Theory and practice of interpolation in the pyramid domain

Mitigating mapping effects

The remaining artifacts in Figure 3a can be attenuated in many ways. For instance, Hung et al. (2004) propose using a sinc interpolation instead of a linear interpolation. Otherwise, if missing-data interpolation is not required, the effects of linear interpolation can be attenuated by inserting a tridiagonal solver within the adjoint in equations (1) and (4). Alternatively, we can recast this attenuation as an inverse problem, in which we want to minimize the energy of the residual vector ${\bf r}$ where

$${\bf r} = {\bf Lm - d},$$ (12)

and minimize $f({\bf m})=\Vert{\bf r}\Vert _2$ iteratively to find the minimum ${\bf\tilde m}={\bf (L'L)^{-1}L'd}$. The results of this iterative formulation can be seen in Figure 4: The plane-waves are recovered without any noise. Note that the iterative solution and the tridiagonal solver approach are equivalent. However, the missing data problem we are trying to solve makes the tridiagonal solver difficult to use.

Theory and practice of interpolation in the pyramid domain