Filter estimation

A PEF (${\bf a}$) by definition is the filter that minimizes the energy when convolved with the data ($\bf d$). To estimate a space-invariant filter, this amounts to applying the fitting goal,

$$\bf 0\approx {\bf D} {\bf a} .$$ (7)

When estimating a space-varying PEF, the number of filter coefficients can quickly become more than the number of data points, creating an underdetermined problem. Crawley et al. (1999) proposed estimating space varying filter with radial patch. The fitting goals become

$$\begin{eqnarray} \bf 0&\approx&{\bf D} \bf A^{-1}\bf p\\ \nonumber \bf 0&\approx&\epsilon \bf A^{-1} ,\end{eqnarray}$$ (8)

where $\bf A^{-1}$ is a preconditioning operator that smoothes in a radial direction (assuming that dips will be more consistent along radial lines, Figure 1.

 

pef Figure 1 Space varying filter composition. A different filter is placed inside each micro-patch. The filter estimation problem is done globally, with the filter coefficient smoothed in a radial direction.