This paper reviews diffraction tomography as it applies to electromagnetic and acoustical imaging. We have discussed the various approximations that are used and shown their effects by computer simulation. In addition results are shown for three reconstruction algorithms and their computational requirements are explored.*
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The three basic methods for the reconstruction of an object from its measured projections , namely the central slice theorem, filtered back projection and the Radon transform, are shown to follow independently and directly from very simple requirements on the object density. It is suggested how the approach developed here may lead to new imaging procedures. A consideration of the backprojected image, which is relatively easily obtained to high quality by analogue methods, reveals a novel interpretation which suggests a new approximate algorithm for object reconstruction. A similar approach is suggested to unify inverse scatter methods, in the weak scattering approximation.
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Tomographic algorithms have been used to generate cross-sectional images of sound velocity in the human body from time-of-flight measurements of ultrasonic pulses. Similar algorithms have been employed to reconstruct optical refractive-index fields, in which optical path lengths are measured interferometrically rather than transit times. Using a ray propagation model, an ultrasonic transit time or optical path length measurement is proportional to the line integral of the ultrasonic (or optical) refractive index over the ray path. Under the assumption that the propagation paths are straight, conventional computerized tomography (CT) algorithms have been used to perform the reconstructions. In real media, ray refraction introduces a time-of-arrival (or optical path length) error in the measurement, leading to image degradation. To date, only iterative techniques based on numerical ray tracing have been proposed to correct for the effects of refraction. In this paper, we present a perturbation approach to this problem which, for relatively small refractive-index fluctuations, requires neither iteration nor ray tracing. Assuming that the average deviation of the refractive index from its mean is on the order of the small quantity e, an expression is derived for the ray trajectory whose departure from a straight line is first order in E. Using this first-order ray path, we obtain a perturbation expansion of the path integral of the refractive index along the refracted ray and derive a path-length correction of order e2 arising from the deviation of the refracted ray from a straight line. This second order correction can then be applied to refraction-degraded time-of-flight or optical path length measurements before submission to a conventional CT algorithm. The result is an improvement in image quality after correction.
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The problem of determining a function from known integrals of that function over hyper-surfaces (curves in 2D case) leads to the notion of the generalized Radon transform. We treat this problem and describe the inversion procedure. It is shown that the problem of inversion of the generalized Radon transform can be reduced to solving a Fredholm integral equation. Also, we consider some applications. In particular, we estimate the error of Shepp-Logan's filter used in Computerized X-ray Tomography, present inversion formulae for Exponential and Attenuated Radon transforms, and consider the example of the Hyperbolic Radon transform.
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The problem of reconstructing a multi-dimensional field from noisy, limited projection measurements is approached using an object-based stochastic field model. Objects within a cross-section are characterized by a finite-dimensional set of parameters, which are estimated directly from the projection measurements using maximum likelihood estimation. The computational structure, performance and robustness of the ML estimation procedure are investigated.
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The frequency-domain Neumann-Born approximation is used to study the determination of the radii and material parameters of spherical flaws. Bandwidth limiting and the effects of noise are studied. An eikonal approximation by Rose and DeFacio was also used. Only the reconstruction using a single scattering angle is addressed. Backscattered amplitudes (θ = 180°) and near forward scattering amplitudes at θ = 30° were not as successful as θ = 60° data. The radii were determined better than material parameters and in the case of a single backscatter with 5% noise and fairly strong scattering no radius was determined.
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Despite its popularity (due to its simplicity) the Born/Rayleigh-Gans approximation has a tiny range of practical applicability. WKB (called LSJWKB here) transformation of the wave equation suggests an apparently widely valid monochromatic quasi-Born inverse scattering approximation, which promises to lead to images which, while suffering from irreducible geometric distortion, exhibit faithful contrast. Virtually all monochromatic images are degraded by speckle artefacts, which can be inhibited by averaging over a wide band of frequencies. We claim here that our recently developed coherent shift-and-add processing of ultra-sonic scattering data provides support for our new interpretation of the LSJWKB-transformed wave equation. Experimental results, demonstrating image enhancement simultaneously with speckle reduction, are presented.
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For most theories of wave propagation in a random medium the statistical quantity of primary interest is the two point correlation function associated with the index of refraction fluctuations. Here we show how this function may be determined in a statistically homogeneous and isotropic medium by measuring two and four point coherence functions after the radiation has propagated through the medium. We shall discuss both plane wave and finite beam initial conditions.
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Wolf's formulation for inverse scattering in the first Born approximation is reviewed and discussed from an experimental point-of-view. Some of the experimental problems associated with obtaining enough data to satisfy the requirements of this theory are described and a method for using symmetry to reduce the data requirements for a class of scatterers is explained. A successful experiment which employed this theory to reconstruct structure in an object with rectangular structure is reviewed. It appears that this theory is extremely difficult to apply to a more general scatterer with no simplifying symmetries.
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This paper presents work using the effective area-function approach to inverse problems in the scattering of elastic and electromagnetic waves. The emphasis in this paper is on the technique of inventing experimental elastic wave scattering data to obtain size and shape information. The procedure of elastic wave data acquisition and some results of the inversion technique are described for a small (~ 1000 μm) ellipsoidal cavity. The inversion is demonstrated for 1 < ka ~ 4 were k is the elastic wave number and a is the effective radius of the scatterer. The scatterers are embedded in the interior of a Ti-alloy sphere.
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The probabilistic approach to the inverse scattering problem is easy to describe: namely, given the results of scattering measurements, to determine the most probable scatterer. We start with a stochastic model of the measurement process in the usual signal-plus-noise form. However, here the signal (i.e., the model of a scattering measurement with no experimental error) is also considered to be random, thereby reflecting our partial lack of a priori knowledge of the nature of the scatterer. In the present treatment, we consider only nonparametric scatterer models (i.e., models involving an essentially infinite number of parameters - at least a number very large compared with the effective number of degrees of freedom in the measurements). We will consider three types of random models of the scatterer: (1) the material property deviations (elastodynamic or electromagnetic) are Gaussian random processes in space, (2) a single property deviation is also a Gaussian random process except for an added positivity bias, and (3) the scatterer has known uniform property deviations in an unknown domain. The inverse scattering problem for all three cases has been solved for the Born approximation using suitable mixtures of analytical and computational approaches. In cases (2) and (3) we employed the conjugate vector technique in order to reduce the computational effort to reasonable size. A special version of case (3), in which internal propagation is negligible, has been treated in the so-called Kirchhoff approximation in the regime of intermediate to high frequencies. A number of results obtained with theoretical synthetic test data will be presented and discussed.
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In this report a general formulation will be used to demonstrate three aspects of inverse scattering theory: the use of a ptioti information, the mathematical structure of inverse scattering theory, and the construction of more complex models from simpler models. First, the periodic and resonance properties of the scattering data are used to determine the periodicity of a structure. A prioti information about this structure enables the use of an iterative method to determine the amplitude of this periodicity. Second, the analytic properties of the scattering data are used to determine the permittivity of an inhomogeneous and dispersive region. A prioti information about the dispersion relation is essential for this method; in principle this mathematical method is "exact". Third, the first two methods are combined to investigate more complicated scattering data produced by approximately periodic or almost physical models.
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By introducing a splitting into up and down going waves, linear wave equations can be rewritten as coupled first-order systems in the up and down going components of the field. Exact local splittings yield uncoupled systems wherever the medium does not vary in the preferred direction, but approximate splittings can be more convenient. Given a slab of an inhomogeneous medium and a splitting, one can define an associated scattering matrix. Invariant imbedding techniques allow one to write a coupled system of differential equations for the operator entries of the scattering matrix, where the differentation is with respect to the location of one of the planes of the slab. One can then deduce the behavior of the reflection operator for small times, which provides a connection between the up and down going fields and the properties of the medium on the edge of the slab. Downward continuation inversion algorithms can then be derived. An example is given for the electromagnetic wave equation. Dissipative effects can be modelled, and are further studied in the companion paper, "The effects of dissipation in one-dimensional inverse scattering problems."
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Consider the problem of determining the permittivity as a function of depth in a one-dimensional medium. If the incident field is a transverse electric plane wave then the permittivity can be determined by using the resulting reflected field in an appropriate inversion algorithm, assuming the conductivity of the medium is known. However, if the conductivity is only known approximately, use of such inversion techniques can result in erroneous estimates of the permittivity. In this paper we examine how the performance of an inversion algorithm is altered by misrepresenting the dissipative effects of the medium. Two cases are of interest:
1. The assumed dissipative effects differ slightly from the actual dissipative effects throughout the medium, e.g., assuming no dissipation in a medium which in fact has small, non-zero conductivity.
2. Ignoring conductivity in a narrow region of the medium. These studies will be carried out using the techniques developed in the companion paper, "Wave splittings, invariant imbedding and inverse scattering."
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Assume that the permittivity ε and the permeability μ depend on z , i.e. ε= ε6(z) , μ = μ(z). For perpendicular polarization the electric and magnetic fields are determined by E_{x} , the x-component of the electric field. For this polarization let μ = E_{x} , μ = μ(z) and c(z) = 1 / /6(z)μ(z) . For parallel polarization the electric and magnetic fields are determined by Hx , the x-component of the magnetic field. For this polarization let μ = Hx and μ = 6(z) and c(z) = 1 / /6 (z)μ (z) Suppose that in the half-space z < 0 both the permittivity and permeability are known, but they are unknown for z > 0. In this paper it is shown how to determine 6(z) and p(z) using the trace method which itself is discussed. The following assumptions are necessary. (1) c(z), μ(z) approach cp1 as z -co and cp1 as z +co where c2'p2 are not known but it must be known that c2 > c1 . (2) certain smoothness assumptions on c(z) and p(z). Suppose a point harmonic source with angular frequency and w is located in the region z < 0 , and suppose p(r,z) is measured at some depth z < 0 for all r for both wi and w2 . Trace methods are then used to recover c(z) and μ(z) for z > 0 , under the assumption that the point source excites only the continuous spectrum. Numerical examples are presented.
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A problem of interest in thin film optics is to nondestructively measure one-dimensional refractive index profiles. We have been investigating the use of measurements of reflectivity magnitude as a function of incidence angle to determine such refractive index profiles. One approach is to use the Gel'fand Levitan algorithm as outlined by Moses and deRidderl. The algorithm requires knowledge of the Fourier transform of the complex reflectivity function. In this paper we present the results of computer testing an iterative technique for generating the complex reflectivity function from its magnitude and from appropriate constraints. The empirical results indicate that the iterative technique is capable of obtaining complex reflectivity estimates that could be within laboratory instrument generated uncertainty.
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The reduced scalar Helmholtz equation for a transversely inhomogeneous half-space sup-plemented with an outgoing radiation condition and an appropriate boundary condition on the initial-value plane defines a direct acoustic propagation model. This elliptic formulation admits a factorization and is subsequently equivalent to a first-order Weyl pseudo-differ-ential equation which is recognized as an extended parabolic propagation model. Perturbation treatments of the appropriate Weyl composition equation result in a systematic devel-opment of approximate wave theories while exact inversions for several nontrivial profiles provide for an analysis of strong refractive and diffractive effects. The analysis, in a natural manner, provides the basis for the formulation and exact solution of an arbitrary-dimensional nonlinear inverse problem appropriate for ocean acoustic, seismic, and optical studies- Moreover, the n-dimensional reduced scalar Helmholtz equation for the transversely inhomogeneous medium is naturally related to parabolic propagation models through (1) the above mentioned n-dimensional extended parabolic (Weyl pseudo-differential) equation and (2) an imbedding in an (n + 1)-dimensional parabolic (SchrOdinger) equation. The first relationship provides the basis for the parabolic-based Hamiltonian phase space path integral representation of the half-space propagator. The second relationship provides the basis for the elliptic-based path integral representations associated with Feynman/Fradkin, Feynman/Garrod, and Feynman/Dewitt-Morette. The path integrals allow for a global perspective of the transition from elliptic to parabolic wave theory in addition to providing a unifying framework in the direct and inverse formulations for dynamical approximations, resolution of the square root operator, and the concepts of an underlying stochastic process and free motion on curved spaces. The wave equation and path integral analysis provides for computational algorithms while foreshadowing the extension to (1) the vector formulation appropriate for elastic media, (2) the bilinear formulation appropriate for acoustic field coherence, and (3) the stochastic formulation appropriate for wave propagation in random media.
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An analytic method for inversion problems of the Fredholm type is developed. This method allows a quantitative discussion of the information content of various sets of experimental measurements. It is shown that if the solution is assumed to have a finite Fourier series representation then the solution may be separated into two independent parts, one given by the data and the other by constraints imposed on the solution.
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Satellite thermal sounding of temperature and moisture plays an important role in current atmospheric observational systems. Next generation systems may utilize surface-based passive observations and their com-bination with both active sensors and satellites. This technology continues to require development of mathematical retrieval algorithms to derive profiles from radiance observations. In this paper, commonly used algorithms and their application to contemporary sounding problems are discussed.
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I will discuss the reconstruction of the shape of an object from the sound-diffraction pattern created when the object is placed in the path of a plane monochromatic sound wave. The values of amplitude and phase on a plane square grid of points spaced by half the sound wavelength are adequate for the reconstruction.
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The angular spectrum of plane waves representation for acoustic fields is discussed and shown to be effective for describing the field radiated by an electroa.coustical transducer. A compact description is demonstrated for the case of a circularly symmetric planar transducer. Using the representation it is possible to inverse propagate measurements of the radiated field to reconstruct the field at the face of the transducer.
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This paper considers the problem of determining an intensity function given a finite set of discrete samples of its Fourier transform. The problem is inherently illposed since there are an infinite number of different intensity functions consistent with the given Fourier transform samples. A more reasonable problem is to infer some linear functional of the intensity function, for example the integral of the intensity function over some subset of its support. Although this quantity cannot be deter-mined exactly, nontrivial upper and lower bounds on its possible values can be determined.
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We consider a number of inverse diffraction problems where different models are compared. Ideal measurements yield Cauchy data , to which corresponds a unique solution. If a convolutional observation map is chosen, uniqueness can no longer be insured. We also briefly examine a non-linear non-invertible observation map , which describes a quadratic detector. In all of these cases we discuss the link between aperture identification and optimal control theory , which leads to regularised functional minimisation. This task can be performed by a discrete gradient algorithm of which we give the flow chart.
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We consider mathematical algorithms for the restoration of object information from finitely many measurements of the object's spectrum, with particular emphasis on the development of linear and nonlinear non-iterative methods that can incorporate prior information about object extent and shape. The linear method presented here generalizes the minimum energy bandlimited extrapolation procedure, which is the closed form limit of Gerchberg-Papoulis iteration in this case. The nonlinear method generalizes the maximum entropy method (MEM) of Burg.
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There are many situations in which only the modulus of a camplex wave function can be measured experimentally; examples include structure determination fram X-ray, optical and electron scattering. Methods for phase recovery in optical problems are discussed and broadly classified into two groups: those based on the analytic properties of the camplex field and those based upon constrained iterative techniques making use of a priori knowledge about the image or object width, for example. The limitations of these methods are described and the importance of prior knowledge about the abject and the field to resolve phase ambiguities is considered.
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Phase retrieval is concerned with the problem of finding the phases of a complex function from its absolute value. The square of the absolute value of a function may be considered as an energy density. Thus phase retrieval can be considered also as the problem of obtaining a field function from an energy density. Usually, in the one-dimensional case, one uses the method of Hilbert transforms on the logarithm of the Fourier transform. However, use of the Hilbert transform requires knowledge of the position of the zeros of the Fourier transform. From an operational point of view it is difficult to obtain the zeros in the complex plane. Moreover, the phase may be unstable with respect to errors in the position of the zeros. The present talk will concentrate on the problem of obtaining phases from energy densities only. A theorem will be given which shows that knowledge of the density in physical space (i.e., x-space) and a certain type of energy in frequency space is sufficient to determine the phase uniquely within a constant. One can also introduce energies in spaces other than in frequency space. An example is the energy associated with the Mellin transform. It is easy to give a meaning to energy density in frequency space as the energy associated with the frequency after the wave goes through a grating. The grating Fourier transforms the wave form. Other spaces, as the Mellin transform space, would need to have a physical interpretation analogous to that for the Fourier transform. Such an interpretation is given for the Mellin transform. Finally we consider generalization to electromagnetic fields in two and three dimensions. Here the phase retrieval problem is replaced by the problem of finding autocorrelation functions from suitable energy densities. A uniqueness theorem is given.
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A method of phase recovery from several intensity measurements is presented. The uniqueness is established for band-limited signals with known cut-off frequency. Intensities are generated by differential operators from a complex amplitude. The Gerchberg-Saxton algorithm can be applied in this method for effective phase reconstruction.
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From the analytic properties of a complex wavefield, the phase ambiguities associated with an intensity distribution can be described by the non self-conjugate zero factors in the product expansion for the scattered field. We examine methods for phase retrieval from limited sampled intensity data and illustrate the possible types of ambiguities. The success of phase retrieval techniques is shown to depend upon having an appropriate model for the function to be recovered and sufficient additional information.
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Holographic reconstruction of optical images is one instance of the inverse scattering problem and holographic interferometry is a method for solving phase retrieval problems. These experimental examples of inverse optical methods can be helpful in formulating inverse problems. The experimental methods also often suggest essential characteristics and fundamental limitations of numerical and theoretical solutions. Inthis paper an experimental study of acoustic waves propagating in water will be described which utilized optical holography and holographic interferometry as measurement tools. It is shown that the acoustic fields can be completely determined experimentally by simultaneous measurements of acoustic pressure and particle velocity which can be obtained from the same whole-field,optical interferometry data.
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This paper summarizes the current experience with phase retrieval at the University of Canterbury. While Fienup's algorithms need no modification for handling computer generated intensities corresponding to images that exhibit strong contrast, they must be extended to obtain satisfactory reconstruction of low contrast images. These same extensions, which have been incorporated into a composite computational procedure, tend also to be necessary when the given data are measured or are reduced from observations. The several steps in the procedure as it presently stands are described and discussed. Other steps, which are needed to widen the class of practically recoverable images and to make the procedure more objective, are suggested.
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