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An Extended Class of Scale-Invariant and Recursive Scale Space Filters
July 1995 (vol. 17 no. 7)
pp. 691-701

Abstract—In this paper we explore how the functional form of scale space filters is determined by a number of a priori conditions. In particular, if we assume scale space filters to be linear, isotropic convolution filters, then two conditions (viz. recursivity and scale-invariance) suffice to narrow down the collection of possible filters to a family that essentially depends on one parameter which determines the qualitative shape of the filter. Gaussian filters correspond to one particular value of this shape-parameter. For other values the filters exhibit a more complicated pattern of excitatory and inhibitory regions. This might well be relevant to the study of the neurophysiology of biological visual systems, for recent research shows the existence of extensive disinhibitory regions outside the periphery of the classical center-surround receptive field of LGN and retinal ganglion cells (in cats). Such regions cannot be accounted for by models based on the second order derivative of the Gaussian. Finally, we investigate how this work ties in with another axiomatic approach of scale space operators (propounded by Lindeberg and Alvarez et al.) which focuses on the semigroup properties of the operator family. We show that only a discrete subset of filters gives rise to an evolution which can be characterized by means of a partial differential equation.

[1] M. Abramowitz and I.A. Stegun,Handbook of Mathematical Functions.New York: Dover Publications, Inc., 1970.
[2] L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel,“Axioms and fundamental equations of image processing,” Archive for Rational Mechanics and Analysis, vol. 123, no. 3 pp. 199–257, 1993.
[3] L. Alvarez,Private Communication. 1993.
[4] J. Babaud, A. Witkin, M. Baudin, and R. Duda, "Uniqueness of the Gaussian Kernel for Scale-Space Filtering," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, pp. 26-33, Jan. 1986.
[5] R.N. Bracewell,The Fourier Transform and its Applications.Tokyo: McGraw-Hill, 1978.
[6] J.L. Crowley,“A representation for visual information,” Technical Report CMU-RI-TR-82-7, Robotic Inst., Carnegie-Mellon Univ., Pittsburgh, Pa., 1982.
[7] D.C. Champeney,A Handbook of Fourier Theorems. Cambridge Univ. Press, 1987.
[8] F. Defever,Private Communication. Dept. Theoretical Physics. K.U.Leuven, Belgium, 1994.
[9] A. Derrington and P. Lennie,“The influence of temporal frequency and adaptation level on receptive field organization of retinal ganglion cells in cats,” J. Physiology, vol. 333, pp. 343‐366, 1982.
[10] C. Enroth–Cugell,J. Robson,D. Schweitzerg–Tong,, and A. Watson,“Spatio-temporal interactions in cat retinal ganglion cells showing linear spatial summation,” J. Physiology, vol. 341, pp. 279–307, 1983.
[11] L.M.J. Florack,“The syntactical structure of scalar images,” PhD thesis, Univ. of Utrecht, The Netherlands, 1993.
[12] I.S. Gradshteyn and I.M. Ryzhik,Table of Integrals, Series and Products. Academic Press, 1965.
[13] E. Hille and R.S. Phillips,“Functional analysis and semigroups,” Am. Math. Soc., Colloquium Publications, vol. 31, 1957.
[14] J.J. Koenderink,“The structure of images,” Biological Cybernetics, vol. 50, pp. 363–370, 1984.
[15] C.-Y. Li,X. Pei,Y.-X. Zhow,, and H.-C. von Mitzlaff,“Role of the extensive area outside the X-cell receptive field in brightness information transmission,” Vision Research, vol. 31, no. 9, pp. 1, 529–1,540, 1991.
[16] C.-Y. Li,Y.-X. Zhow,X. Pei,F.-T. Qiu,C.-Q. Tang,and X.-Z. Xu,“Extensive disinhibitory region beyond the classical receptive field of cat retinal ganglion cells,” Vision Research, vol. 32, no. 2, pp. 219–228, 1991.
[17] T. Lindeberg,“Scale-space for discrete signals,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 3, pp. 234–254, 1990.
[18] T. Lindeberg,“Discrete scale-space theory and the scale-space primal sketch,” PhD thesis, Royal Inst. of Tech nology, Stockholm, Sweden, 1991.
[19] D. Marr,Vision.San Francisco: Freeman, 1982.
[20] E.J. Pauwels,P. Fiddelaers,T. Moons,, and L.J. Van Gool,“An extended class of scale-invariant and recursive scale space filters,” Technical Report KUL/ESAT/MI2/9316, K.U.Leuven ESAT-MI2, Belgium, 1993.
[21] W. Rudin,Real and Complex Analysis.New York: McGraw-Hill, 1981.
[22] A.P. Witkin,“Scale space filtering,” Proc. IJCAI,Karlsruhe, W. Germany, pp. 1, 019–1,023, 1983.
[23] A.P. Witkin,Scale Space Filtering: A New Approach to Multi-Scale Description. S. Ullman, and W. Richards, eds.,Image Understanding 1984,N. J.: NordwoodAblex, 1984.
[24] R.A. Young,“Simulation of the human retinal function with the Gaussian derivative model,” Proc. IEEE Conf. Computer Vision and Pattern Recognition,Miami, Fla., June 1986, pp. 564–569.
[25] K. Yosida,Functional Analysis.Springer Verlag, 1986.
[26] A. Yuille and T. Poggio, "Scaling Theorems for Zero Crossings," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, pp. 15-26, Jan. 1986.

Index Terms:
Scale space, non-Gaussian filters, scale invariance, recursivity, causality, semigroups.
Citation:
Eric J. Pauwels, Luc J. Van Gool, Peter Fiddelaers, Theo Moons, "An Extended Class of Scale-Invariant and Recursive Scale Space Filters," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 7, pp. 691-701, July 1995, doi:10.1109/34.391411
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