Nonconvex Log-Sum Function-Based Majorization–Minimization Framework for Seismic Data Reconstruction

Because of the fact that complete seismic data can have a low rank in the frequency-space (f-x) domain, rank-reduction methods are classical techniques used for seismic data reconstruction. Models that employ nuclear-norm minimization signify convex relaxation methods in traditional rank minimization problems. However, the results obtained after solving the nuclear-norm minimization problem are usually suboptimal because the nuclear norm indicates a loose approximation of the rank function. To overcome the limitations of the nuclear norm, we propose a new method of seismic data reconstruction based on the log-sum function minimization, which is closer to the rank function than the convex nuclear norm. However, the problem based on the log-sum function is nonconvex. Consequently, a majorization-minimization framework has been adopted to solve the associated minimization problem. Numerical experiments performed using synthetic and real data demonstrate that the quality of reconstruction derived from our proposed algorithm is better than that of the singular value thresholding algorithm and the weighted nuclear norm method.