ISAR imaging using a null space ℓ1 minimizing Kalman filter approach

In this paper, we presented a novel compressive sensing (CS) imaging method for inverse synthetic aperture radar (ISAR) exploiting the null space of the sensing matrix. We decompose the solution of the underdetermined linear imaging system into two parts: an observable part that is not unique and a non-observable part lying in the null space of the sensing matrix. We first obtain a weighted least square (WLS) solution/image using the available measurements. Then, the ℓ₁ norm of the unknowns is introduced as an additional nonlinear measurement to the underdetermined imaging system. Unlike existing CS reconstruction methods that estimate the unknowns straightforward, we estimate the null space part using this nonlinear “pseudo” measurement in the framework of Kalman filter with the aim to minimize the ℓ₁ norm of the whole solution. We demonstrate the performance of the proposed method using real ISAR data. The results show that our imaging method can provide better reconstructed images as compared to the Primal-dual ℓ₁ minimization method and the typical greedy type sparse reconstruction method, orthogonal matching pursuit (OMP) method. As the number of measurements increases, our method can work more efficiently while both Primal-dual and OMP methods become slower and in particular, the primal-dual algorithm becomes much slower than the presented imaging method.