Regularization

In the formulation above our model quality is now greatly determined by our choice of regularization operator. Biondi and Vlad (2001) implemented two different approach. The first was simply applying a derivative filter,

$$-\rho$$ 1

along the offset axis. The $\rho$ controls the length of the smoother. For symmetry we can cascade a left derivative $\bf D_l$ followed by a right derivative $\bf D_r$ for the combined regularization operator $\bf A= {\bf D_r} {\bf D_l}$.

Biondo and Vlad's (2001) second choice is more interesting. Vlad and Biondi (2001) describes a very fast implementation of AMO (based on the DMO formulation in the logstretch domain of Zhou et al. (1996)) on a regularly sampled mesh. They suggested instead of minimizing the difference between two offset cubes, to minimize the difference between the two cubes continued to the same offset through AMO. If we now imagine the filter operating on (t,cmp_x,cmp_y) cubes, the right derivative operation becomes

$\bf I$	- $\rho {\bf T_{h_{i+1,i}}}$

where $\bf T_{h_{i,i+1}}$ is the AMO transformation of the (t,cmp_x,cmp_y) at offset i+1 to offset i.