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.

中文翻译：

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在逆散射中使用小波迭代地加权ℓ1-ℓ2范数最小化。

近来，已经利用许多技术来利用小波基础上的散射体稀疏性来解决逆散射问题。在本文中，我们提出了在Born迭代方法（BIM）的设置内迭代加权的$ {\ ell _1} $范数正则化方案，以有效利用小波系数的稀疏性。然后，将此“迭代加权的$ {\ ell _1} $最小化”方法与$ {\ ell _2} $范数最小化一起使用，以实现在不连续点处不会过度平滑的解决方案。所提出的方法是对众所周知的联合$ {\ ell _1} {-} {\ ell _2} $范数最小化技术的扩展。该算法提供的优势在于，重构现在独立于权重的初始选择。该技术说明了以下事实：稀疏度更多地集中在细节小波系数上，而不是它们的粗略对应系数上。通过在BIM的每次迭代中使用几个2D反散射示例，证明了该方法的有效性。