As the approximation of the derivative (equation 40) is not precise, it introduces a bias into the SPM estimation technique. This source of bias can be eliminated by using a finite difference calculation instead of the derivative-based one.
That is, the difference field is:
(42) |
The variance of this difference field is:
(43) | |||
(44) |
Setting
with
,
the expectations can be calculated using equation 14:
(45) | |||
(46) |
(47) |
This enables the individual smoothing parameters,
to be found. That is: