Iteratively reweighted ℓ₁-ℓ₂ norm minimization using wavelets in inverse scattering

Recently, many techniques have been employed to solve inverse scattering problems by exploiting the sparsity of the scatterer in the wavelet basis. In this paper, we propose an iteratively reweighted $ {\ell _1} $ norm regularization scheme within the settings of the Born iterative method (BIM) to effectively leverage the sparsity of the wavelet coefficients. This "iteratively reweighted $ {\ell _1} $ minimization" method is then used along with $ {\ell _2} $ norm minimization in order to achieve solutions that are not over-smoothened at the discontinuities. The proposed method is an expansion of a well-known joint $ {\ell _1} {-} {\ell _2} $ norm minimization technique. The advantage offered by the algorithm is that the reconstruction is now independent of the initial choice of weights. This technique accounts for the fact that sparsity is concentrated more in the detail wavelet coefficients rather than their coarse counterpart. The effectiveness of the method is demonstrated using several 2D inverse scattering examples by employing it in each iteration of the BIM.