Adaptive shaping filters

The first step is to consider non-stationary shaping filters. Experience with missing data problems Crawley et al. (1998); Crawley (1999b) has shown that working with smoothly-varying non-stationary filters often gives better results than working with filters that are stationary within small patches.

With a non-stationary convolution filter, ${\bf f}$, the shaping filter regression equations,

$${\bf A}_1 \, {\bf f} - {\bf a}_2 = {\bf 0},$$ (3)

are massively underdetermined since there is a potentially unique impulse response associated with every point in the dataspace Rickett (1999). We need constraints to ensure the filters vary-smoothly in some manner.

The simplest regularization scheme involves applying a generic data-space roughening operator, ${\bf R}$, to the non-stationary filter coefficients. ${\bf R}$ can be a simple derivative operator, for example. This leads to the set of equations,

$$\begin{eqnarray} {\bf A}_1 \, {\bf f} - {\bf a}_2 & = & {\bf 0} \\ \epsilon \; {\bf R} {\bf f} & = & {\bf 0} \;. \end{eqnarray}$$ (4) (5)

By making the change of variables, ${\bf q}={\bf R} \, {\bf f}$ Fomel (1997b), we get the following system of equations,

$$\begin{eqnarray} {\bf A}_1 {\bf R}^{-1} {\bf q} - {\bf a}_2 & = & {\bf 0} \\ \epsilon \; {\bf q} & = & {\bf 0} \;.\end{eqnarray}$$ (6) (7)

Equations (6) and (7) describe a preconditioned linear system of equations, the solution to which converges rapidly under an iterative conjugate-gradients solver. In practice, I set $\epsilon=0$, and keep the filters smooth by restricting the number of iterations Crawley (1999a).