This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Optimal L1${\cal L}_1$ Approximation of the Gaussian Kernel With Application to Scale-Space Construction
October 1995 (vol. 17 no. 10)
pp. 1015-1019

Abstract—Scale-space construction based on Gaussian filtering requires convolving signals with a large bank of Gaussian filters with different widths. In this paper we propose an efficient way for this purpose by ${\cal L}_1$ optimal approximation of the Gaussian kernel in terms of linear combinations of a small number of basis functions. Exploring total positivity of the Gaussian kernel, the method has the following properties: 1) the optimal basis functions are still Gaussian and can be obtained analytically; 2) scale-spaces for a continuum of scales can be computed easily; 3) a significant reduction in computation and storage costs is possible. Moreover, this work sheds light on some issues related to use of Gaussian models for multiscale image processing.

[1] 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.
[2] R.A. DeVore, B. Jawerth, and B.J. Lucier, “Image Compression through Wavelet Transform Coding,” IEEE Trans. Information Theory, vol. 38, no. 2 (Part II)), pp. 719-746, 1992.
[3] W.T. Freeman and E.H. Adelson, "The Design and Use of Steerable Filters," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, pp. 891-906, 1991.
[4] S. Karlin,Total Positivity, vol. I. Stanford, Calif: Stanford Univ. Press, 1968.
[5] J. Koenderink,“The structure of images,” Biological Cybernetics, vol. 50, pp. 363-370, 1984.
[6] X. Li and T. Chen,“Efficient synthesis of parameterized Gaussian-like filters byapproximation,” Signal Processing, vol. 41, pp. 119-134, 1995.
[7] T. Lindeberg,“Scale-space for discrete signals,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 3, pp. 234–254, 1990.
[8] D. Marr,Vision.San Francisco: Freeman, 1982.
[9] C.A. Micchelli and A. Pinkus,“Some problems in the approximation of functions of two variables andn-widths of integral operators,” J. Approximation Theory, vol. 24, pp. 51-77, 1978.
[10] P. Perona,“Deformable kernels for early vision,” Technical Report MIT-LIDS-P-2039, 1991.
[11] P. Perona,“Steerable-scalable kernels for edge detection and junctionnalysis,” Second European Conf. Computer Vision, vol. 588, Lecture Notes in Computer Sciences, Springer-Verlag, 1992.
[12] E.P. Simoncelli, W.T. Freeman, E.H. Adelson, and D.J. Heeger, “Shiftable Multi-Scale Transforms,” IEEE Trans. Information Theory, vol. 38, no. 2, pp. 587-607, Mar. 1992.
[13] B.M. ter Haar Romeny,L.M.J. Florack,J.J. Koenderink,, and M.A. Viergever,“Scale-space: its natural operators and differential invariants,” Int’l Conf. Information Processing in Medical Imaging, vol. 511, Lecture Notes in Computer Sciences, pp. 239-255,Springer-Verlag, 1992.
[14] A. Witkin,“Scale-space filtering,” in Int’l Joint Conf. Artificial Intelligence, pp. 1,019-1,021,Karsruke, Germany, 1983.
[15] 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:
Gaussian kernel, scale-space, total positivity, ${\cal L}_1$ approximation.
Citation:
Xiaoping Li, Tongwen Chen, "Optimal L1${\cal L}_1$ Approximation of the Gaussian Kernel With Application to Scale-Space Construction," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 17, no. 10, pp. 1015-1019, Oct. 1995, doi:10.1109/34.464565
Usage of this product signifies your acceptance of the Terms of Use.