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Adaptive Determination of Filter Scales for Edge Detection
May 1992 (vol. 14 no. 5)
pp. 579-585

The authors suggest a regularization method for determining scales for edge detection adaptively for each site in the image plane. Specifically, they extend the optimal filter concept of T. Poggio et al. (1984) and the scale-space concept of A. Witkin (1983) to an adaptive scale parameter. To avoid an ill-posed feature synthesis problem, the scheme automatically finds optimal scales adaptively for each pixel before detecting final edge maps. The authors introduce an energy function defined as a functional over continuous scale space. Natural constraints for edge detection are incorporated into the energy function. To obtain a set of optimal scales that can minimize the energy function, a parallel relaxation algorithm is introduced. Experiments for synthetic and natural scenes show the advantages of the algorithm. In particular, it is shown that this system can detect both step and diffuse edges while drastically filtering out the random noise.

[1] F. Bergholm, "Edge focusing,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-9, pp. 726-741, 1987.
[2] M. Bertero, T. A. Poggio, and V. Torre, "Ill-posed problems in early vision,"Proc. IEEE, vol. 76, pp. 869-889, Aug. 1988.
[3] D. P. Bertsekas and J. N. Tsitsiklis,Parallel and Distributed Computations. Englewood Cliffs, NJ: Prentice-Hall, 1989.
[4] J. Besag, "Spatial interaction and the statistical analysis of lattice systems,"J. Royal Stat. Soc., vol. 36, pp. 192-236, 1934, Ser. B.
[5] J. F. Canny, "Finding lines and edges in images," Artificial Intell. Lab., Massachusetts Inst. Technol., Tech. Rep. TM-720, 1983.
[6] J. F. Canny, "A computational approach to edge detection,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 679-697, 1986.
[7] R. Fletcher,Practical Methods of Optimization. New York: Wiley, 1987.
[8] S. Geman and D. Geman, "Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of image,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-6, pp. 721-941, 1984.
[9] J. Hadamard,Lectures on the Cauchy Problem in Linear Partial Differential Equations. New Haven: Yale University Press, 1923.
[10] F. B. Hildebrand,Methods of Applied Mathematics. Englewood Cliffs, NJ: Prentice-Hall, 1965.
[11] E. C. Hildreth, "Implementation of a theory of edge detection,"AI-TR- 579, AI Lab, Mass. Inst. Technol., 1980.
[12] S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi, "Optimization by simulated annealing,"Sci., vol. 220, pp. 671-680, 1983.
[13] D. G. Luenberger, Optimization byVector Space Methods. New York: Wiley, 1969.
[14] D. Marr and T. Poggio, "A Computational theory of human stereo vision," inProc. Roy. Soc. London B, 1979, vol. 204, pp. 301-328.
[15] D. Marr and E. Hildreth, "Theory of edge detection," inProc. Roy. Soc. London, 1980, vol. B-207, pp. 187-217.
[16] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, and A. H. Teller, "Equation of state calculations by fast computing machine",J. Chem. Phys., vol. 21, no. 6, pp. 1087-1092, June 1953.
[17] T. Poggio, H. Voorhees, and A. Yuille, "Regularizing edge detection,"AI Memo, no. 776, AI Lab, Mass. Inst. Technol., Cambridge, 1984.
[18] T. Poggio, H. Voorhees, and A. Yuille, "A regularized solution to edge detection," Tech. Rep. MA, Rep. AIM-833, MIT Artificial Intell. Lab., May 1985.
[19] T. Poggio, V. Torre, and C. Koch, "Computational vision and regularization theory,"Nature, vol. 317, pp. 314-319, 1985.
[20] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling,Numeric Recipes in C-The Art of Scientific Computing.Cambridge, UR: Cambridge University Press, 1988.
[21] F. Sjoberg and F. Bergholm, "Extraction of diffuse edge focusing,"Patt. Recogn. Lett., vol. 7, pp. 181-190, Mar. 1988.
[22] A. N. Tikhonov, "Solution of incorrectly formulated problems and the regularization method,"Soviet Math., vol. 4, pp. 1035-1038, 1963.
[23] V. Torre and T. A. Poggio, "On edge detection,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 147-163, Mar. 1986.
[24] A. Witkin, "Scale-space filtering," inProc. IJCAI, 1983, pp. 1019-1021.

Index Terms:
adaptive scale determination; filter scales; edge detection; regularization method; optimal filter; adaptive scale parameter; energy function; continuous scale space; parallel relaxation algorithm; filtering and prediction theory; pattern recognition; picture processing
H. Jeong, C.I. Kim, "Adaptive Determination of Filter Scales for Edge Detection," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no. 5, pp. 579-585, May 1992, doi:10.1109/34.134062
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