Why projection over convex sets works and how to make it better

Projection over convex sets (POCS) is one of the most widely used algorithms to interpolate seismic data sets. A formal understanding of the underlying objective function and the associated optimization process is, however, lacking to date in the literature. Here, POCS is shown to be equivalent to the application of the half-quadratic splitting (HQS) method to the L₀ norm of an orthonormal projection of the sought after data, constrained on the available traces. Similarly, the apparently heuristic strategy of using a decaying threshold in POCS is revealed to be the result of the continuation strategy that HQS must use to converge to a solution of the minimizer. In light of this theoretical understanding, another methods able to solve this convex optimization problem, namely the Chambolle-Pock primal-dual algorithm, is shown to lead to a new POCS-like method with superior interpolation capabilities at nearly the same computational cost of the industry-standard POCS method.