Smooth Polynomial Approach for Microwave Imaging in Sparse Processing Framework

We developed a novel qualitative imaging algorithm based on a polynomial approximation of the unknown contrast and sparse (
$l_{1}$
) regularization. Contrary to previously published results, we defined polynomial basis functions on subdomains that divide the investigation domain. Moreover, we formulated constraints that ensure the continuity of the contrast on subdomain borders. We showed that the proposed algorithm improved imaging resolution, particularly in multiple target scenarios. We demonstrated that partitioning the investigation domain together with contrast continuity formulation enhanced the numerical stability and reduced the computation time. The obtained results were significantly less sensitive to the regularization parameter values than those obtained using the standard polynomial approximation. Namely, smaller domains allow lower polynomial orders, which are numerically more favorable. Continuity constraints reduce the search space and mitigate the occurrence of false solutions. Another contribution of this study is a novel strategy for regularization parameter selection. We considered different figures of merit and numerical scenarios to study the influence of various parameters involved in the imaging process, such as the polynomial order and number of subdomains. An extensive analysis proved the robustness of the approach against noise. The proposed algorithm was designed for two-dimensional geometry. However, generalization to three-dimensional space is straightforward. The algorithm can also be used with other types of regularization such as the
$l_{2}$
regularization. Potential applications include medical microwave imaging, in which high resolution and noise immunity are vital features.