3D pyramid interpolation

Conclusions

The pyramid domain is a very promising domain for missing-data interpolation. The synthetic examples demonstrate that, with a good initial PEF estimate, we can use the information in the low frequency to interpolate the aliased missing data relatively accurately and interpolate the unaliased missing data fairly well.

One disadvantage of this interpolation scheme is computation cost. First, to get a decent result, the data samples in pyramid domain is an order more than that in $f$-$\bf x$ domain along each spatial axis; in 3D, that amounts to a factor of $100$ or more. In addition, the linearized nonlinear iteration adds a factor of about five in the synthetic test. In other words, we have to do both PEF estimation and data interpolation five times in total. Altogether, we first increase the data size by a factor of $100$, then run about $5$ rounds of data estimation. So the overall computational cost is about $500$ times greater than a conventional PEF based interpolation scheme (e.g. Spitz, 1991). However, with a patching technique, many patches of data can be interpolated simutaneously using parallelized version of this algorithm.

3D pyramid interpolation