P.M. van den Berg

A modified gradient method for two-dimensional problems in tomography

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

Imaging of biomedical data using a multiplicative regularized contrast source inversion method

In this paper, the recently developed multiplicative regularized contrast source inversion method is applied to microwave biomedical applications. The inversion method is fully iterative and avoids solving any forward problem in each iterative step. In this way, the inverse scattering problem can efficiently be solved. Moreover, the recently developed multiplicative regularizer allows us to apply the method blindly to experimental data. We demonstrate inversion from experimental data collected by a 2.33-GHz circular microwave scanner using a two-dimensional (2-D) TM polarization measurement setup. Further some results of a feasibility study of the present inversion method to the 2-D TE polarization and the full-vectorial three-dimensional measurement will be presented as well.

The three dimensional weak form of the conjugate gradient FFT method for solving scattering problems

The problem of electromagnetic scattering by a three-dimensional dielectric object can be formulated in terms of a hypersingular integral equation, in which a grad-div operator acts on a vector potential. The vector potential is a spatial convolution of the free space Greens function and the contrast source over the domain of interest. A weak form of the integral equation for the relevant unknown quantity is obtained by testing it with appropriate testing functions. The vector potential is then expanded in a sequence of the appropriate expansion functions and the grad-div operator is integrated analytically over the scattering object domain only. A weak form of the singular Greens function has been used by introducing its spherical mean. As a result, the spatial convolution can be carried out numerically using a trapezoidal integration rule. This method shows excellent numerical performance. >

Two‐dimensional location and shape reconstruction

A method for reconstructing the location and the shape of a bounded impenetrable object from measured scattered field data is presented. The algorithm is, in principle, the same as that used for reconstructing the conductivity of a penetrable object and uses the fact that for high conductivity the skin depth of the scatterer is small, in which case the only meaningful information produced by the algorithm is the boundary of the scatterer. A striking increase in efficiency is achieved by incorporating into the algorithm the fact that for large conductivity the contrast is dominated by a large positive imaginary part. This fact, together with the knowledge that the scatterer is constrained in some test domain, constitute the only a priori information about the scatterer that is used. There are no other implicit assumptions about the location, connectivity, convexity, or boundary conditions. Some refinements of the algorithm which reduce the number of points at which the unknown function is updated are incorporated to further increase efficiency. Results of a number of numerical examples are presented which demonstrate the effectiveness of the location and shape reconstruction algorithm.

Iterative computational techniques in scattering based upon the integrated square error criterion

An iterative technique is developed to rigorously compute the electromagnetic time- and frequency-domain scattering problems. The method is based upon a wave-function expansion technique (this also includes the integral-representation techniques), in which the electromagnetic field equations and causality conditions are satisfied analytically, while the boundary conditions or the constitutive relations have to be satisfied in a computational manner. The latter is accomplished by an iterative minimization of the integrated square error. For the solution of an integral equation, it is shown how to obtain optimum convergence. Some numerical results pertaining to a number of representative problems illustrate the numerical advantages and disadvantages of the iterative method.

Microwave-tomographic imaging of the high dielectric-contrast objects using different image-reconstruction approaches

Microwave tomography is an imaging modality based on differentiation of dielectric properties of an object. The dielectric properties of biological tissues and its functional changes have high medical significance. Biomedical applications of microwave tomography are a very complicated and challenging problem, from both technical and image reconstruction point-of-views. The high contrast in tissue dielectric properties presenting significant advantage for diagnostic purposes possesses a very challenging problem from an image-reconstruction prospective. Different imaging approaches have been developed to attack the problem, such as two-dimensional (2-D) and three-dimensional (3-D), minimization, and iteration schemes. The goal of this research is to study imaging performance of the Newton and the multiplicative regularized contrast source inversion (MR-CSI) methods in 2-D geometry and gradient and MR-CSI methods in 3-D geometry using high-contrast, medium-size phantoms, and biological objects. Experiments were conducted on phantoms and excised segment of a pig hind-leg using a 3-D microwave-tomographic system operating at frequencies of 0.9 and 2.05 GHz. Both objects being of medium size (10-15 cm) possess high dielectric contrasts. Reconstructed images were obtained using all imaging approaches. Different approaches are evaluated and discussed based on its performance and quality of reconstructed images.

An extended range-modified gradient technique for profile inversion

A method for reconstructing the complex index of refraction of a bounded inhomogeneous object from measured scattered field data is presented. The index and the unknown fields within the object are simultaneously reconstructed in an iterative algorithm. The method is a refinement of earlier work which incorporates a more effective way to update the unknowns at each stage of the iteration. Considerable efficiency in the algorithm is achieved. Some numerical examples are given indicating the limits on the contrasts which can be reconstructed. These limits show that the range of contrasts that may be reconstructed is extended over that achievable with the earlier work.

A total variation enhanced modified gradient algorithm for profile reconstruction

The total variation minimization method for deblurring noisy data is shown to be effective in dramatically increasing the resolution in a modified gradient approach to index of refraction reconstruction from measured scattered field data. Numerical evidence is presented which shows that by including the total variation in the functional to be minimized the reconstructions of piecewise constant profiles are considerably sharpened. The stability of the modified gradient method with respect to noise is apparently also enhanced. Furthermore, the presence of the total variation does not appear to adversely effect the established effectiveness of the modified gradient method in reconstructing smooth profiles.

Three-dimensional imaging of multicomponent ground-penetrating radar data

Scalar imaging algorithms originally developed for the processing of remote sensing measurements (e.g., the synthetic‐aperture radar method) or seismic reflection data (e.g., the Gazdag phase‐shift method) are commonly used for the processing of ground‐penetrating radar (GPR) data. Unfortunately, these algorithms do not account for the radiation characteristics of GPR source and receiver antennas or the vectorial nature of radar waves. We present a new multicomponent imaging algorithm designed specifically for vector electromagnetic‐wave propagation. It accounts for all propagation effects, including the vectorial characteristics of the source and receiver antennas and the polarization of the electromagnetic wavefield. A constant‐offset source‐receiver antenna pair is assumed to overlie a dielectric medium. To assess the performance of the scalar and multicomponent imaging algorithms, we compute their spatial resolution function, which is defined as the image of a point scatterer at a fixed depth using a ...

Iterative methods for solving integral equations

A number of iterative algorithms to solve integral equations arising in field problems are discussed. We describe the essential features of the Neumann Series, overrelaxation methods, Krylov subspace methods, and the conjugate gradient technique. Proofs of convergence of the conjugate gradient method are directly available when the underlying integral operator is self-adjoint, and in this case the method is equivalent to the Krylov method. However, for non-self-adjoint operators the conjugate gradient method requires an implicit symmetrization which results in poorer convergence than that obtained using the Krylov method. Some convergence results are also available for overrelaxation methods for both self-adjoint and non-self-adjoint operators. Relations between all of the methods will be described and numerical performance will be contrasted using a uniform square error criterion. All the methods are treated in the continuous operator form which is especially useful in using the physical setting to arrive at effective preconditioners.