a large field of view

Optical imaging through complex media has many important applications. Although research progresses have been made to recover optical image through various turbid media, the widespread application of the technology is hampered by the recovery speed, requirement on specific illumination, poor image quality and limited field of view. Here we demonstrate that above-mentioned drawbacks can be essentially overcome. The realization of high speed color imaging through turbid media is successfully carried out by taking into account the media memory effect, the point spread function, the exit pupil of the optical system, and the optimized signal to noise ratio. By retrieving selected speckles with enlarged field of view, high quality image is recovered with a responding speed only determined by the frame rates of the image capturing devices. The immediate application of the technique is expected to register static and dynamic imaging under human skin to recover information with a wearable device.

Deconvolution of partially compensated solar images from additional wavefront sensing

A technique for restoring solar images partially compensated with adaptive optics is developed. An additional wavefront sensor is installed in an adaptive optics system to acquire residual wavefront information simultaneously to a solar image. A point spread function is derived from the wavefront information and used to deconvolve the solar image. Successful image restorations are demonstrated when the estimated point spread functions have relatively high Strehl ratios.

Restoration of images degraded by underwater turbulence using structure tensor oriented image quality (STOIQ) metric

Recent advances in image processing for atmospheric propagation have provided a foundation for tackling the similar but perhaps more complex problem of underwater imaging, which is impaired by scattering and optical turbulence. As a result of these impairments underwater imagery suffers from excessive noise, blur, and distortion. Underwater turbulence impact on light propagation becomes critical at longer distances as well as near thermocline and mixing layers. In this work, we demonstrate a method for restoration of underwater images that are severely degraded by underwater turbulence. The key element of the approach is derivation of a structure tensor oriented image quality metric, which is subsequently incorporated into a lucky patch image processing framework. The utility of the proposed image quality measure guided by local edge strength and orientation is emphasized by comparing the restoration results to an unsuccessful restoration obtained with equivalent processing utilizing a standard isotropic metric. Advantages of the proposed approach versus three other state-of-the-art image restoration techniques are demonstrated using the data obtained in the laboratory water tank and in a natural environment underwater experiment. Quantitative comparison of the restoration results is performed via structural similarity index measure and normalized mutual information metric.

Least-squares estimation of imaging parameters for an ultrasonic array using known geometric image features

Ultrasonic array images are adversely affected by errors in the assumed or measured imaging parameters. For non-destructive testing and evaluation, this can result in reduced defect detection and characterization performance. In this paper, an autofocus algorithm is presented for estimating and correcting imaging parameter errors using the collected echo data and a priori knowledge of the image geometry. Focusing is achieved by isolating a known geometric feature in the collected data and then performing a weighted leastsquares minimization of the errors between the data and a feature model, with respect to the unknown parameters. The autofocus algorithm is described for the estimation of element positions in a flexible array coupled to a specimen with an unknown surface profile. Experimental results are shown using a prototype flexible array and it is demonstrated that (for an isolated feature and a well-prescribed feature model) the algorithm is capable of generating autofocused images that are comparable in quality to benchmark images generated using accurately known imaging parameters.

Fast and optimal multiframe blind deconvolution algorithm for high-resolution ground-based imaging of space objects

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.

Temporal response of a random medium from speckle intensity frequency correlations

We reconstruct the temporal response of a random medium by using speckle intensity frequency correlations. When the scattered field from a random medium is described by circular complex Gaussian statistics, we show that third-order correlations permit retrieval of the Fourier phase of the temporal response with bispectral techniques. Our experimental results for random media samples in the diffusion regime are in excellent agreement with the intensity temporal response measured directly with an ultrafast pulse laser and a streak camera. Our speckle correlation measurements also demonstrate sensitivity to inhomogeneous samples, highlighting the potential application for imaging within a scattering medium.

Description and simulation of an active imaging technique utilizing two speckle fields: root reconstructors

Quasi-monochromatic light will form laser speckle upon reflection from a rough object. This laser speckle provides information about the shape of the illuminated object. Further information can be obtained if two colors of coherent light are used, provided that the colors are sufficiently close in wavelength that the interference is also measurable. It is shown that no more than two intensities of two speckle patterns and their interference are required to produce an unambiguous band-limited image of an object, to within an overall spatial translation of the image, in the absence of measurement errors and in the case where all roots of both fields and their complex conjugates are distinct. This result is proven with a root-matching technique, which treats the electric fields as polynomials in the pupil plane, the coefficients of which form the desired complex object. Several root-matching algorithms are developed and tested. These algorithms are generally slow and sensitive to noise. So motivated, several other techniques are applied to the problem, including phase retrieval, expectation maximization, and probability maximization in a sequel paper [J. Opt. Soc. Am. A 19, 458 (2002)]. The phase-retrieval and expectation-maximization techniques proved to be most effective for reconstructions of complex objects larger than 10 pixels across.

Temporal response of a random medium from third-order laser speckle frequency correlations

We demonstrate for the first time that the temporal response of a random medium can be obtained from optical intensity fluctuations. Our method uses third-order intensity correlations of measured speckle patterns from a multiple scattering random medium as a function of optical frequency. In particular, our experimental results for the temporal response extracted from third-order intensity correlations are in good agreement with the predictions of a diffusion model. Our results are valid for waves in random media where the scattered field is described by circular complex Gaussian statistics.

Quasi-elastic light scattering for intermittent dynamics

Dynamic light-scattering techniques provide noninvasive probes of diverse media such as colloidal suspensions, granular materials, and foams. Traditional analysis relies on the Gaussian properties of the scattering process found in most experimental situations and uses second-order intensity-correlation functions. This approach fails in the presence of, among other things, the collective intermittent dynamics found in systems such as granular materials. By extending the existing formalism and introducing higher-order intensity-correlation functions, we show how to detect and quantify the intrinsic dynamics and switching statistics of intermittent processes. We then explore two systems: (1) an auger-driven granular column for which the granular dynamics are controlled and the formalism is tested and (2) a granular heap whose dynamics are a priori unknown but may, now, be characterized.

Effect of CCD Readout Noise in Astronomical Speckle Imaging

A theoretical and experimental comparison of photon-countingcameras and CCD's for use in astronomical speckle imaging wasperformed. Photon-counting cameras able to detect single-photonevents typically exhibit a lower quantum efficiency (QE) and suffersaturation effects at high light levels. In contrast, CCD's offera high QE and virtually unlimited photon-count rate. However CCD'sare limited at lower light levels by noise associated with the readoutprocess. Speckle-imaging performance was quantified by derivationof the signal-to-noise ratio (SNR) of the power spectrum and theKnox-Thompson product to include CCD readout noise. Ground-basedtelescope observations at various light levels were obtained with anadvanced, high-speed, low-noise CCD camera to verify SNRexpressions. The useful operating ranges for these two camera typeswere compared by consideration of the effects of QE, readout noise, andmaximum photon-count rate. Although photon-counting camerascontinued to dominate low-light-level applications, CCD's are shown tooffer significant improvements over photon-counting cameras for a widerange of light levels. Future reductions of readout noise will further improve CCD speckle-imaging performance.

Optoelectronic implementation of the triple correlation

The auto triple correlation has several fundamental advantages over the ordinary autocorrelation of second order. We present an optoelectronic processor for the computation of the auto triple correlation.

Projection speckle spectroscopy for a real-time mode

Projected specklegrams are used for spectroscopy with high spatial resolution under atmospheric turbulence. One-dimensional peak tracking allows one to realize high throughput and simple real-time operation for speckle spectroscopy. Results of simulation experiments are presented, and they confirm the usefulness of the proposed method.

Phase recovery from the bispectrum aided by the error-reduction algorithm

An essential step in bispectral imaging is the recovery of the object's Fourier phase from the bispectral phase. Such reconstruction can be performed by recursive or least-squares (minimization) methods. It is generally accepted that least-squares methods perform better because of their higher noise tolerance. Two weighted least-squares minimization schemes are compared, and the use of the error-reduction algorithm is suggested as a way to overcome the stagnation at local minima common to minimization problems.

Minimum entropy-neural network approach to turbulent-image reconstruction

We investigate a neural net-based algorithm for enhanced imaging through atmospheric turbulence. The concept is based on a standard model of optical turbulence, according to which a short-exposure point-spread function is a random superposition of speckles. This leads to a new method of image processing called the Fourier division approach. The latter requires the taking of two short-exposure images in rapid succession, which are picked up by an image-plane array, divided in Fourier space, and then processed by a minimum entropy-neural net approach. The main task of the processing is to estimate the two short-exposure point-spread functions that characterize the two images. Given these estimates, the two images may now be inverse filtered to produce two sharp object-scene estimates. These have most of the turbulence degradation removed, and are averaged to produce a single output image. The approach shows promise, in computer simulations, of removing nearly all of the turbulence degradation very quickly (currently tens of seconds). A further benefit arises from knowledge of the twoshort-exposure point-spread functions. These should permit identification of the state of turbulence along the imaging line of sight and, in particular, the presence of wind shear.

Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions

Simulating the effects of atmospheric turbulence on optical imaging systems is an important aspect of understanding the performance of these systems. Simulations are particularly important for understanding the statistics of some adaptive-optics system performance measures, such as the mean and variance of the compensated optical transfer function, and for understanding the statistics of estimators used to reconstruct intensity distributions from turbulence-corrupted image measurements. Current methods of simulating the performance of these systems typically make use of random phase screens placed in the system pupil. Methods exist for making random draws of phase screens that have the correct spatial statistics. However, simulating temporal effects and anisoplanatism requires one or more phase screens at different distances from the aperture, possibly moving with different velocities. We describe and demonstrate a method for creating random draws of phase screens with the correct space-time statistics for a bitrary turbulence and wind-velocity profiles, which can be placed in the telescope pupil in simulations. Results are provided for both the von Kármán and the Kolmogorov turbulence spectra. We also show how to simulate anisoplanatic effects with this technique.

Widening the effective field of view of adaptive-optics telescopes by deconvolution from wave-front sensing: average and signal-to-noise ratio performance

A fundamental problem of adaptive-optics systems is the very narrow corrected field of view that can be obtained because turbulence is extended in altitude throughout the atmosphere. The correctable field of view is of the order of 5-10 µrad at visible wavelengths and increases as the wavelength increases. Previous concepts to broaden the corrected field of view have been hardware oriented, requiring multiple wave-front sensor (WFS) measurements to control multiple deformable mirrors. We analyze the average and the signal-to-noise-ratio performance of an image measurement and postprocessing technique that uses simultaneous measurements of a short-exposure compensated image measured in an off-axis direction; an additional WFS measurement is taken in the off-axis direction. Results are presented for infinite-altitude WFS beacons driving both the WFS for the adaptive optics and the WFS looking in the off-axis direction, a variety of seeing and WFS light-level conditions, and off-axis angles from two to six times the isoplanatic angle. This technique improves the average effective transfer function out to a field angle of at least six times the isoplanatic angle while providing a higher signal-to-noise ratio in the spatial frequency domain.

Application of bispectral speckle imaging to near-diffraction-limited imaging in the presence of unknown aberrations

A laboratory experiment that demonstrates near-diffraction-limited imaging of a detailed object in the presence of unknown fixed aberrations in the imaging system is described. A random-phase plate is introduced in a pupil plane of the imaging system to eliminate the effect of fixed aberrations in the system. We employ a bispectral speckle imaging technique to recover the object from speckled images affected by both the random-phase fluctuations induced by the random-phase plate and the fixed aberrations present in the imaging system. For the case where the random phase is assumed to obey Gaussian statistics an approximate form of the bispectral speckle transfer function is obtained with an asymptotic expansion. This approximate form of the transfer function shows the diffraction-limited nature of bispectral speckle imaging when the standard deviation of the random-phase fluctuations is of the order of a wavelength of light. Experimental results are presented for fixed aberrations associated with lens tilt and defocus in the imaging system.

Image restoration using the W-slice method

We propose the use of higher order statistics (HOS)-based methods to address the problem of image restoration. The restoration strategy is based on the fact that the phase information of the original image and its HOS are not distorted by some types of blurring. The difficulties associated with the combination of 2-D signals and their HOS are reduced by means of the Radon transform. Two methods that apply the weight-slice algorithm over the projections are developed. Simulation results illustrate the performance of the proposed methods.

Compensated speckle imaging: theory and experimental results

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.

Zero tracks for blind deconvolution of blurred ensembles

The analytically continued Fourier transform of a two-dimensional image vanishes to zero on a two-dimensional surface embedded in a four-dimensional space. This surface uniquely characterizes the image and is known as a zero sheet. An algorithm is described that employs the zero-sheet concept to blindly deconvolve an ensemble of differently blurred images. To overcome the difficulty of operating within a four-dimensional space, we calculate projections of the zero sheets, known as zero tracks. The zero tracks of each member of the ensemble are superimposed on a plane. The zero tracks that pertain to the original image are similar for every blurred and contaminated image. By contrast those associated with the blurring vary widely across the ensemble. A method of selecting the appropriate zero tracks in order to reconstruct an estimate of the original image is presented. Preliminary results for small positive images suggest that this deconvolution technique may be successful even when the level of contamination is significant.

Scale, translation, and rotation invariant orthonormalized optical/optoelectronic neural networks

We use a higher-dimensional version of the one-dimensional scale, translation, and in-plane rotation invariant transforms of Fang and Hausler [Appl. Opt. 29, 704-708 (1990)] in conjunction with an orthonormalization technique in an optical or optoelectronic resonator neural network. The system is tested by computer simulations that use a number of realistic stored and input images. Type-I (in-class discrimination) and type-II (out-of-class discrimination) false-alarm rates for several distortion types as well as results for individual examples of distorted images are presented. Our results indicate that the two-dimensional transforms exhibit considerably lower type-I false-alarm rates than the one-dimensional ones. They also show that such a configuration is capable of identifying a set of diverse inputs with cluttered and noisy backgrounds.

Performance analysis of the self-referenced speckle-holography image-reconstruction technique

Self-referenced speckle holography (SRSH) is a postdetection turbulence-compensation technique for obtaining diffraction-limited imagery from ground-based telescopes degraded by atmospheric turbulence. In SRSH, image-plane information is used together with wave-front distortion information to reconstruct an estimate of the object spectrum. The wave-front distortion information is obtained from a wave-front sensor in the pupil plane of the telescope. This information is used in a postprocessing environment to estimate the point spread function of the combined telescope and atmosphere. The point spread function is then used to obtain an estimate of the object intensity distribution by deconvolution. We present the results of a detailed performance analysis of SRSH. Performance is quantified in terms of a system transfer function and a system point spread function. The results show how the performance of SRSH is dependent on the sampling intervals and shot noise in the wave-front sensor. The results also indicate how the technique, for a given set of design parameters, responds to changing seeing conditions. For wave-front sensor sampling intervals of the order of a Fried coherence cell size r(0) and adequate light levels, SRSH boosts the high spatial frequencies (those near the diffraction limit of the telescope) to nearly 0.6.

Multipurpose analyzers for photoelectron statistics: implementation and use

The statistical properties of variously scattered laser light can be derived from photocount data through the estimate of different functions. Even if the second-order correlation usually plays the main role, other function (e.g., triggered and nontriggered distributions, moments of various order, and higher-order correlations) may give more appropriate results in many experimental conditions. We present a multifunction analyzer whose working principle is based on the acquisition of a long sequence of interpulse intervals (through a circuitally simple personal-computer front-end interface), which is followed by the off-line calculation of one or more of the functions for which an algorithm is available. Up to 5 × 10(5) photopulse intervals can be recorded at a maximum rate of approximately 2 × 10(5) data points/s. A short description of the algorithms used to calculate the different functions is given together with some useful hints and a table of typical processing times.

Waveform estimation from noisy signals with variable signal delay using bispectrum averaging

A technique based on bispectrum averaging is described for generally recovering the signal waveform from a set of noisy signals with variable signal delay. The technique does not require explicit time alignment of signals and any initial estimate of signal. The technique, however, does not yield estimates of the signal position. A comparison is made of two algorithms for recovering the Fourier amplitude and the Fourier phase from an averaged bispectrum. These algorithms are the recursive method and the least squares method. The methods are numerically investigated using computer generated-data and a physiological signal and noise. The advantages and disadvantages of these different algorithms are discussed. Some experimental results for the evoked potential studies that demonstrate the technique are given. The results show the effectiveness of the technique: various potential applications of the technique might be expected.

Linear reconstruction of compensated images: theory and experimental results

Linear image reconstruction techniques are proposed for postprocessing astronomical images measured with compensated imaging systems. Linear techniques use averaging to overcome the effects of noise and deconvolution to remove system effects. Experimental results from compensated image measurements of four single stars and one binary star at visible wavelengths are reported for the first time, to our knowledge, and a previously derived analytic expression relating the statistics of the compensated optical transfer function to the compensated image spectrum signal-to-noise ratio is verified. The performance of deconvolution on a bright binary star with angular subtense previously estimated to be 0.52 arcsec (2.52 microrad) is demonstrated.

Wave-front reconstruction using a Shack-Hartmann sensor

An analysis of the problem of wave-front reconstruction from Shack-Hartmann measurements is presented. The wave-front aberration is assumed to result from passage of the wave front through Kolmogorov turbulence. Limitations of using Zernike polynomials as an orthogonal basis for wave-front reconstruction are highlighted, and the advantage of using the Karhunen-Loeve functions for computing the higher-order modes of the wave front is shown.

Recovery of planetary images by speckle imaging

Limits to the recovery of planetary images by speckle imaging were investigated by means of a numerical simulation of the image-forming process. Laboratory measurements established that the numerical model correctly represented the process. With this numerical model we studied the observing conditions required to obtain useful planetary data for Neptune and Io, and we learned which factors are important for the successful recovery of images. We also showed that our methods of amplitude and phase recovery do not contribute to image degradation, which is essentially caused by photon noise alone. From this conclusion we were led to two simple models of speckle imaging that involve neither details of the recovery procedure nor of the atmospheric behavior (except for the size of the seeing disk). Both models are governed by a dimensionless ratio that involves all parameters relevant to the process of image restoration. The models can be used by any observer, and they have been used by us to predict the performance of speckle imaging for T-tauri stars and for a Mars cloud study.

Speckle imaging of satellites at the U.S. Air Force Maui Optical Station

Results are presented from a series of experiments in which the U.S. Air Force Maui Optical Station's 1.6-m telescope and a bare CCD speckle camera system were used to image satellites at distances of up to 1000 km. A brief overview of the image reconstruction algorithms is presented. The choice of the experiment site and various imaging parameters are described. Power spectra and power spectral signal-to-noise ratio curves that result from imaging several point stars are compared with theory. Reconstructed images of several binary stars are shown as a base-line assessment of our technique. High-quality image reconstructions of an Earth-satellite, the Hubble Space Telescope, are presented. The results confirm that speckle imaging techniques can be used with a bare CCD imaging system to provide a powerful and flexible method for imaging objects of moderate magnitude.