A unified theoretical framework based on the scalar wave equation is presented for transmission tomography of weakly inhomogeneous objects embedded in a known uniform background medium The developed formulation differs from the usual treatments of diffraction tomography in that it is not limited to applications employing plane-wave illumination and planar measurement boundaries and can be extended to weakly inhomogeneous objects embedded in non-uniform backgrounds and to the class of strongly scattering objects The new formulation is based on the mathematical operations of propagation and backpropagation which are generalisations of the conventional tomographic operations of projection and backprojection to the case of diffracting wavefields The projection and backprojection operations are then obtained from the generalised operations in the zero wavelength limit A class of reconstruction algorithms are developed within the new formulation that are generalisations, to the case of diffraction tomography, of the class of iterative ART (algebraic reconstruction technique) algorithms of conventional (X-ray) tomography The reconstruction algorithms developed (called generalised ART algorithms) are shown to be applicable to situations where the tomographic data are sparse (limited-view problem) and to the incorporation of a priori information into the reconstruction process

Reconstructive tomography with diffracting wavefields

On distingue 2 problemes dans le traitement du signal; le premier concerne la reconstruction d'une image a partir de donnees mesurees transformees de facon appropriee qui echantillonnent le plan de frequences spatiale le long d'arcs circulaires; le second probleme est celui de desenroulement de phase des profils mesures

Signal Processing for Diffraction Tomography

The author addresses the problem of reconstructing a bandlimited signal from a finite number of its unevenly spaced sampled data. A Fourier analysis of the available unevenly spaced sampled data is presented. The result is utilized to develop an interpolation scheme from the available data. Conditions for accurate reconstruction are examined. Algorithms to implement the reconstruction scheme are discussed. The method's application to one-dimensional and two-dimensional reconstruction problems is shown. >

Band-limited interpolation from unevenly spaced sampled data

Inverse scattering solutions by a sinc basis, multiple source, moment method--Part III: Fast algorithms.

The purpose of this paper is to reappraise the linearizing methods frequently used to solve inverse scattering problems. We describe inversion algorithms based on the Born and the Rytov approximations and the nature of the distortions obtained in the reconstructions when using them. We present extensions of these methods, namely, the distorted-wave Born and the distorted-wave Rytov approximations, which incorporate prior knowledge about part of the scattering structure. A method for inverting scattered field data using these distorted-wave approximations is described, which retains the computational simplicity of the Born and the Rytov techniques. Some examples of their use with simulated and real data are given. A further extension of our distorted-wave formalism, which leads to improvements of the reconstructed image, is suggested. This entails a spectral estimation procedure also based on the incorporation of prior knowledge about the scatterer. This spectral estimation procedure can be useful for interpolation of scattered field data as well as resolution enhancement.

Image estimation from scattered field data

Presented are algorithms for reconstructing two-dimensional functions from discrete and finite information available about their Fourier transform. The procedure is based upon an interpolation technique in the Fourier domain. The concept is applied to straight-path and diffraction tomography where Fourier domain information is available on a finite number of rotated contours. Finally, the resultant Fourier domain based methods are compared with their corresponding spatial domain based methods, i.e., filtered backprojection and backpropagation in terms of accuracy of reconstruction and computational efficiency.

Fourier Domain Reconstruction Methods with Application to Diffraction Tomography

This paper presents a new view of the problem of image reconstruction in the spatial Fourier domain based on the generalized projections of a two-dimensional image. The algorithms that are presented are derived from the exact continuous relations between the Fourier transforms of a spatially limited image and those of its generalized projections. A Fourier domain reconstruction formula is derived that is a natural consequence of these relations giving values of the two-dimensional transform on a regular grid useful for inverse FFT. This approach is contrasted with the more ad hoc methods of Fourier domain interpolation and with the spatial domain approaches such as the backprojection and backpropagation methods.

Image reconstruction from frequency domain data on arbitrary contours

The exact solution of the scattering of a plane ultrasonic wave by an isotropic and homogeneous cylinder can be found via a closed-form series solution. Thus, weakly scattering cylinders provide building blocks of useful models for studying the properties of diffraction tomography methods. Over the past few years we have demonstrated the merits of diffraction tomography based on the Born and Rytov transformations using soft cylinders.

Theoretical and Numerical Results on Scattering from Soft Cylinders with Implications in Diffraction Tomography

This paper addresses the problems of image reconstruction in fan beam straight-path tomography from complete and incomplete data. A method for restoring the parallel beam projections of an object from its fan beam projections is introduced. The scheme utilizes FFT routines for this purpose. It is shown that the approach reduces the required amount of the collected fan beam data by half. Moreover, a noniterative method of image reconstruction, when arbitrary segments of the object projections cannot be detected, is presented. The technique exploits the functional properties of the object projections, in conjunction with a priori information available about the test object, to restore the unknown data.

M. Soumekh

Papers

Fourier Domain Reconstruction Methods with Application to Diffraction Tomography

Image reconstruction from frequency domain data on arbitrary contours

Theoretical and Numerical Results on Scattering from Soft Cylinders with Implications in Diffraction Tomography

A method of image reconstruction in fan beam tomography