In the previous chapter, the equations governing the three-dimensional scattering phenomena involving dielectric and conducting materials were derived. A preliminary distinction between *direct* and *inverse* scattering problems has been introduced.

Since this book is devoted to microwave imaging techniques, which are essentially short-range imaging approaches, the scattering equations constitute the foundation for the formulation and the development of the various reconstruction procedures.

The inverse scattering problem considered here belongs to the category of inverse problems (Colton and Kress 1998), which includes many very challenging problems encountered in several applications, including atmospheric sounding, seismology, heat conduction, quantum theory, and medical imaging.

From a strictly mathematical perspective, the definition of a problem as the inverse counterpart of a direct one is completely arbitrary. To this end, it is helpful to recall the following well-known sentence by J. B. Keller (Keller 1976) quoted by Bertero and Boccacci (1998, pp. 1–2): “We call two problems inverses of one another if the formulation of each involves all part of the solution of the other. Often, for historical reasons, one of the two problems has been studied extensively for some time, while the other has never been studied and is not so well understood. In such cases, the former is called a direct problem, while ...

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