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. 488-499, May, 1995.

E. P. Simoncelli and H. Farid, Steerable wedge filters for local orientation analysis, IEEE Trans. Information Theory vol. 5, no. 9, pp. 1377-1382 , 1996.

E. P. Simoncelli and W. T. Freeman, The steerable pyramid: a flexible architecture for multi-scale derivative computation, 2nd Annual IEEE Intl. Conference on Image Processing, Washington, DC. October, 1995. MERL-TR95-15.

P C. Teo and Y. Hel-Or, A Computational Group-Theoretic Approach to Steerable Functions<\cite>, STAN-CS-TN-96-33, Department of Computer Science, Stanford University, April 1996.

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