Seismic data interpolation using nonlinear shaping regularization

Connection with projection onto convex sets

If we define:

$$\mathbf{d}'_n =\mathbf{d}_n+\mathbf{B}[\mathbf{d}_{obs}-\mathbf{Md}_n]$$ (9)

$$\mathbf{S} = \mathbf{d}_{obs} + (\mathbf{I}-\mathbf{M})\mathbf{ATA}^H[\mathbf{d}'_n]$$ (10)

and takes $\mathbf{B}=\mathbf{I}-\mathbf{M}$ , then equation 6 turns to:

$$\begin{split}\mathbf{d}_{n+1} &= \mathbf{d}_{obs} + (\mathbf{... ...\mathbf{I}-\mathbf{M})\mathbf{ATA}^H[\mathbf{d}_n]. \end{split}$$ (11)

The last line of equation 11 is the formulation of what we call POCS. We derive POCS from the framework of nonlinear shaping regularization and thus prove that POCS is no more than a special nonlinear shaping regularization iteration.