ABSTRACT
This paper reviews the collaborative research of Kleinman and Van den Berg with respect to the inverse scattering problem of the determination of the shape, the location and the constitutive parameters of a local inhomogeneity from measurements of the scattered field when a monochromatic wave is incident upon the inhomogeneity. Since the inverse scattering problem is nonlinear, an algorithm for its solution is iterative in nature and each iteration requires the solution of a forward or direct problem. In order to avoid a full solution of the forward problem in each iteration, the Modified Gradient method was developed, in which a cost functional was minimized such that the unknown fields and contrast are updated simultaneously. This cost functional consists of the superposition of the mismatch of the measured field data with the field scattered by an object with a particular contrast function and the error in satisfying consistency in the interior of the object. In these relations integral operators act on contrast sources being the products of the unknown fields and unknown material contrast. Further advantage of this structure has been taken by introducing the Contrast Source Inversion method that is based on a dual minimization of the cost functional by developing updates for the unknown contrast sources (instead of the fields) and the contrast. This inversion algorithm exhibits the best features of the modified gradient method, successfully reconstructing a variety of contrasts and fairly insensitive to noise. However, it exhibits additional properties which surpass the modified gradient method.
