## 1. Introduction

[2] A central problem in target identification, non-destructive testing, medical imaging and numerous other areas of application concerns the determination of the shape, location and constitutive parameters, such as complex index of refraction or local sound speed, of a object or local inhomogeneity from measurements of the scattered field, when the object is illuminated successively by a number of known incident electromagnetic or acoustic waves. This problem is nonlinear and ill-posed, but during the years useful reconstruction algorithms have been developed. Comprehensive overviews of several results are given by *Habashy et al.* [1994], *Chew* [1999], *Lesselier and Duchêne* [1996], *Colton et al.* [2000] and *Sabatier* [2000] and the results of testing various inversion algorithms against experimental data can be found in the special section in the Inverse Problems journal organized by *Belkebir and Saillard* [2001]. Most of these algorithms make use of the domain integral equation for the field inside the scattering object as well as the related integral representation for the field outside the object.

[3] Our starting point is the contrast source inversion method [*van den Berg and Kleinman*, 1997], where the unknown contrast sources and the unknown contrast are updated by an iterative minimization of an appropriate cost functional. Inspired by the successful implementation of the minimization of total variation and other edge-preserving algorithms in image restoration and inverse scattering, previously *van den Berg et al.* [1999] have investigated the use of these image-enhancement techniques as an extra regularization. The drawback of adding a regularization term to the cost functional is the presence of an artificial weighting parameter in the cost functional, which can only be determined through considerable numerical experimentation and a priori information about the desired profile. Therefore, we take the regularization as a multiplicative constraint and show that the weighting parameter is now completely prescribed by the error norm of the data equation and the object equation. This multiplicative regularization technique is originally introduced by *van den Berg et al.* [1999], using the *L*^{1}- and *L*^{2}-norms total variation [*Rudin et al.*, 1992] as constraints. Subsequently, *van den Berg and Abubakar* [2001] have shown that a weighted *L*^{2}-norm total variation factor improves the reconstruction results significantly. The advantage is that the updating parameters in the contrast source inversion method can be determined explicitly, without the usual line minimization. A possible drawback of the introduction of a multiplicative regularization factor is the danger of increasing the nonlinearity of the inversion problem, viz., the loss of convexity of the cost criterion for updating the contrast.

[4] In this paper we have modified slightly the updating scheme of the contrast. This modification allows us to derive a convexity criterion of the multiplicative cost functional using the weighted *L*^{2}-norm of the total variation as regularization factor. In our numerical experiments we observe that the present implementation of the contrast update leads always to a convex problem. Moreover the modification of the contrast updating scheme shows a further improvement of the reconstructions. In the numerical experiments using both synthetic and experimental data, we illustrate the reconstruction results with three different regularization factors, the *L*^{1}-, *L*^{2}-, and weighted *L*^{2}-norm regularization factor, respectively. The numerical results clearly indicate that the use of the latter one yields a very robust inversion scheme.