Why POCS works, and how to make it better

Projection Over Convex Sets (POCS) is one of the most widely used algorithms to interpolate seismic datasets. A formal understanding of the underlying objective function and the associated optimization process is however lacking to date in the literature. We show that POCS can be interpreted as the application of the Half-Quadratic Splitting (HQS) method to the $L_0$ 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 employ to converge to a solution of the minimizer. In light of our new theoretical understanding, we suggest that other methods able to solve this convex optimization problem, such as Chambolle-Pock Primal-Dual algorithm, may lead to a new POCS-like method with superior interpolation capabilities at nearly the same computational cost of the industry-standard POCS method.