Sparse Aperture Autofocusing and Imaging Based on Fast Sparse Bayesian Learning From Gapped Data

Sparse aperture (SA) autofocusing and imaging is a hot research problem in the signal processing field and has been widely used. Under SA, the absence of echoes destroys the coherence between the pulses, which then affects the autofocusing accuracy of the imaging, leading to defocus of the image. In this article, a novel SA autofocusing and imaging algorithm based on sparse Bayesian learning (SBL) is proposed, which uses a fast SBL algorithm to achieve SA high-resolution imaging and the minimum Tsallis entropy algorithm to realize autofocusing. As is known to all, SBL has strong robustness and high precision. Unfortunately, the direct calculation of the inversion and multiplication operations involved in each iteration of SBL results in significant computational costs. In the proposed fast SBL algorithm, the matrix required to be inverted has a special structure. The inverse matrix can then be represented by Gohberg–Semencul (G–S) factorization. Also, almost all operations except for G–S factorization during each iteration can be completed by fast Fourier transform (FFT) or inverse FFT (IFFT), which greatly reduces the amount of computation by several orders of magnitude. In each SBL iteration, the minimum Tsallis entropy algorithm is used for estimating the phase error, which has better noise sensitivity and obtains the images with the best focused degree. Finally, the effectiveness and high efficiency of the proposed fast algorithm are verified by experimental results obtained by simulation and measured data.