Effects of data collection schemes and systems on the imaging performance of electromagnetic inverse problems

Abstract

Although electromagnetic inverse problems are, as is canonically understood, ill-posed mathematical problems, there are several possibilities that arise with respect to the practical design of the imaging system with which scattered field data is collected that can ameliorate the problem and significantly improve the imaging performance. The standard techniques for dealing with the ill-posedness of the problem aim at reducing modelling error and regularizing the mathematical inverse problem. This includes the creation of simplified systems that are amenable to a manageable numerical inversion model; both in terms of the achievable model accuracy as well as with respect to the required computational resources. Of course, the reduction of modelling errors is important as large errors between the numerical model and the actual system contribute to the inability of the inversion algorithm to converge to the true solution (i.e., modelling error manifests itself as systematic non-random noise and the instability of the inverse problem results in convergence to non-unique, non-true, solutions of the inverse problem). Data calibration techniques and numerical methods of regularizing the mathematical problem have been well studied in the past, and are indeed necessary to arrive at useful solutions. In this work we focus on some available system design options that can promote better convergence and accuracy of the converged solution. In particular, the following options will be considered: 1) the use of resonant metallic chambers of various shapes; 2) the collection of different field components within the chamber; 3) the use of several immersion media; and 4) the use of dynamic bound-aries to establish not only diverse incident field data, but also to diversify the effective Green's function of the inverse problem. With regard to the first option, the way one incorporates the boundary conditions of the chamber into the inversion algorithm will be delineated. It will be shown that several options are available with respect to the formulation of the data equation and regularization terms. Under the second option, we will show how collecting the tangential magnetic field on the surface of the chamber walls can provide advantages with respect to modelling error and the amount of data that can be collected. Under the third option, we show how the inversion algorithm can be made quasi-independent of the immersion medium that is used (when, for example, in breast imaging, such a design parameter is available) and we will show how for the same frequency the use of different immersion media will provide a variety of diverse data that interrogate the object of interest with different wavelengths. Finally, with the use of dynamic boundaries that can be turned off and on, we show how a different incident field can be produced for the exact same transmitter antenna location. For the most part we focus on biomedical imaging applications where all of these options are available, as well as on a novel stored-grain imaging application that is implemented within grain bins.