# AN UNIFIED THEORY FOR STEERABLE AND QUADRATURE FILTERS

### Kai Krajsek, Rudolf Mester

#### 2006

#### Abstract

In this paper, a complete theory of steerable filters is presented which shows that quadrature filters are only a special case of steerable filters. Although there has been a large number of approaches dealing with the theory of steerable filters, none of these gives a complete theory with respect to the transformation groups which deform the filter kernel. Michaelis and Sommer (Michaelis and Sommer, 1995) and Hel-Or and Teo (Teo and Hel-Or, 1996; Teo and Hel-Or, 1998) were the first ones who gave a theoretical justification for steerability based on Lie group theory. But the approach of Michaelis and Sommer considers only Abelian Lie groups. Although the approach of Hel-Or and Teo considers all Lie groups, their method for generating the basis functions may fail as shown in this paper. We extend these steerable approaches to arbitrary Lie groups, like the important case of the rotation group SO(3) in three dimensions. Quadrature filters serve for computing the local energy and local phase of a signal. Whereas for the one dimensional case quadrature filters are theoretically well founded, this is not the case for higher dimensional signal spaces. The monogenic signal (Felsberg and Sommer, 2001) based on the Riesz transformation has been shown to be a rotational invariant generalization of the analytic signal. A further generalization of the monogenic signal, the 2D rotational invariant quadrature filter (Ko¨ the, 2003), has been shown to capture richer structures in images as the monogenic signal. We present a generalization of the rotational invariant quadrature filter based on our steerable theory. Our approach includes the important case of 3D rotational invariant quadrature filters but it is not limited to any signal dimension and includes all transformation groups that own an unitary group representation.

#### References

- B ülow, T. and Sommer, G. (2001). Hypercomplex signals - a novel extension of the analytic signal to the multidimensional case. IEEE Transactions on Signal Processing, 49(11):2844-2852.
- Danielsson, P. E. (1980). Rotation-invariant linear operators with directional response. In Proc. Int. Conf. Pattern Recognition, Miami, FL.
- Felsberg, M. and Sommer, G. (2001). The monogenic signal. IEEE Transactions on Signal Processing, 49(12):3136-3144.
- Freeman, W. and Adelson, E. (1991). The design and use of steerable filters. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(9):891-906.
- Granlund, G. H. and Knutsson, H. (1995). Signal processing for computer vision. Kluwer.
- Köthe, U. (2003). Integrated edge and junction detection with the boundary tensor. In Proc. of 9th International Conference on Computer Vision, volume 1, pages 424-431, Nice, France.
- Michaelis, M. and Sommer, G. (1995). A Lie group approach to steerable filters. Pattern Recognition Letters, 16:1165-1174.
- Perona, P. (1995). Deformable kernels for early vision. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(5):488-499.
- Simoncelli, E. and Farid, H. (1996). Steerable wedge filters for local orientation analysis. IEEE Transactions on Image Processing, 5(9):1377-1382.
- Simoncelli, E. P., Freeman, W. T., Adelson, E. H., and Heeger, D. J. (1992). Shiftable multiscale transforms. IEEE Transactions on Information Theory, 38(2):587-607.
- Teo, P. and Hel-Or, Y. (1996). A common framework for steerability, motion estimation and invariant feature detection. Technical Report STAN-CS-TN-96- 28, Stanford University.
- Teo, P. and Hel-Or, Y. (1998). Lie generators for computing steerable functions. Pattern Recognition Letters, 19(1):7-17.
- Wigner, E. (1959). Group Theory and its Application to Quantum Mechanics of Atomic Spectra. Academic Press, New York.
- Yu, W., Daniilidis, K., and Sommer, G. (2001). Approximate orientation steerability based on angular gaussians. IEEE Transactions on Image Processing, 10(2):193-205.

#### Paper Citation

#### in Harvard Style

Krajsek K. and Mester R. (2006). **AN UNIFIED THEORY FOR STEERABLE AND QUADRATURE FILTERS** . In *Proceedings of the First International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP,* ISBN 972-8865-40-6, pages 48-55. DOI: 10.5220/0001377100480055

#### in Bibtex Style

@conference{visapp06,

author={Kai Krajsek and Rudolf Mester},

title={AN UNIFIED THEORY FOR STEERABLE AND QUADRATURE FILTERS},

booktitle={Proceedings of the First International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP,},

year={2006},

pages={48-55},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001377100480055},

isbn={972-8865-40-6},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the First International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP,

TI - AN UNIFIED THEORY FOR STEERABLE AND QUADRATURE FILTERS

SN - 972-8865-40-6

AU - Krajsek K.

AU - Mester R.

PY - 2006

SP - 48

EP - 55

DO - 10.5220/0001377100480055