Steerable filters example
2nd derivative of Gaussian
Example of steerable filters.
Top line: three rotations of the second derivative of a Gaussian.
That filter has three (complex) frequencies in polar angle,
so a linear combination of three copies of the
filter are sufficient to synthesize all rotations of the filter (see references).
Middle: Zone plate test image. Bottom: By the linearity of
convolution, the output, filtered to any orientation, can be
synthesized as a lineaer combination of the outputs of the basis
filters.
Synthesized filter and output
Rotated version of 2nd derivative of Gaussian, obtained as a linear
combination of the basis filters above. The output of the zone plate
to that filter was obtained as the same linear combination of the
outputs to the basis filters.
Architecture for applying steerable filters
Steerable quadrature pair
A steerable quadrature pair allows for continuous control of the filter's
phase and orientation, useful for contour analysis and
enhancement. Seven basis filters span the space of all orientations
and phases of this filter.
References

W. T. Freeman and E. H. Adelson,
The design and use of steerable filters,
IEEE Trans. on Pattern Analysis and Machine Intelligence,
vol. 13, no. 9, pp. 891  906, September, 1991.
MIT Vision and Modeling Group TR 126.

G. H. Granlund and H. Knutsson, Signal processing for computer
vision, Kluwer Academic Publishers, 1995.

P. Perona,
Deformable kernels for early vision,
IEEE Trans. on Pattern Analysis and Machine Intelligence,
vol. 17, no. 5, pp. 488499, May, 1995.

E. P. Simoncelli and H. Farid,
Steerable wedge filters for local orientation analysis,
IEEE Trans. Information Theory
vol. 5, no. 9, pp. 13771382 , 1996.

E. P. Simoncelli and W. T. Freeman,
The steerable pyramid: a
flexible architecture for multiscale derivative computation, 2nd
Annual IEEE Intl. Conference on Image Processing, Washington, DC.
October, 1995.
MERLTR9515.

P C. Teo and Y. HelOr,
A Computational GroupTheoretic Approach to Steerable
Functions<\cite>, STANCSTN9633, Department of Computer
Science, Stanford University, April 1996.

J. W. Zweck and L. R. Williams
Euclidean group invariant computation of stochastic completion
fields using shiftabletwistable functions,
December, 1999.