A tomographic time-domain reconstruction algorithm for solving the inverse electromagnetic problem is described. The application we have in mind is dielectric breast cancer detection but the results are of general interest to the field of microwave tomography. Reconstructions have been made from experimental and numerically simulated data for objects of different sizes in order to investigate the relation between the spectral content of the illuminating pulse and the quality of the reconstructed image. We have found that the spectral content is crucial for a successful reconstruction. The work has further shown that when imaging objects with different scale lengths it is an advantage to use a multiple step procedure. Low frequency content in the pulse is used to image the large structures and the reconstruction process then proceed by using higher frequency data to resolve small scale lengths. Good agreement between the results obtained from experimental data and simulated data has been achieved.

Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.

Microwave imaging is an interesting and growing research field with a number of medical applications. This paper is based on the first series of experimental results using an iterative gradient algorithm based on the finite difference time domain (FDTD) method and synthetic pulses. Using our method, the permittivity and conductivity of an object are reconstructed layer by layer by minimizing a functional consisting of the difference between the measured and calculated electric field surrounding the object. This is done by surrounding the object with a number of antennas which are all in turn transmitting and receiving. The dielectric profiles used in the calculations are then iteratively updated until the functional is minimized. Results are presented demonstrating the ability to detect metallic and dielectric material in air and water.

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An adjoint field approach is used to formulate a general numerical framework for Fisher information-based sensitivity analysis in electrical impedance tomography. General expressions are given for the gradients used in standard least-squares optimization, i.e. the Jacobian related to the forward problem, and it is shown that these gradient expressions are compatible with commonly used electrode models such as the shunt model and the complete electrode model. By using the adjoint field formulations together with a variational analysis, it is also shown how the computation of the Fisher information can be integrated with the gradient calculations used for optimization. It is furthermore described how the Fisher information analysis and the related sensitivity map can be used in a preconditioning strategy to obtain a wellbalanced parameter sensitivity and improved performance for gradient-based quasi-Newton optimization algorithms in electrical impedance tomography. Numerical simulations as well as reconstructions based on experimental data are used to illustrate the sensitivity analysis and the performance of the improved inversion algorithm in a four-electrode measurement set-up.