Electrical impedance imaging systems1,2,3 make electrical measurements on the surface S of a body B, and from these measurements reconstruct an approximation to the electrical properties of the interior. Mathematically, the problem can be formulated as follows. The electric potential u satisfies the equations

We introduce a modified technique for restoring an image that has been distorded by a linear system whose impulse response function is itself random, and in the presence of detection of a long tailed noise. A robust non-parametiic function estimation is introduced. The estimate is motived from the theory of M-estimation and of adaptive kernel estimation regression function.

An introductory discussion of wavelets is presented for optics. By replacing the normed space of finite energy images with a phase-space (with twice as many variables) and by adding dilation invariance to the translation; a fiducial function called the mother wavelet is obtained, giving an enrichment of Fourier optics. This paradigm, wavelet optics, is discussed and the application to inverse scattering is indicated.

The inverse problems of image reconstruction in Very Long Baseline Interferometry are very particular myopic deconvolution problems. The real and imaginary parts of the logarithm of the transfer function lie in the ranges of two remarkable operators: the amplitude aberration operator B? and the phase aberration operator B?. The projections onto the ranges of B? and B? are very simple operators which can be written in closed form. When the data are “spectral quantities” such as visibility measurements, the constraints on the transfer function (the calibration constraints) can therefore be easily imposed. In the case of homogeneous arrays, this leads to a better efficiency of the calibration techniques developed by Schwab and Cornwell. When the data are “higher-order spectral quantities” such as those provided, for example, by averaged complex closure terms, it is easy to extract from these data a spectral information quite similar to that given by visibility measurements. This modifies the very principle of the reconstruction methods used today in such situations, for example, the method of Readhead and Wilkinson. This new approach results from the particular algebraic structures of interferometry in higher-order spectral analysis. To exhibit these structures we introduce four compilation operators, among which: the phase compilation operator C? (the phase closure operator of order p > 3), and the alternate amplitude compilation operator C? (the amplitude closure operator of even order p > 4). All these operators have remarkable properties. For example, the generalized inverses of the closure operators of order p are given by very simple backprojection relations. According to these remarkable algebraic structures, a higher-order spectral information ? (with the appropriate subscript) yields a spectral information ? defined up to a function lying in the range of B. In the corresponding reconstruction process, all the closure data are taken into account with the same weight (not only those corresponding to a set of independent closure relations). The interest of homogenous arrays is thus reinforced. It is also clear that the quality of a higher-order spectral information ? only depends on that of its projection onto the range of the phase or amplitude closure operator. This remark is essential for the choice of the closure order. All these nice “homological” structures, and their implications in optical interferometry, have been taken into account in our image-reconstruction algorithms.

A general approach to the problem of reconstructing a non-negative function from finitely many linear functional values is to select from among all data consistent functions that one closest to a prior estimate, according to some measure of directed distance. Minimizing the cross entropy (MCE), or more generally, minimizing the Kullback-Leibler distortion, are examples of this approach. Shore and Johnson have characterized the MCE approach for the reconstruction of probability densities in terms of axioms of probabilistic inference. Many of the applications of these methods involve the reconstruction of functions that are not essentially probabilistic in nature, such as energy distributions in space, x-ray attenuation functions, and so on. The properties of the reconstructions, as approximations of the true solution, are not easily determined from these axioms. Because the basic problem is how to approximate one function by others, we adopt axioms that govern the approximation theoretic behavior of the directed distances. The first three axioms merely impose reasonable behavior on the directed distances. The fourth, directed orthogonality, characterizes a wide class of directed distances, including the Kullback-Leibler and the Itakura-S alto distortion measures. The addition of a fifth axiom, invariance to scale, uniquely characterizes the Kuilback-Leibler distortion, which reduces to MCE (for probability densities) when the prior estimate is constant.

A method for maximum-entropy image reconstruction from projections is presented. Two image entropies are studied under the assumption that image-density distributions over image-element array follow a multinomial random process. The multinomial distribution process reflects those physical aspects of images: (1) image density is non-negative, (2) image-density distributions are conserved, (3) image-density distributions are spatially correlated. One image entropy assumes a uniform spatial correlation and resembles Shannon's entropy that has been widely used in image-processing problems. The other assumes non-uniform spatial correlations among nearby image elements and resembles the cross entropy introduced by Kuilback and Leibler. The non-uniform correlations among nearby elements are modeled in a similar manner as Markov random field does, i.e., image density changes slowly within a homogeneous region and rapidly at the boundaries of the homogeneous regions. The projections are assumed as measured by a linear imaging system. The measurement constraints are incorporated into the maximum-entropy method by employing a set of Lagrange parameters. Each Lagrange parameter is related to a constraint of a datum. The solutions that maximize the image entropies subject to the measurement constraints are given, as determined respectively by a superposition of the Lagrange parameters. The Lagrange parameters are then estimated iteratively using a steepest descent technique. Comparative study on the image entropies is carried out using computer generated noise-free and noisy projections. Improved results are obtained by use of the image entropy assuming the non-uniform correlations. The non-uniform correlated image entropy needs little more computation time and memory.

The Kirillov quantization procedure is used to explain coherent optical holography and to describe its extension to the superresolution method of imaging by incoherent -to-coherent conversion.

We consider the problem of reconstructing remotely obtained images from image-plane detector arrays. Although the individual detectors may be larger than the blur spot of the imaging optics, high-resolution reconstructions can be obtained by scanning or rotating the image with respect to the detector. As an alternative to matrix inversion or least-squares estimation the method of convex projections is proposed.

Image restoration procedures are commonly unstable in the presence of noise, and some technique for restoring stability becomes essential. The methods of regularization theory are particularly appropriate for this purpose. A specific type of regularized solution is introduced in the general context of image reconstruction. A super-resolution problem is then considered from the point of view of the computational tasks involved, with particular reference to the estimation of certain key parameters and to implementations which increase the efficiency of the calculations. Parameter estimation is performed by weighted cross-validation. The improvement in efficiency is achieved through the exploitation of symmetries or cyclic properties inherent in the reconstruction operator. The concept of displacement rank is introduced and estimates made of the computational burden associated with various classes of regularized reconstruction matrices.

We present an inversion algorithm based on a recently developed inversion method referred to as the Renormalized Source-Type Integral Equation approach. The objective of this method is to overcome some of the limitations and difficulties of the iterative Born technique. It recasts the inversion, which is nonlinear in nature, in terms of the solution of a set of linear equations; however, the final inversion equation is still nonlinear. The derived inversion equation is an exact equation which sums up the iterative Neuman (or Bom) series in a closed form and; thus, is a valid representation even in the case when the Bom series diverges; hence, the name Renormalized Source-Type Integral Equation Approach.

We will do three things: Show how to infer elementarily from timedomain precursors some frequency-domain information. Second, make observations on energy velocity that may be surprising. Third, generalize Sommerfeld's theorem about the speed of light.

An inverse scattering theory is used to design gradient-index optical waveguides capable of transmitting images without degradation. The propagation characteristics on N modes each carrying a pixel of the image are specified by a transverse rational reflection coefficient. The Gelfand-Levitan-Marchenko inverse scattering theory is used to obtain the unique solution of the permittivity profile of the waveguide from the reflection coefficient.

A method for reconstructing the index of refraction of a bounded inhomogeneous object of known geometric configuration from measured far field scattering data is presented. This work is an extension of recent results on the direct scattering problem wherein the governing domain integral equation was solved iteratively by a successive relaxation technique. The relaxation parameter was chosen to minimize the residual error at each step. Convergence of this process was established for indices of refraction much larger than required for convergence of the Born approximation. For the inverse problem the same technique is applied except in this case both the index of refraction and the field are unknown. Iterative solutions for both unknowns are postulated with two relaxation parameters at each step. They are determined by simultaneously minimizing the residual errors in satisfying the domain integral equation and matching the measured data. Numerical results are presented for the dielectric slab. The algorithm is shown to be effective in cases where the iterative solution of the direct problem is rapidly convergent.

This paper describes the detection and identification of buried objects, at shallow depths, from the complex-valued reflectance of electromagnetic waves that are radiated and received by an antenna. The antenna is scanned over an area to search for the object. Because object depth is shallow, the antenna is placed in the object's nearfield. The measured distribution of reflectance is processed to form an image which gives shape because diffraction spreading is slight in the nearfield. The reflectance over the object's center is processed by inversion to yield object depth, thickness, and refractive index from multiple frequency measurements; the object is assumed homogeneous as is the contiguous region.

Phase space and functional (path) integral methods, so successfully applied today in quantum and statistical physics, are applied to both direct and inverse wave propagation problems in extended inhomogeneous environments. The modeling is provided by the n-dimensional scalar Helmholtz equation. Both solution representations and numerical algorithms are presented.

Phase constrast microscopy is famous for allowing direct visualization of an object phase specimen. That is, the output intensity image is linear in the conjugate phase object. However, it is well known that this ideal result is only an approximation that works well for small phase values.1

From the recent developments in Laser Scanning Tomography it appears that dark field microscopy methods make it possible to observe very small microprecipitates which are bound to grow in the bulk of semi conductor materials or epitaxial layers. Improvements in such investigations rely on reducing the effective "sectioning' thickness of the optical observation in order to be able to measure the axial position of the point sources with submicron precision. This is possible owing to the fact that the recorded individual images of the point sources (considered as Point Spread Functions) are separable diffraction patterns. This communication aims at presenting the first attempt to explore this typical reverse problem of "micro ranging". It will be shown using a series of images from precipitates in Indium Phosphide bulk material that numerical interpolation can lead to the relative position of each particle with a convenient precision. Optical perturbations such as spherical aberration and coma are also to be taken into account if the optical system is not already corrected.

An algorithm for the solution of a nonlinear inverse scattering problem in polar coordinates in ultrasonic tomography is presented. The scattering and diffraction of ultrasonic waves propagating through an inhomogeneous, nondispersive medium, f, may be described by the nonlinear integral equation The unknown f is to be determined from measurements of the scattered field u8 = u – ui on ?D Since k = (k, ?) contains information on frequency and direction of propagation of the incident wave both multiple frequencies and multiple angles of incidence information are used in measurements of the scattered field. These integral equations, which are given in two-dimensional rectangular coordinates, are converted to polar coordinates. Then they are written as a nonlinear operator equation T'(w) = 0 which is solved using Newton's method, with the derivative of the operator being the Frchet derivative I!W). The linear operator equations in each step of Newton's method are solved numerically by using piecewise linear function approximations and then solving the resulting linear discrete equations.

One major character of the limited angle inverse scattering problems is the sparsity of the information contained in the measurement data. The sparsity of information content in the limited angle inverse scattering problems makes it difficult to reconstruct the object functions by using the conventional inversion techniques. To overcome the above difficulty, a boosting procedure is employed to obtain the maximum amount of information for an arbitrary predefined experiment set-up. The numerical simulations are performed in order to ascertain the role of the boosting procedure in the limited angle inverse scattering problems. The results of the computer simulations for two-side view tomography demonstrate that by applying the boosting procedure, the quality of the reconstruction and the speed of the convergence are improved significantly. Furthermore, for subsurface detection where both the sources and sensors are confined on one side of the object, the reconstruction has become possible after applying the boosting procedure.

Numerical methods recently developed to invert scattered field data are presented in this paper. Based on the extension to the well-known first-order Born approximation for linearized inverse scattering, these methods allow a wider class of more strongly scattering objects to be imaged. The distorted-wave approximation is described and the further approximations enabling it to be used for inverse scattering problems are given. These algorithms are evaluated using both numerical simulations and experimental data collected from known scattering objects.

A method of three-dimensional inverse scattering based on image projections is derived and implemented for deterministic and random weak scatterers. For each illumination of the object, the measured scattered field is directly backpropagated onto a single plane in the image space. The backpropagated field evaluated on the plane is defined as the image projection because it closely approximates the straight line projection of the object for both high frequency incident waves and low spatial frequency scatterers. The object can be reconstructed by using conventional tomographic algorithms such as the Fourier slice method. This approach permits practical inversions using a limited set of illumination directions. For random weak scatterers, we show that the second order statistics of the medium can be determined from the image projections. We illustrate the method with numerical examples for both deterministic and random scatterers.

A reciprocity theorem relating the coherence and spatial distribution of a slow randomly time varying, quasihomogeneous scatterer and its scattered field is discussed and applied to the problem of imaging the spatial distribution of the scatterer by use of inverse scattering techniques.

It is shown that multiple blind deconvolution and phase retrieval in three or more dimensions are overdetermined problems if the image or Fourier amplitude, respectively, is measured continuously (or, equivalently, is sampled at the Nyquist rate). A sampling scheme is derived that removes the overdeterminancy and allows a measure of the redundancy to be defined. This indicates that phase retrieval and deconvolution are more stable, with respect to noise and partial information, with increasing dimensionality.

Advances in antenna diagnostic methodologies have been very significant in recent years. In particular, microwave holographic diagnostic techniques have been applied very successfully in improving the performance of reflector and array antennas. These techniques use the knowledge of the measured amplitude and phase of the antenna radiated fields and then take advantage of the existing Fourier transform relationships between the radiated fields and the effective aperture or current distribution to eventually determine the reflector surface or array excitation coefficients anomalies. In this paper an overview of the recent developments in applying microwave holography is presented. The theoretical, numerical and measurement aspects of this technique is detailed by providing representative results.

An analytical time-domain expression is derived for the early time impulse response for smooth, convex, perfectly conducting scatterers under the physical optics approximation for the bistatic case. The physical optics bistatic early time impulse responses can still be interpreted as cross-sectional areas of the scatterer similar to the work of Kennaugh for the monostatic case. A crude polarization correction to the leading edge of the physical optics impulse response is obtained for the bistatic case, leading to a simple asymptotic relation between the specular principal curvature difference and certain co-polarized phase terms in the bistatic scattering matrix. Applications to direct scattering are discussed. Profile reconstruction from bistatic data with a priori knowledge of the validity range of physical optics in the time domain is proposed and tested with the sphere.

Bayesian methods of image reconstruction promise better, more appealing images. With the introduction of the EM algorithms for emission and transmission tomography, it is possible to incorporate Bayesian priors in a natural way. Priors tend to accelerate convergence of the algorithms and turn underdetermined systems of likelihood equations into overdetermined systems. Two types of priors have been suggested. One penalizes abrupt changes in estimated values for neighboring pixels. These smoothing priors are based on Gibbsian interaction terms. The second type of prior actually steers each pixel estimate toward a predetermined value. Because the smoothing priors require fewer assumptions, they probably are more defensible at the present time. Integration of emission tomography with other imaging modalities may eventually make the second type of prior the method of choice in emission tomography. Combination of the two types of priors is also possible and requires only minor adjustment of the EM algorithms.

The attenuated Radon transform mathematically represents the measured projections in single photon emission computed tomography (SPECT) for an ideal detector with a delta geometric response function and no detected scattered photons. As a special case of the attenuated Radon transform, the exponential Radon transform is defined for a constant attenuator by modifying the measured projections through a transformation which places the detector at the center of rotation. Several papers have presented analytical spectral decompositions of the Radon transform; however, no analytical decomposition of the exponential or the attenuated Radon transform has been derived. Here an eigenanalysis of the exponential Radon transform is compared with that of the Radon transform using the Galerkin approximation to estimate the spectral decomposition. The condition number of the spectrum increases with increased attenuation coefficient which correlates with the increase in statistical error propagation seen in clinical images obtained with low energy radionuclides.

The frequency speciral characteristics, bias and variance of images reconstructed from real Positron Emission Tomography (PET) data have been studied. Feasible images obtained from statistically based reconstruction methods have been compared to Filtered Backprojection (FBP) images. Feasible images have been described as those images that are compatible with the measured data by consideration of the Poisson nature of the emission process. The results show that the spectral characteristics of reconstructions obtained by statistically based methods are at least as good as those obtained by the FBP methods. With some exceptions, statistically based reconstructions do not exhibit abnormal amounts of bias. The most significant difference between the two groups of reconstructions is in the image variance, where the statistically based methods yield substantially smaller variances in the regions with smaller image intensity than the FBP images.

Since the time domain data observed in the magnetic resonance imaging represent the spatial frequency data in the k-space, all the imaging schemes in MRI can be interpreted as the data acquisition procedures as uniformly as possible on the k-space with some form of trajectories. Among the various k-space scanning methods, spiral scanning has some interesting properties such as the rapid scanning and localization. In this paper, characteristics of the spiral scanning will be examined and its applications to high speed MR imaging as well as volume localization for the spectroscopic application will be reported.

A method to reconstruct motion using magnetic resonance (MR) imaging is presented. Magnetic resonance imaging tagging is used to create intensity gradients within objects, where the intensity would otherwise be largely constant. Since the tag field decays with T1, a modified optical flow algorithm is developed and implemented. Object parameters T1, T2, and D0 are required to be known across the image at the time the tag field is created; motion is then determined by computing optical flow between pairs of images and estimating the implied motion reference map at each pixel site.

Several techniques have been proposed for rapidly acquiring the Fourier domain data required for magnetic resonance (MR) imaging. These fast-scan techniques sample data on nonuniform grids in the Fourier space of the object. This work explores the image reconstruction errors that occur when images are reconstructed as if the data were measured on the ideal uniform grids. It is also shown how interpolation can be used to correct these reconstruction errors and produce images of acceptable image quality.

A technique is described for the simultaneous acquisition of NMR data using two independent receiver coils surrounding the same region of tissue which enable the acquisition of all data necessary for an image in a reduced number of acquisitions. This results in a 50% reduction in minimum scan time, but potentially the technique may be extended to multiple receivers with a proportionally higher time savings. The algorithm and imaging procedures are described and example images are described which illustrate the reconstruction. Signal to noise and artifacts are discussed.

Biomagnetic imaging is the determination of electrical current flows, internal to the body, from measurements of the external magnetic fields these currents produce. In this paper we concentrate on factors that influence the use of mathematical estimation algorithms. It is these algorithms that produce the current images. We will ignore problems of data acquisition, assuming that all measurements are perfect and plentiful. We will further restrict our attention to formulations of magnetic-field generation and current-density reconstruction for linear, isotropic, shift invariant media.

An array of SQUID biomagentometers may be used to measure the spatio-temporal neuromagnetic field produced by the brain in response to a given sensory stimulus. A popular model for the neural activity that produces these fields is a set of current dipoles. We present here a common linear algebraic framework for three common spatio-temporal dipole models: i) moving and rotating dipoles, ii) rotating dipoles with fixed location, and iii) dipoles with fixed orientation and location. Our intent here is not to argue the merits of one model over another, but rather show how each model may be solved efficiently, and within the same framework as the others. In all cases, we assume that the location, orientation, and magnitude of the dipoles are unknown. We present the parameter estimation problem for these three models in a common framework, and show how, in each case, the problem may be decomposed into the estimation of the dipole locations using nonlinear minimization followed by linear estimation of the associated moment time series. Numerically efficient means of calculating the cost function are presented, and problems of model order selection and missing moments are also investigated. The methods described are demonstrated in a simulated application to a three dipole problem.

In the integral equations that arise in potential theory it is necessary to evaluate the solid angle subtended at a point r on the surface by each element of area of the surface. If the surface is approximated by a set of triangles then those triangles that contain the point r as a vertex require special attention, and this is referred to as the "auto solid angle" problem. We give an exact formula for evaluating the auto solid angle, and a prescription for apportioning it amongst the vertices of these triangles.

Human thought and language has its biological substrate in the neural machineiy of the cerebral cortex. The cerebral hemispheres are composed of approximately one hundred separate regions that perform specialized information processing functions. When activated, synchronized electrical currents in these regions can produce magnetic fields that are measureable from the surface of the scalp, a procedure known as magnetoencephalography (MEG). To study the physiology of language perception, we have recorded MEG evoked by auditorily presented phonemic and nonphonemic stimuli. Problems arise in interpreting such data, as multiple spatially-distributed cortical generators probably contribute to the measured magnetic fiekL Two approaches may be taken to resolving this problem: principal components analysis of temporally separated fields and residual field analysis. Both approaches require experimental verification before they can be applied widely. Despite this limitation, MEG data are already capable of detecting abnormalities of cortical language localization, as ifiustrated in a case of displaced language cortex resulting from an arteriovenous malformation.

A method for neuromagnetic imaging of impressed current density in complex volume conductors using current source density calculations is investigated. The method is applied to phantom experiments with one and two curmnt dipoles in two different volume conductor phantoms. The results indicate that strong filtering effects, inherent in the Maxwell equations, pretend volume current effects in the images. The method yields better localization and spatial resolution of single and multiple sources compared with conventional total current density imaging.

A method is proposed in which the locations and electric current vectors of multiple biomagnetic current dipoles can be estimated without a priori information regarding the exact number of dipoles. In this method, an additional constraint is used in the cost function in conjunction with a constraint regarding degree of matching between estimated dipole parameters and a measured magnetic field. Two constraints, a constraint on the sum of current magnitude and that on the sum of each dipole's contribution on measured data, are proposed as the additional constraint in this paper. Computer simulation clearly shows that çither constraint can be effectively used to suppress the ambiguity caused by the lack of information regarding the exact number of dipoles. Since the cost function becomes highly nonlinear, the simulated annealing algorithm1,2 is essential to search for the minimum of the cost function.

Ionic flow associated with neural activation of the brain produces a magnetic field that can be measured outside the head in a magnetically unshielded room using a highly sensitive neuromagnetometer based on a superconducting quantum interference device (SQUID). Reconstruction of images portraying the tomographic distribution of neural generators (assumed to be current dipoles) of the neuromagnetic field, a modality that we have termed "neuromagnetic imaging" or NMI, represents a powerful noninvasive method of dynamic functional imaging dependent upon brain structure and activity. Reconstruction in NMI, i.e., the inverse problem, however, has no unique solution and requires incorporation of modeling constraints for practical implementation. Results of several phantom and test-object studies and a preliminary human study to develop the method of NM! under various modeling constraints are presented.

Biomagnetism is a promising new tool for the noninvasive 3D -localization of electrophysiological activity. For reconstructing current distributions from biomagnetic fields the quasi-stationary limit of the theory of electromagnetic fields can be used [1]. This means that, once one has recorded a time series of magnetic field data at multiple sensor positions, one can pick out the magnetic field distribution at any arbitrary time instant. The momentary source current distribution can then be reconstructed from this field regardless of what happens with the sources before or after this time instant. Though this is an important feature in the evaluation of biomagnetic data, taking into account apriori- and physiological knowledge can improve the biomagnetic localization result significantly. The sources to be reconstructed are electrophysiological processes, and the time course and propagation of these activities obviously depends in a causal way on their previous history. So in fact we have much more information than the quasi-static magnetic field map alone, an information which can be used to improve the reliability of the reconstruction result. Biomagnetic signal variation in time and space can be used for the improvement of data evaluation algorithms and in simulation studies for biomagnetic system optimization. In this paper we will focus on four fields, which will be discussed in detail in the following sections: - use of source model simulations as function of time to optimize frequency passband and data acquisition rate with respect to source reconstruction accuracy - half-automated validation of the reconstruction result by comparison of the time course of the equivalent source with physiological criteria - separation of an electric activity of interest from simultaneous processes - sensitive pattern recognition algorithm to recognize and average similar events

Recently-developed, high-resolution Superconducting QUantum Interference Device (SQUID) magnetometers can provide magnetic images of small objects with a signal-to-noise ratio and spatial resolution that are an order of magnitude better than is achievable with conventional SQUID magnetometers. The MicroSQUIDtm magnetometer has 4 channels, each of which is a differential magnetometer with a 3 mm diameter pick-up coil located less than 1.5 mm from the room-temperature region outside the Dewar. The system noise is approximately 100 fT/Hz1/2. We present magnetic field data recorded from several two-dimensional current distributions, and from the remanent magnetization in rocks, photocopier images, and magnetic contamination in metallic tubes. The system is capable of locating to better than 10 µm a straight wire carrying a 20 µA peak-to-peak sine wave current. The system can detect growth currents in a fertile chicken egg, and signals from a voluntarily-activated single motor unit in the human thumb. Magnetic fields produced by two-dimensional current distributions can be converted into current density images using Fourier-transform spatial filtering algorithms. This approach allows us to examine the difference between the imaging and localizing resolution of a magnetometer, and allows us to devise apodized pick-up coils with increased spatial resolution.

Biomagnetic data are analysed with methods which do not involve point source descriptions. A vector signal transformation, previously defined, is generalised to reveal localised features at different scales in developing chick embryos in ovo. A powerful, but computationally intensive method is applied to a range of evoked MEG signals. Post-inversion temporal filtering is finally used to reveal a weak signal which seems to be generated at a surprisingly distant source. In short this work demonstrates the versatility and power of modern biomagnetic techniques in detecting, sizing and imaging the impressed current density generated by biological function in a completely non-invasive fashion.

Impedance tomography is one of the most recent imaging techniques of the human body and it needs specific reconstruction techniques. So we have developed an algorithm which, using peripheral potential values, allows the calculation of the potential distribution at all points of a domain ?, without the knowledge of the conductivity. This algorithm using optimal control techniques thus allows the detection of perturbations in an homogeneous domain. Results in 2 and 3D are presented after the theoretical analysis of the problem to be solved.

Two different iterative methods for the reconstruction the quantitative distribution of the complex permittivity in an inhomogeneous body, for application in active microwave tomography, are presented. The first method is based on simulated annealing, and the second one consists of successive perturbations of a linearized system.

It is now twenty years since the pioneering work of Antoine Labeyrie gave the technique of stellar speckle interferometry to the astronomical community. Subsequently, the field has exploded with many new insights, ideas and innovations. This paper reflects upon and discusses these developments. In particular, the extremely powerful speckle imaging techniques such as triple correlation analysis or speckle masking are reviewed. Image reconstruction algorithm developments are considered. The recent developments in the area of imaging using pupil masking, aperture synthesis and phase-closure techniques are also discussed.

The atmosphere of the earth restricts the resolution of conventional astrophotography to~1 arc- sec. Much higher resolution can be obtained by interferometric speckle techniques. Bispectrum analysis1-3 (also called speckle masking) of many speckle interferograms (short-exposure photographs; exposure time ~0.05sec) yields diffraction-limited images with, for example, 0.02arcsec resolution for a 5-m telescope. After the first processing steps of speckle masking the object bispectrum O(3)(u,v) is obtained up to the cut-off frequency of the telescope.

The triple correlation algorithm employing a minimum of computation is evaluated as a candidate process for near-real-time, quick-look, stellar image reconstruction at the telescope. Computational efficiency is gained by exploiting the binary nature of the photon-list images and by using a small number of high SNR bispectrum subplanes. When compared to the Knox-Thompson algorithm extended to 2, 4 and 6 subplanes, the fidelity of reconstruction by the TC process is better and less object dependent, even in the minimum-computation case of 2 planes.

Several algorithms based upon a weighted least squares methodology are presented for phase reconstruction from the bispectrum. Results from applying these algorithms to both simulated and field data are presented and compared.

We present results of a horizontal path imaging experiment using a 0.5-m telescope focused on a target 1.2 km away. We consider an extended representation of a satellite with grey scale and size variations. We imaged at 0.7 µm and found an average atmospheric degradation factor of D/ro = 17. We used a slow read-rate, bare CCD detector; thus, we had to deal with additive measurement noise. Our image reconstruction algorithms are based on the complex bispectrum, and we have demonstrated diffraction-limited imaging down to light levels approaching a few photons per speckle per resolution area. We have paid careful attention to the effects of additive noise on the reconstruction process and have shown that they can be adequately overcome. We present some new algorithms based on conjugate gradients and least squares. We demonstrate that these algorithms can improve image quality.

A fast approach to the estimation of bispectra of two-dimensional objects at high light levels is described. By radially slicing each two-dimensional specklegram to produce a set of one-dimensional signals the number of calculations required becomes 0(N3) instead of 0(N4) for (N x N)-sized specklegrams. The speed-up factor is 175 when N = 512, for example. Computer simulations of this algorithm are described and their results are summarized. These results show that a high quality result is feasible if each reconstructed slice can be accurately centered. Some approaches to solution of this problem are discussed.

Reconstruction of astronomical images has been performed by phase retrieval methods from autocorrelation or, equivalently, power spectrum data estimated by applying the stellar speckle interferometry (S.S.I.) technique to photon-limited, turbulence degraded, short exposure images. We present results obtained from real data, and also from computer simulated data. We make comparisons of the results depending both on the phase retrieval method used, and on whether we impose the available information in the object space (autocorrelation), or in the spatial frequency space (power spectrum).

In speckle imaging of astronomical objects, the atmospheric point spread function is commonly estimated by imaging a point source (unresolved single star) near the object of interest either before or after the time of observation. In circumstances where no reliable PSF is available to deconvolve from the data, ‘blind deconvolution’ may be of use. Both direct and indirect or ‘implicit’ blind deconvolution schemes have been applied to (photon-limited) data of binary stars and the implications of these results for referenceless speckle imaging are addressed.

Second-, third-, and fourth-order intensity correlations measured in the field in the pupil plane are used to construct the amplitude and phase of the two-dimensional mutual coherence function. Information about the noncoherent object is derived by a two-dimensional spatial Fourier transform of the mutual coherence function. A computer simulation of the Fourier domain laser speckle patterns is used to provide data from which the expected second-, third-, and fourth-order intensity correlations are computed. These correlations are used in the program for the explicit reconstruction of the phase. In addition, the signal-to-noise ratio (SNR) is discussed with reference to the measured integrated intensity, ?0TI(t)dt, as compared to the theoretically assumed instantaneous intensity,I(t). The study of the SNR for the second-, third-, and fourth-order intensity correlations involves higher-order intensity correlations. With the assumed Gaussian statistics of the wave amplitude, the analytical expressions for the higher-order correlations are algebraically complex. The SNR for the third-order case is discussed. For further development, symbolic manipulation programs (e.g., DERIVE, MATHEMATICA, or MACSYMA) will be used. The discussion of the signal-to-noise ratio applies to intensity correlation interferometry (low light levels) for which the integration time, T, is large compared to the coherence time, ?c, that is, T >> ?c. We will consider the case for laser speckle interferometry for which ?c » T in our follow-up work.

A novel concept is described for generation of a high-resolution target images, even when viewing through atmospheric turbulence, based on exploitation of laser speckle statistics. A laser transmitter projects a moving sinusoidal (spatially) modulated irradiance pattern on the target. The irradiance of the backscattered speckle pattern is measured at a set of small apertures distributed over some region on the ground. By developing the correlation of the time-varying signal measured at various pairs of apertures, it is possible to obtain measurement values from which the target image can be developed. In this paper we define the details of the concept and then proceed to calculate a bound for the performance of this technique using an approximation to an estimator of the needed correlation. The estimator is unbiased over photon statistics, speckle statistics, and scintillation statistics. The shear-speckle technique requires large numbers of data frames to achieve reasonable performance at moderate spatial frequencies. However, the good signal-to-noise ratios obtained for moderate numbers of frames at low spatial frequencies suggest that shear speckle might be used to augment other imaging techniques which have trouble at lower spatial frequencies.

Active optical systems have potential for both long range discrimination and pointing and tracking missions. The narrow beamwidth and high angular resolution of optics provides advantages which can make optics the sensor of choice for these missions. The large aperture required to achieve high angular resolution presents several problems for conventional optical systems. For imaging with sub-meter resolution at a range of several thousand kilometers, apertures greater than one meter in diameter are required. Apertures of this size are difficult to steer rapidly to image many targets per second. In addition, fabrication of large primary mirrors is more difficult for large aperture sizes. Finally, the weight of large mirrors must scale approximately as D3 to maintain the mirror figure without active correction; weight equals cost for a space-based system. Active correction requires complex control systems and a beacon or other means for determining correct actuator position.

Image correlation techniques, such as bispectrurn analysis, are now widely used in astronomy and other fields to obtain high resolution images. We report on two new methods for obtaining such images from intensity correlation functions. Both methods treat the correlation functions as convolutions and attempt to deconvolve these functions while using a priori information to constrain the results. The first method is an iterative algorithm based on deconvolution schemes recently proposed by us1,2. The algorithm constrains any reconstructed image to be positive and forces its associated spatial correlation spectrum to be consistent, within noise estimates, with the spectrum of the measured correlation function. The second method is an implementation of the Monte-Carlo optimization technique known as simulated annealing. This method minimizes the difference between the measured correlation spectrum and that associated with the deconvolved image. The two proposed methods are describe and results are presented for both computationaly simulated data and astronomical data obtained at the Wyoming Infrared Observatory.

New algorithms are summarized for recovering an object's intensity distrubution from the second- or third-order autocorrelation function, or equivalently, the Fourier magnitude or bispectnim, of the intensity.

In many imaging situations the quality of the image is degraded by phase errors. In this paper we describe an algorithm for correcting phase errors. It is applicable to cases in which the phase- error-degraded complex Fourier transform of the aberrated image is available; these include imaging with heterodyne sensors or with interferometric sensors. The phase-error correction algorithm is a variation on the iterative Fourier transform (phase retrieval) algorithm. It uses a support constraint on the object, making it useful for imaging bright objects on dark backgrounds. It can be extended to include uncertainties in both the modulus and the phase of the Fourier transform.

Retrieval of the phase of a complex analytic function when only the amplitude is known has application to many real-world problems. For example, phase retrieval has been used in astronomical speckle imaging [1] and is being considered for laser speckle imaging systems.[2,3] In the latter case, the computational requirements may limit the number of images which can be obtained in a given time. This paper describes a modification to the iterative Fourier transform algorithm (IFTA) as described by Fienup [4] which may reduce the computation to reconstruct an image to a given quality by as much as a factor of three.

We review our previous work in which we have shown that imaging by an ideal optical interferometric array, which suffers only from the Poisson noise of photoelectron counting, is essentially insensitive to how light beams are split and recombined. Any physically large, monolithic array such as those planned for space will however also suffer from mechanical noise in its structure. We show that inclusion of this technical noise amounts to an effective decorrelation that degrades sensitivity. We calculate this decorrelation factor for an otherwise ideal nC2 array and make some pertinent comments about its effect on other sorts of arrays as well. Finally, we review our work on ground-based arrays which suffer sharp reductions in the sensitivity compared to ideal arrays, particularly at low count rates.

Diffraction-limited imaging of an incoherent object observed through fixed unknown aberrations is demonstrated using rotational shear interferograms. Reconstruction algorithms were inspired by radio-astronomy self-calibration methods. They require no reference point source other than the object itself to calibrate the effects of aberrations.

Non-coplanar sampling of the visibility function measured by interferometric arrays leads to difficulties in imaging wide-fields. Unlike the case for co-planar sampling or small fields of view, the relationship between sky brightness and the visibility is not a simple two-dimensional Fourier transform, and so the usual methods of image reconstruction cannot be applied. We describe and analyze some of the many schemes which have been advocated to overcome this problem. The most promising is based upon an observation by Clark that if the sky brightness is thought of as lying on a surface embedded in a three dimensional space, a Fourier relationship does hold.

Simultaneous deconvolution of the Stokes parameter images I, Q, and U from a shift invariant point spread function is achieved by extending the maximum entropy algorithm of Cornwell and Evans1 to deal with linear polarization. This algorithm is used to image very long baseline interferometer (VLBI) total intensity and linear polarization data. Unresolved features can be removed from the data via other methods prior to maximum entropy deconvolution, or the data can be tapered with a Gaussian beam to smooth any point sources.

This paper describes a technique for imaging coherently illuminated objects using visibility measurements averaged over a set of speckle realizations of the pupil plane field. A technique for removing the effects of receiver position and external phase errors is also discussed. Computer simulation results which illustrate these concepts are given for the case of a simple one-dimensional object.

The use of an integrated optical circuit (IOC) for synthesizing a multi-element aperture is presented in this paper. The IOC is augmented with additional optical and electrical signal processing devices to provide complex (quadrature) measurements of an optical field. The principal issue addressed in this paper is the method by which phase information is recovered and the precision of these measurements.

A twenty-five element, two dimensional array of one millimeter diameter fibers, each coupled to a photomultiplier tube, has been used to image a variety of targets with a resolution close to that obtainable with a monolithic optic of equivalent total aperture. In addition to imaging the targets, which were illuminated by coherent ultraviolet light at 353 nm, the array was used to measure both translational and rotational target motions. An optical local oscillator signal supplied to the array allowed phase-tracking across the multiple channels which permitted simultaneous imaging and Doppler-shift velocity measurements. Results for millimeter-size targets at a range of six meters from the receiver are presented.

It has been suggested that high resolution images of laser illuminated objects can be digitally synthesized from measurements of the wavefront slope (gradient) associated with the backscattered laser-speckle field. We describe the image synthesis procedure and present images reconstructed from computer simulated laser-speckle fields. Noise was added to simulated wavefront-difference measurements to illustrate the effect on the imagery. We also describe a Shack-Hartmann type of wavefront sensor that was designed and built at the Weapons Laboratory and initially used to investigate the distribution of ray directions in a speckle field. Imaging results obtained with the sensor in the laboratory are presented and we describe an adaptation of the basic imaging technique that can be used to image coherently illuminated objects through optical phase distortions.

In order for a phased-array telescope to achieve its resolution potential, the individual telescopes and beamcombining optics must be precisely aligned. Misalignments could be measured directly with laser interferometers. We present an alternative, based on Gonsalves’ phase-diversity concept, in which misalignments are inferred from the collected imagery. Once the misalignments have been estimated they can be used to actively correct the system or to construct a Wiener-Helstrom filter to deblur the collected imagery. The total process is referred to as multiple-plane measurement for aberration correction (MMAC). In this paper we present simulations that demonstrate the use of MMAC in estimating both piston and tilt misalignments in the presence of noise. Measurements of the sensitivity of MMAC to certain systematic errors are presented and a subframing technique is demonstrated.

Results in postdetection turbulence corrected imaging are presented. Theory, laboratory results at low light levels and what we believe are the first successful astronomical images are described.

Image formation of orbital objects through the turbulent atmosphere using the Knox-Thompson algorithm is investigated. With the total observing time fixed by satellite dynamics, it is found that the percentage of the diffraction limit obtainable varies over most of its range as (r0/D)3/2. With ro roughly proportional to wavelength, longer observing wavelengths become attractive to maximize the obtainable resolution. Longer wavelengths are also preferred for daytime observations as the sky background decreases with increasing wavelength in the visible and near infrared. Observations at longer wavelengths suffer, however, from a decrease in the maximum obtainable resolution due to diffraction and the lack of efficient photoemissive materials beyond 0.9 ?m. The concept of a speckle imaging system spatial frequency cut off is developed and equations are derived to allow its calculation, including contributions from detector and sky background noise. The maximum system resolution as a function of wavelength is found to maximize, and equations are developed for the computation of the optimum observing wavelength.

We report in this Communication the current progress in development at the Nice University of the technique of Probability Imaging based on a complete statistical analysis of the speckle pattern observed at the focus of a large telescope. The technique, which was first proposed for the imaging of double and multiple stars, is here generalized to the image reconstruction of any general extended astronomical object. Making use of a technique of orthonormal expansion currently employed in statistical optics, a mathematical expression for the characteristic function of any order of the speckle pattern is given as the inverse of the determinant of a matrix whose elements are defined by the spatial correlation function of the amplitude of the point source speckle pattern and a diagonal matrix representing the astronomical object. Recent results obtained for the bright infrared double star ? Aqr and simulations of the implementation of the algorithm for optical photon-counting detectors with clipping problems are given. The use of the technique for synthesized apertures is also considered.

We present an experiment aimed at diffraction-limited imaging of astronomical sources in the 2-5 ?m range. It is based on a rotation shearing interferometer that produces fringes in the pupil plane that can be scanned through the zero optical path difference. The two-dimensional interferograms are recorded with an infrared camera looking at the recombined pupil image. The modulus and the phase of the Fourier transform of the object intensity distribution are derived from these interferograms. The main advantage of this technique is its constant transfer function that makes it independent of seeing variations and instrumental aberrations. We describe the experiment set-up and discuss some simulation results which illustrate the operation of the interferometer. We present astronomical data recently obtained at the 4.20m William Herschel Telescope of the Royal Greenwich Observatory in La Palma, and results on the circumstellar shell star NML Cygni.

Post Detection Turbulence Compensation has been proposed as a technique to obtain diffraction limited imaging through the atmosphere. The technique consists of two simultaneous measurements, one made at a point conjugate to the pupil plane of an optical system and one at it's focal plane. The optical transfer function of the atmosphere is constructed from the pupil plane measurement. This is then deconvolved from the focal plane image. The result is an image with resolution at or near the diffraction limit of the imaging system. The process is highly non-linear and as a result the quality of the restored images is difficult to estimate analytically. In an attempt to understand the characteristics of this process we have developed computer models to simulate speckle holography image compensation. This paper will show results of computer simulations showing reconstructed image quality versus light levels.

Images are recorded with a CCD camera through a laboratory simulated 9-aperture stellar interferometer. Aperture rotation is used to improve the M,V plane coverage. From these images, objects are reconstructed using the CALTECH VLBI data reduction package. The result is compared to the original object. The dynamic range of the reconstruction process is studied under various experimental conditions

The principles of image formation using a dilute, multiple- telescope interferometer will be considered. In particular, the requirements for an instrument capable of providing a unique reconstruction of object Fourier phases and instrumental phase errors from a single 'snapshot' of data will be examined. The roles of redundancy in the telescope array, the imposition of the positivity requirement in data inversion, the quality of the instrument point spread function and the stability of the data inversion will be taken into account.