Diffraction Tomographic Algorithm Research Articles

A generalized diffraction tomography algorithm

Available diffraction tomography algorithms are based on Fourier transform techniques and require either plane-wave illumination in a uniform background medium or far-field illumination combined with paraxial approximations. In this paper a generalized diffraction tomography algorithm is introduced that can handle both irregularly spaced measurement data, nonuniform background models, and general aquisition geometries. Using data from water tank experiments, the method’s ability to yield high-quality reconstructions of geometry and velocity, as long as the weak-scattering assumption is satisfied, is demonstrated.

Generalized ART algorithm for diffraction tomography

The reconstruction problem in diffraction tomography is addressed for cases where noise-free scattering data are available for a limited number of view angles (discrete set of incident wave directions). The solution to the inverse problem (reconstruction problem) is obtained in the form of an iterative method known in the literature as Kaczmarz's method and is found to be a generalization of the algebraic reconstruction technique (ART) of conventional computed tomography (CT) to diffraction tomography (DT). The generalized ART algorithm is shown to generate a reconstruction that is consistent with the data and that has minimum L2 norm along all such solutions. A computer simulation is presented comparing the performance of the algorithm with the filtered backpropagation algorithm.

Scattered Wave Inversion by Image Projections

A three-dimensional diffraction tomography algorithm based on image projections is implemented. For each view, 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. The object is then reconstructed by parallel slices using conventional straight ray tomographic techniques. This approach permits practical three-dimensional reconstruction using a limited number of views. The reconstructions made with image projections are of comparable quality to ideal diffraction-limited images. By backpropagating the field prior to filtering, curved or misaligned recording surfaces can be used. The limits on the image projection technique for multiple object systems are explored. A diffuse structure is reconstructed.

Scattered wave inversion by image projections

A three-dimensional diffraction tomography algorithm based on image projections is implemented. For each view, 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. The object is then reconstructed by parallel slices using conventional straight ray tomographic techniques. This approach permits practical three-dimensional reconstruction using a limited number of views. The reconstructions made with image projections are of comparable quality to ideal diffraction-limited images. By backpropagating the field prior to filtering, curved or misaligned recording surfaces can be used. The limits on the image projection technique for multiple object systems are explored. A diffuse structure is reconstructed.

Numerical study of higher-order diffraction tomography via the sinc basis moment method

Behavior and limitations of the sinc basis method have been discovered during simulation studies. An iterative row-action method (ART) is compared with a full-matrix, least-squares solution (QR decomposition) and the problem of multiple solutions is discussed. Knowledge of the spatial distribution of scattered field energy can be used to speed up convergence. The first iteration is shown to be equivalent to the first Born solution. A fundamental limitation of present diffraction tomography algorithms, concerning phase shift through the scatterer, is found and related to clinical values. Effects of improvements on the initial guesses for the object function and internal field are studied. Lossy cylinder and contrived multicomponent reconstructions yield additional understanding of the phase shift problem. A sequence of simulations investigates estimated solution movement in hyperspace.

Numerical study of higher-order diffraction tomography via the sinc basis moment method.

Behavior and limitations of the sinc basis method have been discovered during simulation studies. An iterative row-action method (ART) is compared with a full-matrix, least-squares solution (QR decomposition) and the problem of multiple solutions is discussed. Knowledge of the spatial distribution of scattered field energy can be used to speed up convergence. The first iteration is shown to be equivalent to the first Born solution. A fundamental limitation of present diffraction tomography algorithms, concerning phase shift through the scatterer, is found and related to clinical values. Effects of improvements on the initial guesses for the object function and internal field are studied. Lossy cylinder and contrived multicomponent reconstructions yield additional understanding of the phase shift problem. A sequence of simulations investigates estimated solution movement in hyperspace.

Geophysical diffraction tomography

Regions of the earths subsurface to be imaged tomographically frequently exhibit large variations in refractive index so that the assumption of straight ray paths is invalid and often leads to inferior reconstructions. The use of curved raypaths in diffraction tomography, however, significantly complicates the reconstruction problem. We develop an algorithm for diffraction tomography which involves iterations on successive ray tracing and tomographic inversion. Singular value decomposition as well as conjugate gradients provide alternative approaches to the inversion step at each iteration. Each procedure can give good results but singular value decomposition can be costly to apply. In diffraction tomography, partial singular value decomposition using a very small number of singular values and singular vectors at each inversion step can yield very accurate reconstructions in a few iterations.

Geophysical imaging with arbitrary source illumination

Geophysical diffraction tomography is a technique for quantitative, high-resolution, subsurface imaging. This approach to imaging is a generalization of the conventional backprojection algorithm of X-ray tomography accounting for the diffraction effects that result from longer wavelength seismic or electromagnetic waves necessary for geophysical remote sensing. A diffraction tomography algorithm is presented for a configuration in which a finite number of sources of arbitrary character are distributed along one line and a finite number of receivers are distributed along a line having an arbitrary orientation with respect to the source line. Since most geophysical sources may be reasonably represented as point sources, the two-dimensional form of the algorithm is implemented for cylindrical-beam (a point source in two dimensions) illumination. Numerical experiments are performed to investigate a range of source-receiver configurations. It is found that parallel source and receiver arrays, a cross-borehole configuration, provide better image quality than orthogonal arrays.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On double integrals over spheres

Formulae involving double integrals over spheres arise naturally in inverse scattering problems since the scattered data are measured in the space R*S2*S2. The authors derive a relation between differential forms on the space Sn-1*Sn-1, and those on the space I*Sn-2*Sn-1, where I is a real interval. Specifically, d xi d eta =sinn-2 theta d theta d psi d nu ( xi , eta ) in Sn-1*Sn-1 and ( theta , psi , nu ) in I*Sn-2*Sn-1. This allows them to derive the results of John relating the iterated spherical mean of a function to its spherical mean in a simple way; to obtain new inversion formulae for the Fourier and Radon transforms; to extend formulae for linearised inverse quantum scattering and diffraction tomography to the multifrequency case; and also to establish a relation between multifrequency diffraction tomography and seismic migration algorithms.

Imaging performance of generalized holography

Generalized holography is understood as a heuristic solution of an inverse-scattering problem based on the time-harmonic formulation of Huygens’s principle. It is checked against a scalar two-dimensional numerical-scattering experiment: Simulated data for a cylindrical object with arbitrary elliptical cross section are inserted into the algorithm, and the results are critically discussed. We find that generalized holography yields only sparse improvement over conventional holography—the images obtained are only poor solutions to the inverse-scattering problem—but, on the other hand, its mathematical consequences, such as the Porter–Bojarski integral equation, may serve as an intuitive means to predict the performance of holographic schemes for certain specialized canonical inverse-scattering problems; once the limitations are known, considerable advantage can be drawn for practical applications, and improved extensions are readily proposed yielding very general diffraction tomography algorithms.

Experimental study on geophysical diffraction tomography

Most of the diffraction tomography experiments that can be found in the literature focused on medical or NDE applications. With medical or NDE applications, these experiments allowed us to measure the scattered wave field in all desired. directions. This full coverage of the targets by insonifying sonic waves, however, is impossible for objects of geophysical interests. Diffraction tomography experiments with geophysical applications have not been done yet. In order to experimentally test the applicability of diffraction tomography with geophysical applications, the authors conducted an experiment with a target immersed in water and with a source‐receiver geometry similar to the source‐receiver geometry of the offset VSP techniques currently being used by exploration geephysicists in the field. We measured the scattered wave field by this poor coverage but realistic source‐receiver arrangement. With scattered wave field data, we reconstructed the images of our targets by a diffraction tomography algorithm. The results of our experiment are encouraging. Objects of simple geometry can be successfully imaged. In theoretical studies of diffraction tomography, most reconstruction algorithms developed so far are based on two assumptions: (1) weak scattering approximation and (2) linear behavior of scattered wave field assumption. These two assumptions impose limits on the applications of diffraction tomography and the authors took an experimental approach to delineate these limits.

Reflection mode diffraction tomography.

A reconstruction algorithm is presented which possesses a simple scanning geometry and promises higher resolution than conventional transmission mode diffraction tomography algorithms. This broad-band reflection mode algorithm inherently lacks a certain amount of low frequency information but an estimate of the information is recovered by spectral extrapolation. The resolution of the algorithm will be shown to be limited by the bandwidth and physical size of the single plane wave transducer, as well as the Born approximation.

Reflection mode diffraction tomography

A reconstruction algorithm is presented which possesses a simple scanning geometry and promises higher resolution than conventional transmission mode diffraction tomography algorithms. This broad-band reflection mode algorithm inherently lacks a certain amount of low frequency information but an estimate of the information is recovered by spectral extrapolation. The resolution of the algorithm will be shown to be limited by the bandwidth and physical size of the single plane wave transducer, as well as the Born approximation.

Limitations of Imaging with First-Order Diffraction Tomography

In this paper, the results of computer simulations used to determine the domains of applicability of the first-order Born and Rytov approximations in diffraction tomography for cross-sectional (or three-dimensional) imaging of biosystems are shown. These computer simulations were conducted on single cylinders, since in this case analytical expressions are available for the exact scattered fields. The simulations establish the first-order Born approximation to be valid for objects where the product of the relative refractive index and the diameter of the cylinder is less than 0.35 lambda. The first-order Rytov approximation is valid with essentially no constraint on the size of the cylinders; however, the relative refractive index must be less than a few percent. We have also reviewed the assumptions made in the first-order Born and Rytov approximations for diffraction tomography. Further, we have reviewed the derivation of the Fourier Diffraction projection Theorem, which forms the basis of the first-order reconstruction algorithms. We then show how this derivation points to new FFT-based implementations for the higher order diffraction tomography algorithms that are currently being developed.

Distortion in Diffraction Tomography Caused by Multiple Scattering

In this paper, we have first presented a new computational procedure for the calculation of the "true" forward scattered fields of a multicomponent object. By "true" we mean fields that are not limited by the first-order approximations, such as those used in the first-order Born and Rytov calculations. Although the results shown will only include the second-order fields for a multicomponent object, the computational procedure can easily be generalized for higher order scattering effects. Using this procedure we have shown by computer simulation that even when each component of a two-component object is weakly scattering, the multiple scattering effects become important when the components are blocking each other. We have further shown that when strongly scattering components that are large compared to a wavelength are not blocking each other, the scattering effects can be ignored. Both these conclusions agree with intuitive reasoning. Since all the currently available diffraction tomography algorithms are based on the assumption that the object satisfies the first-order scattering assumption, it is interesting to test them under conditions when this assumption is violated. We have used the scattered fields obtained with the new computational procedure to test these algorithms, and shown the resulting artifacts. Our main conclusion drawn from this computer simulation study is that even when object inhomogeneities are as small as 5 percent of the background, multiple scattering can introduce severe distortions in multicomponent objects.

A filtered backpropagation algorithm for diffraction tomography

A reconstruction algorithm is derived for parallel beam transmission computed tomography through two-dimensional structures in which diffraction of the insonifying beam must be taken into account. The algorithm is found to be completely analogous to the filtered backprojection algorithm of conventional transmission tomography with the exception that the backprojection operation has to be replaced by a back propagation process whereby the complex phase of a field measured over a line outside the object is made to propagate back through the object space. The algorithm is applicable to diffraction tomography within either the first Born or Rytov approximations. Application of the algorithm to three-dimensional structures is also discussed.