Derivative Filters
When computing multi-dimensional derivatives, one often relies on the
linear-algebraic properties of multi-dimensional derivatives that
allow differentiation in arbitrary directions via linear combinations
of separable axis derivatives. For example, the derivative operator
in the direction of unit vector u is ux d/dx +
uy d/dy. We have designed a set of compact separable
filters that optimize for this property. We have also developed
extensions of this formulation to both higher dimensions and
higher-order directional derivatives. The resulting filters are
significantly more accurate than those commonly used.
Below are first- and higher-order derivative filters of various sizes,
where p denotes the lowpass filter, and dn denotes the
n-th order derivative filter.
Matlab code (tar.gz)
(Collaborative work with Eero Simoncelli)
|