The diagonalized contrast source approach: an inversion method beyond the Born approximation

This paper deals with the imaging and inversion of the constitutive material properties of bounded objects embedded in a known background medium. The inversion utilizes measurements of the scattered field due to the illumination of the objects by a set of known single-frequency wave fields. We present two inverse scattering methods that approximately recast the full nonlinear inversion into a number of linear inversion steps. The two methods are as computationally efficient as the constrained Born inversion but provide reconstruction results that are far superior and are almost comparable in quality to the ones obtained using a full nonlinear iterative inversion. The linear inversion steps follow what is referred to as the source-type integral equation approach in which the contrast sources are inverted for in the first step from the linear data equation. In this step, we employ a local (diagonal) approximation of the operator that relates the contrast sources to the incident field. Once the contrast sources have been determined, the total internal wave fields follow from a direct application of the object equation. In the third and final step, the contrast function is estimated from either the constitutive relation (first method) or from solving the data equation again, this time in terms of the contrast profile (second method). The computational cost required by the first method is comparable to that of a constrained Born inversion. With only twice the computational cost, the second method invariably gives an almost perfect image. We will demonstrate the two methods for a number of representative synthetic examples, both in two and three dimensions.