Theory and practice of interpolation in the pyramid domain

Conclusion

The pyramid transform creates a domain where linear events in $(t,x)$-space are linear in $(\omega ,u)$-space. This property offers many opportunities for interpolation schemes based on prediction-error filters. First, one filter can predict all frequencies. Second, this pef can be estimated from all frequencies. Finally, the filter estimation should be more robust to the noise present in data (although not shown here).

One challenge with the pyramid transform is the remapping between the $(\omega ,x)$ and $(\omega ,u)$ domains which can introduce artifacts. We propose mitigating these effects by making the pyramid axis $u$ very dense and using a simple linear interpolation for the transform. One consequence of this proposal is that many empty bins appear in the pyramid domain. Realizing that missing data will add even more empty locations, we introduce a non-linear algorithm that both interpolates missing data (regularly or irregularly-spaced data) and fills the empty bin locations (those resulting from the transform).

Our synthetic and field data experiments prove that the proposed algorithm works and that the pyramid domain is a sensible complement to our existing interpolation toolbox. Although not presented here, the extension to 3-D would be straightforward. Interpolating aliased or irregularly-space data does not require any change of the algorithm. However, we notice that smaller $\delta u$'s are required when de-alasing is needed.

Comparing the pyramid transform to other domains for missing data interpolation goes beyond the scope of this paper, but more work needs to be done to understand how the proposed algorithm fairs when compared to more popular techniques such as $FX$ interpolation.

The pyramid transform could benefit other applications. For instance, the irregular sampling case could be treated with a combination of nonuniform Fourier transform and pyramid transform. The signal/noise separation problem could be easily recast in the pyramid domain where only 1-D (for 2-D data) and 2-D (for 3-D data) projection filters are necessary [Soubaras (1994)].

Theory and practice of interpolation in the pyramid domain