Consider taking the spatial derivative of the smoothed field. That is,
constructing three derivative fields,
, where:
(15) |
Now the partial derivative of the smoothing filter is given by:
(16) |
Therefore, the covariance can be computed as in the last section:
The latter two integrals are given by equation 13. The former integral is:
Rewriting the last part using
gives:
Therefore, combining equations 20, 22 and 27 gives: