Several proposed techniques of focal plane imaging through turbulence are described and compared. The techniques are: shift-and-add, exponential filter, Knox-Thompson (cross-spectrum) and triple correlation, Particular attention is given to the performance of the methods at extremely low light levels of a few detected photons per frame.
The atmosphere of the earth restricts the resolution of conventional astrophotography to about 1 arcsec. Much higher resolution can be obtained by using speckle methods. The speckle masking method (triple correlation method) yields images of general astronomical objects with diffraction-limited resolution, for example, 0.03 arcsec resolution for a 3.6-m telescope. True images are obtained since speckle masking reconstructs both the modulus and the phase of the object Fourier transform. Therefore, speckle masking is a solution of the phase problem in speckle interferometry. The limiting magnitude of speckle masking is about 20m. Speckle spectroscopy is a speckle method that yields objective prism spectra with diffraction-limited angular resolution. Finally, speckle masking can recon-struct high-resolution images from optical long-baseline interferograms. A 1-km telescope array on the earth would yield images with 10 arcsec resolution. With a 40-km array in space a fantastic resolution of 10-6 arcsec can be achieved at λ-200 nm. We show speckle masking observations of NGC 3603 and Eta Carinae and computer simulations of optical aperture synthesis.
The retrieval of phase from Fourier intensity data makes possible the reconstruction of diffraction-limited images despite severe phase errors or total loss of phase information. Phase retrieval is described in three contexts: (1) for active, coherent imaging, (2) for passive, incoherent imaging and (3) for incoherent imaging from multiple realizations of coherent intensity data.
The 6.86 m Multiple Mirror Telescope (MMT) has been operated as a coherently cophased interferometric imaging array since 1983. PSF observations at wavelengths from 410 to 850 nm reveal characteristics which must be controlled to achieve reliable imaging at resolutions less than 75 nanoradians (0.015 arc seconds). The MMT has been used to acquire differential speckle images of Alpha Orionis in the H-alpha line and a nearby "continuum" bandpass at 656.9 nm. Broadband images of a 12th magnitude geosynchronous communications satellite, from which shape and size measures have been obtained with 100 nm resolution, give evidence of the capability of the MMT in speckle imaging applications. We have recently extended the cophasing capability to infrared wavelengths by providing reflective pathlength compensation capability.
Reconstructed images of the asteroid 4 Vesta reveal dark and bright patterns which can be followed across the disk as the asteroid rotates. We find reasonable agreement with its visible lightcurve by assigning relative albedo 0 to the three darkest features and relative albedo 2 to the three brightest features. From power spectrum analysis of sets of images at two different epochs we obtain Vesta's triaxial principal diameters, the orientation of its pole, and its rotational rate and sense. The images confirm the dominant role of surface structure, particularly the dark spots, in determining vesta's lightcurve signature.
We have applied the bispectrum algorithm to one-dimensional infrared speckle data for retrieval of diffraction limited phases. We use a standard weighting averaging technique to combine the multiple phase estimates contained in the bispectrum. We have also used the bispectrum modulus to obtain visibility amplitudes. Results, including images, are presented for three different binary stars. Simulated data has also been used to study the behaviour of the algorithm under different signal-to-noise conditions as well as a study of phase recovery for different defocus conditions. We find that the algorithm is sensitive, at a low level, to focus changes especially for noisy data. Comparison of the bispectrum phases is made with those obtained from Knox-Thompson cross-spectra for all cases.
We have implemented four phase retrieval techniques for one-dimensional infrared specklegrams: bispectrum analysis, Knox-Thompson, phase-gradient, and simple-shift-and-add. Special emphasis has been placed on the problems of phase retrieval using these algorithms. We have developed a detailed statistical analysis for each technique and have computed error bars for the retrieved object phases. The performance of these four techniques has been compared using simulated point source and extended object data both observed under different signal-to-noise conditions. We make recomendations as to which method produces the best precision in the final result with respect to computation time and ease of implementation. Since each algorithm, especially the bispectrum, becomes more formidable when applied to two-dimensional data this study is both timely and useful.
The phase-gradient algorithm (PG) determines the phase of a stellar object's spatial spectrum by estimating the derivatives of the phase directly from the speckle images, and then integrating. A photon-address version of the PG has been successfully implemented for use at very low light levels. Computation of image-domain, cross-correlation functions is simply a matter of generating the 2D histograms of address differences, and bias terms are eliminated by omitting all photon self-correlation products. In a series of comparisons with a photon-address, Knox-Thompson algorithm, PG proved to have greater computational efficiency, to be more robust and to yield images with lower noise.
We show that the matched filter proposed by Ribak for extending the weighted-shift-and-add method does in fact reduce photon-statistics dominated specklegrams. The iterative method for arriving at the matched filter, as originally proposed by Ribak, does not converge in the case of photon-noisy specklegrams for objects with more than one maxima. Methods for making the procedure more "artificially intelligent" are discussed.
Centroiding is investigated as a simple and computationally fast technique of image reconstruction, at low light level, of a randomly translating image. The detected frames are sorted by their number of photons, centroided, separated averages performed and then compared with the usual way of centroiding frames. An algorithm for retrieving the phase for one-dimensional centroided imaging is presented and computer simulated data is used to test the theory and the reconstruction technique.
A technique for finding the most probable object to have resulted in a set of images taken through the turbulent atmosphere is compared to speckle interferometry . The technique involves taking a Karhunen-Loeve decomposition of the set of distortion functions and approximating the probability distributions of the coefficients of the Karhunen-Loeve basis. The probability of an object giving rise to an image is defined as the product of the probabilities of the presumed independent coefficients of the basis vectors. A computer simulation was performed and parameter estimation was performed using the new technique and standard speckle interferometry. The maximum probability technique generates parameter estimates that have standard deviations over an order of magnitude smaller than the speckle interferometry estimates.
Several methods of image reconstruction from turbulently-distorted focal-plane intensity patterns are compared from the viewpoint of noise sensitivity, processing requirements, 4nd ability to deal with extended objects. Computer simulation of the Shift-and-Add method , with and without weighted deconvolution, yields poor results against extended targets. Computer simulation of the speckle interferometry method is carried out as far as correlogram construction; the experimentally meausred noise sensitivity of the phase-retrieval procedure leads to poor predicted image quality. A complete end-to-end simulation of the speckle masking method is shown in detail; the extended target image is reconstructed clearly and unambiguously.
Recent advances in deconvolution have allowed positive images to be deconvolved without prior knowledge of either of the images comprising the convolution. We incorporate these methods into the shift-and-add principle by exploiting the property of the basic shift-and-add image that it is a (noisy) convolution of the true image of an object with some unknown point-spread-function. This allows an estimate of the true image to be extracted from the shift-and-add image. The computational efficiency of basic shift-and-add is preserved during data gathering since extensive computation is only applied to a single image. Results are presented of a computer simulation of the technique, indicating that it can remove the "ghosts" which are present in the basic shift-and-add image of a multiple star.
A program is described whose goal is the demonstration of triple correlation speckle imaging at a rate comparable to video bandwidth. To accomplish this, a Connection Machine Model CM-2 will be used to process the data using a fast version of the triple correlation algorithm. Two approaches to a fast algorithm are discussed, one based on computerized tomography and the other related to the direct phase methods of crystallography. Combination of a fast version of the algorithm and a massively parallel computer has the potential to meet the goal of video rate imaging.
The University of Maryland has developed instruments using Amplitude Interferometry (Al) for various types of groundbased telescope observations in order to support a program in Very High Resolution AstroPhysics. An initial version of the Amplitude Interferometer the Single (pair of) Aperture Amplitude Interferometer (SAAI), has been used in a longterm program of observations, primarily on Mt. Wilson and Palomar Mtn. A new more powerful instrument, the MultiAperture Amplitude Interferometer (MAAI) has been developed and fabricated and is being prepared for telescope testing.
Images can be reconstructed from incoherent holograms recorded with a rotation shearing interferometer. Reconstruction is possible through strong optical aberrations if holograms are also taken either of a reference source through the same shear, or of the same source through a different shear.
The Van Cittert-Zernike theorem from the theory of partial coherence provides a Fourier relationship between the intensity distribution of a plane noncoherent object and the spatial coherence function of the light in a plane illuminated by the object. Although the spatial coherence function has the dimensions of optical intensity, it is not directly accessible to measurement. Information about the real and imaginary parts or the absolute value and the phase argument of the spatial coherence function permits us to derive the intensity distribution of the noncoherent planar object. In this paper, two methods of obtaining information on the complex-valued spatial coherence function are discussed. In the first method, we review briefly the determination of the cosine and the sine of the phase argument using triple- and quadruple-intensity correlations. Combined with the intensity interferometry of Hanbury Brown and Twiss which measures the square of the absolute value of the normalized coherence function, this provides sufficient information to derive the intensity distribution of the noncoherent object. In the second method we show how a hologram of a noncoherent object, referred to as a r hologram, is formed by encoding the complex-valued spatial coherence function on a photographic film. This record is made possible by means of a self-referencing interferometer. Such a record behaves much like a hologram; it permits reconstruction of the noncoherent source object with the aid of the same interferometric arrangement. This procedure opens new possibilities of noncoherent-object information processing such as matched filtering and spatial-frequency filtering.
We consider the effect of turbulence in the atmosphere on measurement of correlations of the intensity of radiation received from an incoherent source. The measurements considered are made on the ground and the source is located well above the turbulent atmosphere. General expressions will be presented which give the two-point correlation of intensities in terms of the source-intensity distribution and the turbulence correlation function. We will give simplified results when isoplanicity is a valid assumption, but will also show a first-order correction to account for non-isoplanicity. Numerical results will be presented for both weak and strong turbulence effects. Finally, we will briefly discuss the three-point correlation function.
When dealing with a relatively bright astronomical object it is possible to obtain nearly diffraction limited images despite turbulence effects, by post detection processing of a nonstatistical nature, i.e., unlike the Knox-Thompson/Labyrie speckle imaging methods. The key to this concept lies in forming a set of narrow spectral band short exposure images of the target object while at the same time passing the balance of the white light from the target through a short exposure wavefront sensor. The wavefront sensor is of the Hartmann type and uses an array of lensletts to form a corresponding array of short exposure images of the target. Wavefront tilt on each lenslett, which manifests itself in the recorded array of images as a shift in the position of the image, can then be assembled into an estimate of the total wavefront distortion at the time of the short exposure. This allows the OTF (n.b. not just the MTF but rather the OTF) for the short exposure narrow spectral band image to be calculated, so that the recorded short exposure narrow band image can be compensated for the effect of wavefront distortionpost detection compensation. It is calculated that for a target of magnitude rn = 7.5 or brighter, the wavefront sensor data will have a sufficient signal-to-noise ratio. To overcome a signal deficiency problem in the post detection compensated image a number of such images would have to be added. To determine the number that have to be added, we have to multiply the number of such narrow band short exposures that would have to be combined if there were no turbulence induced wavefront distortion by the factor (D/ro)2.
Active optical imaging sensor concepts lye a number of outstand-ing technology problems. First, to achieve angularly-resolved images at long ranges, extremely large optics are required. These optics may be needed in large quantities. Second, in order to achieve a robust sys-tem which can handle a large number of objects, both the transmitter and receiver must have greater angular agility than mechanical means can provide. For a conventional angle-angle imaging system, receiver agility is the more difficult problem, due to the larger diameter optics required. Finally, laser sources will be required which can provide coherent illumination at sufficient power levels to operate at the desired ranges.
Imaging correlography is a technique for constructing high-resolution images of laser illuminated objects from measurements of backscattered (non-imaged) laser speckle intensity patterns. In this paper, we investigate the possibility of implementing an imaging correlography system with sparse arrays of intensity detectors. The theory underlying the image formation process for imaging correlography is reviewed, emphasizing the spatial filtering effects that sparse collecting apertures have on the reconstructed imagery. We then demonstrate image recovery with sparse arrays of intensity detectors through the use of computer experiments in which laser speckle measurements are digitally simulated. It is shown that the quality of imagery reconstructed using this technique is visibly enhanced when appropriate filtering techniques are applied. A performance tradeoff between collecting array redundancy and the number of speckle pattern measurements is briefly discussed.
Correlographic image reconstruction, using pupil-plane intensity patterns generated by a coherently-illuminated object, has been simulated O and has successfully reproduced the density mask used to drive the simulation. We describe a set of experiments in which a physical object was illuminated, and detector-plane intensity readouts were used as the input to correlographic reconstruction software. The effects of dark current bias, finite analog-to-digital conversion error, and additive noise are evaluated. The correlogram construction process includes inward extrapolation for smoothing of the D.C. spike. The phase retrieval process includes annealing and median filtering. As compared with the simulated results, the experimental results show greater difficulty in convergence and a lower fidelity in the reconstructed image.
A correlographic image reconstruction technique, using pupil-plane intensity pattern generated by a coherently illuminated diffuse object, has been developed. The technique consists of evaluating the pupil-plane intensity autocorrelation function and taking the Fourier transform of it. The resulting speckle power spectral density contains a term proportional to the autocorrelation of the object intensity in addition to a δ-function term. The object intensity autocorrelation (i.e. correlogram) is used to calculate the modulus of the object field, which is used as the seed for the phase-retrieval algorithm. The mathematical basis for this technique is derived in the paper and the results of computer simulation are also presented. The present technique is similar to that proposed by Idell and Fienup but contains a significant modification, and the difference manifests itself in the mathematical formulation and the computer simulation results.
The close relationship between Fourier phase retrieval and blind deconvolution is discussed. In this paper we show how advances made in phase retrieval can be successfully applied to solving problems requiring blind deconvolution. Firstly we describe how Fienup's iterative algorithms can be used as the first stage in a deconvolution strategy. In contrast to earlier direct techniques, which all appear to be very susceptible to noise, the deconvolution algorithm presented herein is capable of image recovery in the presence of appreciable noise. Secondly we discuss an extension to our zero-and-add technique which incorporates the greatly increased informational content in the zeros of multi-dimensional, as opposed to one-dimensional, images. It appears that the concept of zero-sheets can be invoked to improve the robustness of zero-and-add.
Three filter methods with proved phase-retrieval uniqueness are compared using the Gerchberg-Saxton-Misell error reduction algorithm. Defocusing, differential, and exponential filter methods are used in the one-dimensional simulations. Stagnation of the convergence process occurs more often with differential and exponential filters than with the defocusing filter and it is more likely for objects containing higher frequencies.
We show that the Gerchberg-Saxton iterative phase recovery algorithm can generate usefully accurate image phase distributions, when the magnitude of the image's Fourier transform is given (or measured) very accurately but only a moderately accurate version of the image magnitude is available. The performance of our modified Gerchberg-Saxton algorithm is illustrated with two examples for which the data were computer-generated: the first, being one-dimensional, is relevant to acoustic microscopy, while the second relates to deducing, from the magnitude of its radiation pattern, the forms of the imperfections in the two-dimensional aperture distribution of a radio antenna.
A method was recently proposed that allowed a continuous image distribution to be estimated from limited Fourier magnitude samplesl. This method affects the minimisation of a phase dependent cost function by an iterative procedure. We report here further investigations into this approach, particularly its rate of convergence.
Phase retrieval performed by the iterative Fourier-transform algorithm has been demonstrated through computer simulations for complex-valued objects with a known region of support. An active imaging system could artificially create an object with a known compact support by coherently illuminating a finite region of an extended object or scene. This points to the potential for novel coherent-imaging systems that form fine-resolution imagery utilizing far-field Fourier intensity measurements in conjunction with a priori knowledge of the illumination pattern. In this paper we describe the effects on phase retrieval of some of the measurement realities that would be encountered in an actual system. For example, any real illumination pattern will have tapered edges due to diffraction, and the effect of varying amounts of taper on algorithm convergence and quality of reconstruction are pre-sented. A modification to the algorithm, using an expanding support constraint, was developed to avoid stagnation problems associated with tapered illumination. In addition, the effects of a variety of errors in the data, both random and systematic, are presented. These errors include additive noise, quantization errors, and detector saturation and biasing.
The relevant issues in the design of laboratory experiments to demonstrate phase retrieval from Fourier intensity (far-field speckle pattern) data from a laser-illuminated diffuse object are considered. Based on the results of these considerations, a demonstration phase retrieval experiment was conducted. A diffuse object was coherently illuminated, Fourier intensity data was collected and an image was reconstructed by the iterative Fourier transform algorithm using the Fourier intensity data and an a priori known triangular image support constraint. The intensity of the complex-valued reconstructed image compares favorably to a conventional image with the same spatial frequency bandwidth, thus providing an experimental demonstration of the phase retrieval method.