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- Inverse Scattering
- Regularization and Data Inversion
- Geophysical and Related Inverse Problems
- Image Restoration and Reconstruction
- Tomography
- Phase Retrieval
- Tomography
- Phase Retrieval
- Regularization and Data Inversion
- Image Restoration and Reconstruction
- Tomography
- Regularization and Data Inversion

Inverse Scattering

Wave-field splitting, invariant imbedding, and phase space methods in direct and inverse scattering

Louis Fishman

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Wave field splitting, invariant imbedding, and phase space methods reformulate the Helmholtz wave propagation problem in terms of an operator scattering matrix characteristic of the modeled environment. The subsequent equations for the reflection and transmission operators are first-order in range, nonlinear (Riccati-like), and, in general, nonlocal. The reflection and transmission operator equations provide the framework for constructing inverse algorithms based on, in principle, exact solution methods.

Invariant imbedding method and inverse source problems

Zhiming Sun,
James P. Corones

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This paper studies the direct scattering and inverse source problems for an one-dimensional inhomogeneous slab. The method used is the time domain wave splitting and invariant imbedding technique. For the case when the internal source j is a product, i.e., j(x,s) equals D(x) i(s)Y, a new current scattering operator J that maps the function i(s) into the scattered waves at the boundaries of the slab is defined. A system of coupled nonlinear integrodifferential equations for the current scattering operator kernel J(x,s) and the reflection operator kernel R(x,s) is derived. The inverse source problem solved in this paper is recovering the source space distribution function D(x) from the given permittivity profile and current scattering operator kernel J(0,s) for 0 <EQ s <EQ 1. Numerical results of the computation of the J kernel and the reconstruction of D(x) are presented.

Nonuniqueness in direct and inverse electromagnetic scattering theory

Brian DeFacio,
S. H. Kim

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General statements of impossibility can be important in science and engineering. Ambiguities in inverse problems are cases of non-uniqueness where classes of different objects give the same response. A strong ambiguity is one which no additional data will remove the non- uniqueness, whereas a weak ambiguity is one which can be removed by additional data. In direct scattering theory, different potentials with one or more trapped modes may give the same R(k) or the e

^{i(alpha})R(k) where (alpha) is a real parameter at all wave-numbers k. In three-dimensional direct scattering theory, different material media and sources J, p give the same scattering matrix at all times (or wave numbers) at all scattering angles and all incident angles. Examples of strong ambiguities will be given including one where a temporal relaxation of a homogeneous body is equivalent to a totally different time-independent homogeneous body. Weak ambiguities will be presented including both examples of incident scatters. The conditions on the scatterers at spatial infinity and their trapped mode bound-state structure will be given.
Numerical determination of potentials

William Rundell,
Paul Sacks

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In this paper we will describe some computational methods for the recovery of potentials under a variety of types of data measurements. We will look at both inverse Sturm-Liouville problems on a finite interval and inverse scattering problems on the line. The unifying approach to all of this is the fact that many of these types of problems can be solved by converting the given spectral or scattering data into boundary data for a certain hyperbolic partial differential equation. In all cases the problem is an overdetermined one and it is precisely this fact that allows us to recover the potential. Further, we will show that this translated problem can be solved in a numerically stable way, and indeed this approach leads to an excellent, as well as a unifying, scheme for the reconstruction of the potential. There is a classical method of solving many of the above types of problems and the definitive formulation is due to Gel'fand, Levitan and Marchenko some forty years ago. Our approach has many similarities, and indeed the same starting point, but the crucial difference is while the original scheme reduced the problem to a Fredholm integral equation, we will exploit instead an equivalent hyperbolic partial differential equation.

Inverse eigenvalue problem for a circular membrane

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The problem of determining the density distribution of a circularly symmetric elastic membrane, fixed on the edge, from its frequencies of free vibration, is examined. An asymptotic analysis implies that a density function that is a small perturbation from a homogeneous membrane is determined, up to a finite number of parameters, by two infinite spectra of appropriate angular orders. An algorithm for reconstructing the density from the spectra is presented. The results are illustrated by numerical simulations.

Cepstral filtering for recovery of object from scattered field data

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In this paper, we calculate information about the structure of strongly scattering objects with permittivity fluctuations having scales comparable with the illuminating wavelength. This is the situation for which small wavelength approximations or weakly scattering approximations are not valid. We formulate the problem as one of recovering an object function from a set of noisy versions of itself, which is a problem that frequently arises in imaging. We employ homomorphic filtering and differential cepstral filtering and show that information about the scattering object can be obtained.

Regularization and Data Inversion

New data inversion formula in confocal scanning microscopy

Christine De Mol,
Michel Defrise

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Whereas the resolving power of an ordinary optical microscope is determined by the classical Rayleigh distance, significant super-resolution, i.e. resolution improvement beyond that Rayleigh limit, has been achieved by confocal scanning light microscopy. Furthermore is has been shown that the resolution of a confocal scanning microscope can still be significantly enhanced by measuring, for each scanning position, the full diffraction image by means of an array of detectors and by inverting these data to recover the value of the object at the focus. We discuss the associated inverse problem and show how to generalize the data inversion procedure by allowing, for reconstructing the object at a given point, to make use also of the diffraction images recorded at other scanning positions. This leads us to a whole family of generalized inversion formulae, which contains as special cases some previously known formulae. We also show how these exact inversion formulae can be implemented in practice.

Iterative image reconstruction algorithms based on cross-entropy minimization

Charles L. Byrne,
James Graham-Eagle

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The multiplicative algebraic reconstruction technique (MART) is an iterative procedure used in reconstruction of images from projections. The problem can be viewed as one of finding a nonnegative approximate solution of a certain linear system of equations y equals Px. In the consistent case, in which there are nonnegative solutions of y equals Px, the MART sequence converges to the unique nonnegative solution for which the Kullback-Leibler distance KL(x,x

^{0}) equals (summation) x_{j}log(x_{j}/x^{0}_{j}+ x^{0}_{j}-x_{j}is minimized, where x^{0}> 0 is the starting vector for the iteration. When x^{0}is constant the sequence converges to the maximum Shannon entropy solution, at Lent has shown. The behavior of MART in the inconsistent case is an open problem. When y equals Px has no nonnegative solution the full MART sequence [z^{k}, k equals 0,1,...] does not converge, while the 'simultaneously updated' version, SMART, converges to the nonnegative minimizer of KL(Px,y). In every example we have considered, the subsequences [z^{nI+i}, i fixed, n equals 0,1,...,] consisting of those iterates associated with completed cycles (I is the number of entries in y) do converge, but to distinct limits, which we denote z^{(infinity},i.). Unlike most other reconstruction algorithms, if the new limiting projection data [Pz^{(infinity},i)_{i}] is used in place of the original data y and the algorithm repeated, we do not recapture [Pz^{(infinity}i.)_{i}]; this suggests that the MART algorithm as usually presented may be but part of a complete algorithm involving feeding back the new projection values until convergence. In all our simulations this expanded version of MART has converged, and the limit is the same as SMART; that is, the nonnegative minimizer of KL(Px,y). Both the MART and relaxed MART algorithms can be obtained through the alternating minimization of certain weighted Kullback- Leibler distances between convex sets. Orthogonality conditions in the form of Pythagorean- like identities play a useful role in the proofs concerning convergence of these algorithms.
Fast image reconstruction based on the regularized pseudoinverse of the imaging operator

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A new algorithm for the restoration of extended images, the Regularized Pseudoinverse Deconvolution (RAPID) algorithm, is proposed. The algorithm consists of expanding the regularized pseudoinverse of the imaging operator into a sequence of terms which can be easily implemented using Fourier processing techniques. The first term of the expansion is closely related to generalized Wiener filtering if the point spread function is shift-invariant. The other terms in the expansion are correction terms which are small when the point spread function is shift-invariant, as is the case with many imaging systems. Even when the point spread function of the imaging system is space-variant, such as with a partially obscured imaging system or a system with severe aberrations, the correction terms are both few in number and easily implemented.

Two-dimensional image analysis using the wavelet transform

Charles R. Thompson,
Brian DeFacio

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The wavelet transform is used to give a phase-space approach to the study of two-dimensional images. The compression of the Daubechies - 4 (D4) wavelet is studied and a new hybrid wavelet-linear interpolation method is shown to enhance the compression of reconstructions. Computer generated white noise is added and two methods of image enhancement, which are based on the discrete wavelet transform, are used to improve resolution and sensitivity of the noisy images. The frequency scales in phase-space dominate both the 'contrast' and the edge detection behaviors of the images.

Information-to-noise improvement in the frequency domain using the wavelet transform

Charles R. Thompson,
Brian DeFacio

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Some finite energy, causal, real-valued signals are frequency analyzed using a fast Fourier transform (FFT). Computer generated additive white noise is combined with time-domain samples, and the analysis is repeated. It is shown that the combination of noise and aliasing can hide information when high frequency noise is folded over an interesting part of the frequency domain. Aliasing cannot be avoided in exploratory physics and engineering experiments and aliased noise can drown out real peaks in the signal frequency spectrum. Time-domain filtering techniques are discussed, but they cannot always recover these lost peaks. A new method is introduced in which the wavelet transform and its inverse are used to suppress high frequency noise prior to Fourier analysis. This method is useful in improving on experimental results when aliasing of high frequency noise presents a problem.

Multichannel image identification and restoration based on the EM algorithm and cross-validation

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In this paper we address some of the main shortcomings of multi-channel (MC) linear restoration filters. The problem of restoring a MC image and simultaneously estimating the MC power spectrum of the image and the noise, required by linear minimum mean squared error (LMMSE) filters is investigated, using the expectation-maximization (EM) algorithm. Second, the problem of estimating, the regularization parameters and operator, required by regularized least-squares (RLS) MC restoration filters is investigated using the cross-validation (CV) function. Furthermore, a novel representation of MC signal processing is introduced. This notation leads to a more natural extension of single-channel (SC) signal processing algorithms to the MC case and yields a new class of matrices which we call semi-block- circulant (SBC) matrices. The properties of these matrices are examined and a family of new efficient algorithms is developed for the computation of the MC EM and CV functions.

Regularized image reconstruction using neural networks

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Iterative methods have long been studied in order to reconstruct images from limited noisy spectral data or low pass filtered noisy images; they rely on minimizing a well-defined energy function. Such methods can be implemented on Hopfield neural networks, as a direct result of comparing energy function parameters. Consequently, a fully parallel (neural) processor can be programmed to implement a reconstruction algorithm. We have studied the properties of these neural solutions and show that they provide a regularized and apodized result with some attractive and interesting properties.

Geophysical and Related Inverse Problems

Nonlinear inverse scattering method for multiple suppression and source signature estimation

Arthur B. Weglein

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A nonlinear inverse scattering procedure for multiple suppression is presented. A method for removing the combined effects of the source signature and instrument response is also described. The ability to remove these effects is an important factor in determining the real- world efficacy of this multiple suppression procedure.

Inverse source problem in the presence of external sources

Arthur B. Weglein,
Anthony J. Devaney

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This paper presents a brief review of the various integral equation formulations that have been employed for the inverse source problem for the inhomogeneous scalar Helmholtz equation. It is shown that these formulations apply only in cases where either the data are prescribed on a closed surface surrounding the unknown source or where the unknown source lies entirely on one side of an open measurement surface. A generalized integral equation is derived that applies to the more general case where unknown sources can exist on both sides of an open measurement surface. This latter problem arises in geophysical remote sensing and the derive integral equation offers an approach to this class of problems not offered by currently employed techniques.

Field studies in geophysical diffraction tomography

Alan J. Witten,
S. Scott Stevens,
Wendell C. King,
et al.

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Geophysical diffraction tomography (GDT) is a quantitative, high-resolution technique for subsurface imaging. This method has been used in a number of shallow applications to image buried waste, trenches, soil strata, tunnels, synthetic magma chambers, and the buried skeletal remains of seismosaurus, the longest dinosaur ever discovered. The theory associated with the GDT inversion and implementing software have been developed for acoustic and scalar electromagnetic waves for bistatic and monostatic measurements in cross-borehole, offset vertical seismic profiling and reflection geometries. This paper presents an overview of some signal processing algorithms, a description of the instrumentation used in field studies, and selected imaging results.

Imaging algorithms for geophysical applications of impedance tomography

John E. Molyneux,
Alan J. Witten

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The methods of impedance tomography may be employed to obtain images of subsurface electrical conductivity variations. For practical reasons, voltages and currents are usually applied at locations on the ground surface or down a limited number of boreholes, but almost never over the entire surface of the region being investigated. The geophysical inversion process can be facilitated by constructing algorithms adopted to these particular geometries and to the lack of complete surface data. In this paper we assume that the fluctuations in conductivity are small compared to the background value. The imaging of these fluctuations is carried out exactly within the constraints imposed by the problem geometry. Several possible arrangements of injection and monitoring electrodes are considered. In two dimensions these include: Cross-line geometry, current input along one line (borehole) and measurements along a separate parallel line. Single-line geometry, injection and monitoring using the same borehole. Surface reflection geometry, all input and measurement along the ground surface. Theoretical and practical limitations on the image quality produced by the algorithms are discussed. They are applied to several sets of simulated data, and the images produced are analyzed.

Simultaneous inversion of permittivity and conductivity employing a nonperturbative approach

Tarek M. Habashy,
M. L. Oristaglio,
A. T. de Hoop

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We present a new inversion algorithm for the simultaneous reconstruction of permittivity and conductivity that recasts the nonlinear inversion as the solution of a coupled set of linear equations. The algorithm is iterative and proceeds through the minimization of two cost functions. At the initial step, the data is matched through the reconstruction of the radiating or minimum-norm scattering currents; subsequent steps refine the nonradiating scattering currents and the material properties inside the scatterer. We give recipes for constructing basis functions for the nonradiating components of the scattering currents. The method is illustrated through the reconstruction of a large contrast square cylinder from multiple-source measurements at a single frequency.

Solution of a scalar inverse-scattering problem: qualitative and quantitative ultrasonic imaging in fluid media

Patricia Grassin,
Bernard Duchene

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Here, we aim at imaging 3-D fluid objects concealed within a homogeneous fluid medium and illuminated by a time-harmonic source, from single-frequency multiview data. Two techniques are considered: (1) a diffraction tomography algorithm with Fourier domain interpolation, which provides qualitative images in a very efficient way; (2) a Newton-Kantorovich algorithm which builds up an iterative solution of a non-linear ill-posed inverse scattering problem, through a local linearization and a regularization procedure. Both synthetic and experimental data are considered.

Eddy-current imaging of defects in a conductive half-space

Rodrigo de Oliveira Bohbot,
Dominique Lesselier,
Bernard Duchene,
et al.

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Eddy currents generated within damaged, conductive structures are cast in a wave formalism so that coil-recorded fields appear radiated by Huyghens sources within the defect. This leads to wave-imaging techniques, with limitations due to strong decay with depth of the magnitude of the probing field in the structure. Here, we aim at imaging 2-D anomalies in a metal half- space from the anomalous electromagnetic field above, for a time-harmonic source. We discuss (1) a direct Diffraction Tomography algorithm such that attenuation is accounted for via introduction of imaginary frequencies; (2) an iterative simulated annealing method which builds up a two-valve solution (zero or one, whether an elementary cell consists of air or metal) associated with the global minimum of a cost function linked with the discrepancy between existing and retrieved defect.

Generalized cross-validation applied to a Newton-type algorithm for microwave tomography

Anne Franchois,
Christian Pichot

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The non-linear inverse scattering problem, occurring in the field of microwave tomography for biomedical applications, is solved with the Newton-type iterative algorithm. This method allows the reconstruction of the complex permittivity of strongly inhomogeneous objects. An ill-conditioned system of linear equations is obtained at each iteration and is regularized using Tikhonov's method. The choice of regularization is crucial, specially when random error is present in the data. We investigate the Generalized Cross Validation method for choosing the regularization parameter. Numerical examples are given in the case of 2D TM inverse problems.

Image Restoration and Reconstruction

New results in projection-type algorithms of use in image processing and reconstruction

Henry Stark,
Elwood T. Olsen

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In image restoration the two main goals are noise smoothing and restoration of sharp details. We consider the noise smoothing problem from the point-of-view of convex projections and introduce several normed derivative constraints and their associated projectors.

Signal processing of ultrasonic tomographic data

George A. Tsihrintzis,
Anthony J. Devaney

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We process a set of data measured with a prototype ultrasonic scanner developed by the Norwegian company Norwave Development AS, Oslo, Norway in collaboration with the American company A.J. Devaney Associates, Boston, Massachusetts. In particular, we apply signal processing algorithms, recently developed by the authors to locate known test objects in a fluid background. Possible applications of the research include locating and identifying cancerous tumors in human tissue. These and other avenues for future research are discussed in the paper.

Imaging with evanescent waves in diffraction tomography

Alon Schatzberg,
Anthony J. Devaney

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The possibility of incorporating evanescent wave components of the scattered field into diffraction tomographic reconstruction algorithms offers new options to improve the resolution in reconstructive diffraction tomography. Evanescent wave information has however not been incorporated into diffraction tomographic reconstruction algorithms due to special difficulties that they present. Here we present a generalization of the filtered backpropagation algorithm that incorporates evanescent wave data to improve the resolution limit beyond the usual half wavelength limit.

Quantitative image recovery and restoration from scattered field data at 10GHz

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Numerical methods developed to invert scattered field data are applied in this paper to data collected from a styrofoam target. The imaging methods used are based on the distorted wave 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. Also, the distorted-wave approximation is described and the procedure enabling it to be used for high resolution imaging of weakly scattering features is presented.

Submicron optical ranging of nanometric particles in semiconductor materials

Michel Castagne,
Jean-Pierre Fillard,
Jean-Marc Lussert,
et al.

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Semiconductor materials (bulk material as well as epilayers) contain grown-in defects such as nanometric microprecipitates. These particles play a dramatic role in the electrical specifications of the devices. A new method of investigating the vertical position of these defects in the IC or OEIC layered structures is proposed. It stems from the 3D observation of defocused Point Spread Functions arising from the illumination of these small scattering defects. The calculation of the Fourier transform is numerically performed and the z position of the point sources is deduced from the Fourier Phase Shift. This method is capable of sub- micron measurement. Numerical simulations as well as experimental illustrations are presented which confirm the potential of the method.

Reconstruction of a three-dimensional object from its conoscopic hologram

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Conoscopic holography is a method for recording holograms with incoherent light, first presented in 1985. Its applications range from 3D microscopy to 3D satellite imaging and include robotics. The Point Spread Function (PSF) is a Gabor Zone Pattern, which is known to have zeros in Fourier space. We present an experimental technique to obtain an invertible PSF with an experimental image reconstruction, and an original algorithm to find the object shape, validated with both simulations and first experimental results.

Tomographic reconstruction of strongly refracting fields and identification of light rays simultaneously

Ke Liu,
Yong-Shen Xiong,
Jian Lu,
et al.

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A direct 3-D method is proposed in this paper for reconstructions of strongly refractive index fields. The basic idea is to reconstruct a refractive index field and to identify the ray path simultaneously. The 3-D ray tracing method and the curved ray algebraic inversion method based on many-knot interpolating spline technique and the solving method for linear equations by bidiagonalizing matrices are presented to realize the idea. The simulation experimental results are also provided.

Tomography

Efficient scanning in fan-beam tomography

Frank Natterer

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We describe an efficient arrangement of rays in a fan beam CT scanner which needs considerably less (by a factor of 2-4) rays than in conventional fan beam scanning to obtain a comparable image quality.

Experimental investigation of simultaneously recorded shadowgraphs and images through a high-velocity turbulent flow

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Experiments were conducted by Teledyne Brown Engineering (TBE), that simultaneously recorded flow field shadowgraphs and imaging data, to investigate the relationship between high velocity turbulence and aero-optic image distortion. A laboratory based Dual Nozzle Aero-Optic Simulator (DNAOS) was used to produce a turbulent flight level aero-optic environment similar to that encountered by a hypersonic vehicle. An Nd:YAG laser, operating at 1.064 micrometers , was expanded to 5 mm, collimated, and directed through the turbulent flow field to serve as a point source at infinity. On the opposite side of the flow field, the beam was split into two components and directed towards two 60 micrometers square pixelated, 128 X 128 CCD array cameras. One camera had a bare focal plane and was used to record the turbulence induced scattering field (shadowgraph), while the other had a 3.4 m focal length lens to image this field, producing a point-spread-function (PSF) on the CCD array. A 50 nsec duration laser pulse at a frequency of 92.5 Hz (frame rate of the CCD cameras) was recorded by each of the cameras and the data was digitized by a high speed data acquisition system. The shadowgraphs and imaging data were compared frame-for-frame to determine the similarities between the flow field events and the image distortion. Based on this analysis, a procedure has been proposed to numerically transform shadowgraphs to obtain pseudo-images that could be compared to experimentally recorded images.

Tomographic reconstruction from partial angular views using Gibbsian models

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In some applications of tomographic reconstruction of 2D or 3D fields, the presence of physical constraints allows measurement of projections only within limited angular views; in these cases, a satisfactory resolution may be obtained only with the addition of a priori information about the structure to be estimated. Besides deterministic constraints, some form of probabilistic knowledge is often available, which could help to eliminate a large class of unlikely solutions in the inversion process. However, despite their elegant simplicity, some proposed Gaussian models have shown to be inadequate to satisfactory model objects in actual environments. For these reasons, the paper proposes a flexible probabilistic model based on Gibbs Random Field (GRF) which can be used for tomographic reconstruction. The theoretical framework of the method is described and the performance of the algorithm are illustrated through some simulated experiments.

Analytic inversion of a general model in diffuse tomography

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The idea of diffuse tomography has been recently introduced as a way of modeling an imaging problem using photons with very low energy. It can be seen as a far reaching extension of the standard tomographic problem where photons are assumed to travel in a straight line. Although any real-life application will require the solution to the three-dimensional problem, we start with the two-dimensional problem.

Reconstruction of embedded absorbers in random media with applications in noninvasive 3D imaging of skin lesions

Prashanth Kini,
Atam P. Dhawan

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An optical backscatter measuring instrument called a Nevoscope is being developed to non- invasively determine 3D characteristics of skin lesions with applications in the diagnosis of malignant melanoma. Optical images are obtained by transilluminating the lesion and imaging backscattered radiation emanating from the skin surface. Such emission profiles contain information about the absorption characteristics of sub-surface structures. It is conceivable that such profiles can be used to reconstruct structural information of inhomogeneities such as mole embedded in the skin. Monte Carlo simulations of photon migration are performed to simulate the radiation pattern of backscattered radiation imaged at the surface of the medium when a light source is placed directly on the surface of the medium. In particular, simulations are performed on media with single embedded absorbers. This is the simplest model of a mole embedded in human skin. Some reconstruction algorithms based on the difference in emission profiles is the presence and absence of the absorber, are tested. A prototype Nevoscope presently being used for visual examination is described. Heuristic reconstruction schemes using images obtained from the Nevoscope are also presented.

Iterative reconstruction of near-infrared absorption images

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The successful development and clinical use of instruments that perform real-time near-infra red spectroscopy of transilluminated tissue has led to a widespread interest in the development of an imaging modality. The most promising approach uses picosecond laser pulses input on an object (Omega) , and measures the development of light intensity as a function of time at points on the boundary (partial)(Omega) . The imaging problem is to reconstruct the absorption and scattering coefficients inside (Omega) . We have proposed the following method for the reconstruction algorithm: A Forward model is developed in terms of the Green's Function of the Diffusion Approximation to the Radiative Transfer Equation. Given a perturbation of the image, the Jacobian of the Forward model can be derived. Inversion of the Jacobian then gives a perturbation step for a subsequent iteration. Previously we have derived an analytical expression for the Green's Function in certain simple geometries, and for a homogeneous initial image. We have now developed a Finite Element method to extend this to more general geometries and inhomogeneous images, with the inverse of the system stiffness matrix playing the role of the Green's Function. Thus it is now possible to proceed past the first iteration. The stability of the reconstruction is presented both for the time-independent case where the data is the absolute intensity on the boundary (partial)(Omega) , and for the time-dependent case where the data is the mean time of arrival of light.

Phase Retrieval

Phase retrieval in inverse scattering

Mohammad H. Maleki,
Anthony J. Devaney

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This paper discusses application of phase retrieval in inverse scattering, using optical diffraction tomography as an example. We consider two algorithms for recovering the phase of a scattered field from intensity measurements. The first algorithm is iterative and uses one intensity measurement and support information of the object to retrieve the phase. The second algorithm is non-iterative and uses two intensity measurements at different distances from the object to recover the phase by solving a pair of coupled algebraic equations. The second algorithm requires that the object by weakly scattering, a condition also required by the usual assumption of the Born or the Rytov approximations in diffraction tomography. The paper includes computer simulations of the two phase retrieval algorithms and compares the results to reconstruction obtained from full simulated data and a direct intensity-only reconstruction algorithm.

Bayesian signal reconstruction from Fourier-transform magnitude and x-ray crystallography

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A Bayesian binary signal reconstruction problem which includes noisy magnitude of Fourier transform measurements and a Markov random field a priori model was solved. The solution is analytical and is based on the spherical model and small noise asymptotic approximations. Parameters in the solution are used for data adaptation. The work is motivated by the phase retrieval problem in x-ray crystallography where the signal is the periodic electron density in the crystal. In crystallography, the signal is known to be invariant under the actions of some space group symmetry (e.g., division of the repeat unit of the crystal into two halves with one half the mirror image of the other half). The cited references have been extended in three different ways to incorporate this additional information. In addition, a numerical optimization in the cited references has been improved by the use of analytical gradients which can be rapidly computed using FFT based formulae.

Fourier spectrum extrapolation and enhancement using support constraints

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The use of support constraints for improving the quality of Fourier spectra estimates is discussed in this paper. It is shown that superresolution is an additive phenomena which is a function of the correlation scale induced by the support constraint and is independent of the bandwidth of the measured Fourier spectrum. It is also shown for power spectra that support constraints, due to the enforced correlation of power spectra, reduce the variance of measured power spectra. These theoretical results are validated via computer simulation in the area of speckle interferometry, with very good agreement shown between theory and simulation.

Finite-detector effects in speckle imaging

Robert W. Johnson,
George J. M. Aitken

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When astronomical speckle images are truncated by the finite size of the image detector, important errors occur in the estimated power spectrum and the Knox-Thompson cross- spectra. The errors and their effect on image reconstruction are calculated for an 8-component and a 3-component star model. The distortions and negative artifacts seen in the reconstructions of the star ADS 11344 are explained by image truncation. A method of estimating an approximate error bound given the telescope, atmospheric seeing, detector size, and sampling rate is outlined.

Point-source subpixel positioning from Fourier phase-shift retrieval

Jean-Pierre Fillard,
Michel Castagne,
Jean-Marc Lussert

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A new method has been recently proposed which aims at measuring the relative location of point light emitters from 2D or 3D diffraction patterns; this typical inverse problem of interpretation of the scattering field can be solved with sub-pixel accuracy. It is based on the retrieval of the Fourier phase versus frequency dependence of the calculated Optical Transfer Function of the optical system. The method is favorably compared to the classical centroid method, especially in the case of locally oscillating fields. Computer implementation is proposed which allows fast sub-pixel evaluation of the z coordinates of a collection of numerous point sources. This method applies to various situations such as robotics, precision tracking, astronomy or more specifically sub-micron ranging of nanometric sized scattering particles in a microscope field. Details are given of how to obtain significant information and how to use numerical procedures in experimental frameworks. Evaluation of the accuracy of the method and algorithms is considered.

Phase determination using transform and contrast-variation methods in neutron diffraction studies of biological lipids

David L. Worcester,
Helmut Kaiser,
R. Kulasekere,
et al.

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Biological lipid bilayer structures have been studied which give neutron diffraction data to better than 0.3 nm resolution. The general problem of phase retrieval in these one- dimensional, centrosymmetric cases was investigated using transform method together with deuterium incorporation (contrast variation) in order to establish experimental procedures of general use for such neutron scattering studies. The findings demonstrate procedures which, in many cases, quickly provide the experimental data necessary for reliable phase retrieval.

Tomography

Time-series analysis of event-driven evoked potentials

O. H. Stanley M.D.,
M.P. J, Wright,
A. A. Pike M.D.,
et al.

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This paper describes a method which aims to identify the stimulus related EEG components which make up the average evoked potential and to obtain confidence intervals for component parameters, such as power and peak amplitude. It is hoped that this will define EEG components that correspond to discrete brain sensory processes and which can be compared between individuals.

Phase Retrieval

Theory of the amplitude-phase retrieval in any linear-transform system and its applications

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This paper is a summary of the theory of the amplitude-phase retrieval problem in any linear transform system and its applications based on our previous works in the past decade. We describe the general statement on the amplitude-phase retrieval problem in an imaging system and derive a set of equations governing the amplitude-phase distribution in terms of the rigorous mathematical derivation. We then show that, by using these equations and an iterative algorithm, a variety of amplitude-phase problems can be successfully handled. We carry out the systematic investigations and comprehensive numerical calculations to demonstrate the utilization of this new algorithm in various transform systems. For instance, we have achieved the phase retrieval from two intensity measurements in an imaging system with diffraction loss (non-unitary transform), both theoretically and experimentally, and the recovery of model real image from its Hartley-transform modulus only in one and two dimensional cases. We discuss the achievement of the phase retrieval problem from a single intensity only based on the sampling theorem and our algorithm. We also apply this algorithm to provide an optimal design of the phase-adjusted plate for a phase-adjustment focusing laser accelerator and a design approach of single phase-only element for implementing optical interconnect. In order to closely simulate the really measured data, we examine the reconstruction of image from its spectral modulus corrupted by a random noise in detail. The results show that the convergent solution can always be obtained and the quality of the recovered image is satisfactory. We also indicated the relationship and distinction between our algorithm and the original Gerchberg- Saxton algorithm. From these studies, we conclude that our algorithm shows great capability to deal with the comprehensive phase-retrieval problems in the imaging system and the inverse problem in solid state physics. It may open a new way to solve important inverse source problems extensively appearing in physics.

Regularization and Data Inversion

Multiscale filtering approach to focus blur removal

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Detection and subsequent removal of focus blur in digital images is often undertaken using expensive computational solutions. An alternative approach using simple gradient-based edge detection masks of varying sizes applied locally is presented here. Results indicate that this approach is able to provide significant improvement in the perceived sharpness of blurred edges.

Image Restoration and Reconstruction

Image restoration using nonlinear optimization techniques for partially compensated imaging systems

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This paper shows the development of the imaging restoration problem entirely in the frequency domain, then solves for the analytical solution. The analytical solution is found to be ill-posed, thus, a good approach for the solution is via nonlinear optimization. The image recovery problem is thus formulated in the context of nonlinear optimization using Fourier data. Several examples are shown using unconstrained optimization techniques with the incorporation of the conjugate gradient algorithm. These examples basically are the inverse filter solutions. Three main diagnostic metrics are shown and discussed as possible candidates for evaluation of the results.

Tomography

Layer-stripping approach for recovery of scattering media from time-resolved data

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This paper studies the reconstruction of the absorption properties of a dense scattering medium from time-resolved data. A Progressive Expansion (PE) Algorithm, similar to a layer-stripping approach, has been developed. The method progressively evaluates increasing depths within the medium by successively considering signals entering the detector at increasing time following an incident pulse. In order to reduce the propagation of reconstruction errors occurring at shallower depths, an overlapping scheme is introduced which uses readings from several consecutive time intervals in the reconstruction. In each overlapping time interval, the region under consideration is solved using a perturbation approach recently described by our group. The proposed algorithm is applied to several inhomogeneous media containing simple structures. Two sets of data have been tested: one calculated according to the perturbation model; and the other by Monte Carlo simulations. The results show that the PE method, when combined with proper overlapping, can make effective use of the time-resolved data. Compared to our previous results with steady-state data, the present methods can probe deeper below the surface and produce a more accurate estimate.

Regularization and Data Inversion

Coherent optical tomographic microscope

Anthony J. Devaney,
Alon Schatzberg

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A new microscopic procedure for imaging three-dimensional semitransparent specimens with coherent illumination is described. The procedure employs multiple 'views' of the specimen, obtained by illumination via monochromatic plane waves having different directions of propagation, to generate a high quality image of the three-dimensional complex index of refraction profile of the specimen using the reconstruction algorithms of diffraction tomography (DT). The theory and hardware implementation of such a 'tomographic' microscope are described and the concept is compared to conventional optical microscopy, confocal scanning microscopy, and to other tomographic based optical microscopic concepts that have been employed in the past.