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Scaling Theorems for Zero Crossings of Bandlimited Signals
March 1996 (vol. 18 no. 3)
pp. 309-320

Abstract—Scale-space filtering is the only known method which provides a hierarchic signal description method by extracting features across a continuum of scales. One of its important characteristics is that it demands the filtering involved does not create generic features as the scale increases. It has been shown in [4], [5], [6] that the Gaussian filter is unique in holding this remarkable property. This is in essence the so-called scaling theorem. In this paper, we propose two scaling theorems for band-limited signals. They are applicable to a broader class of signals and a bigger family of filtering kernels than in [4], [5],[6]. An in-depth discussion of our theorems and the previously published ones is also given.

[1] A. Papoulis, Signal Analysis.New York: McGraw-Hill, 1977.
[2] A.P. Witkin, "Scale-space filtering," Proc. Eighth Int'l Joint Conf. Artificial Intelligence,Karlsruhe, Germany, pp. 1,019-1,022, 1983.
[3] A. P. Witkin, "Scale space filtering: a new approach to multi-scale description," S. Ullman and W. Richards eds., Image Understanding 1984.Norwood, N.J.: Ablex, 1984.
[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] A. Yuille and T. Poggio, "Scaling Theorems for Zero Crossings," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, pp. 15-26, Jan. 1986.
[6] L. Wu and Z. Xie, "Scaling Theorems for Zero-Crossings," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, pp. 46-54, Jan. 1991.
[7] A.L. Yuille and T. Poggio, "Fingerprints theorems for zero crossings," J. Opt. Soc. Am. A, vol. 2, no. 5, pp. 683-692, May 1985.
[8] J.J. Clark, "Singularities of contrast functions in scale space," Proc. First Int'l Conf. on Computer Vision,London, pp. 491-495, 1987.
[9] A.J. Jerri, "The Shannon sampling theorem_its various extensions and applications: A tutorial review," Proc. IEEE, vol. 65, no. 11, pp. p. 1,565, Nov. 1977.
[10] A. Rosenfeld and M. Thurston, "Edge and curve detection for visual scene analysis," IEEE Transactions on Computers, vol. 20, pp. 562-569, 1971.
[11] D. Marr and E. Hildreth, "Theory of edge detection," Proc. Royal Society of London, vol. B207, pp. 187-217, 1980.
[12] J.J. Koenderink, "The structure of images," Biological Cybernetics, vol. 50, pp. 363-370, 1980.
[13] V. Torre and T.A. Poggio, "On Edge Detection," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, no. 2, pp. 147-163, Mar. 1986.
[14] D. Slepian, "On bandwidth," Proc. IEEE, vol. 64, no. 3, pp. 292-300, 1976.
[15] R.N. Bracewell, The Fourier Transform and Its Applications, 2nd revised ed. New York: McGraw-Hill, 1986.
[16] J.Y. Shi, H.T. Tsui, and J.G. Li, "Fejér filtering for multiscale signal decomposition," Proc. IEEE-SP Int'l Symp. Time-Frequency and Time- Scale Analysis,Victoria City, Canada, pp. 563-566, 1992.
[17] M. Bertero, T.A. Poggio, and V. Torre, "Ill-posed problems in early vision," Proc. IEEE, vol. 76, no. 8, pp. 869-889, 1988.
[18] K. Chandrasekharan, Classical Fourier Transforms.New York: Springer-Verlag, 1989.
[19] G.I. Barenblatt, Dimensional Analysis. Gordon and Breach, 1987.
[20] B.F. Logan, Jr., "Information in the zero-crossings of bandpass signals," Bell System Technical J., vol. 56, pp. 487-510, 1977.
[21] J.G. Daugman, "Pattern and motion vision without Laplacian zero crossings," J. Opt. Soc. Am. A, vol. 5, no. 7, pp. 1,142-1,148, 1988.

Index Terms:
Scaling theorems, zero crossings, Gaussian kernels, scale space, multiscale analysis, signal descriptions, bandlimited signals, Whittaker-Shannon sampling theorem, quadratic forms.
Citation:
Vo Anh, Ji Yu Shi, Hung Tat Tsui, "Scaling Theorems for Zero Crossings of Bandlimited Signals," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 3, pp. 309-320, March 1996, doi:10.1109/34.485558
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