This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Singularity Theory and Phantom Edges in Scale Space
September 1988 (vol. 10 no. 5)
pp. 720-727

The process of detecting edges in a one-dimensional signal by finding the zeros of the second derivative of the signal can be interpreted as the process of detecting the critical points of a general class of contrast functions that are applied to the signal. It is shown that the second derivative of the contrast function at the critical point is related to the classification of the associated edge as being phantom or authentic. The contrast of authentic edges decreases with filter scale, while the contrast of phantom edges are shown to increase with scale. As the filter scale increases, an authentic edge must either turn into a phantom edge or join with a phantom edge and vanish. The points in the scale space at which these events occur are seen to be singular points of the contrast function. Using ideas from singularity, or catastrophy theory, the scale map contours near these singular points are found to be either vertical or parabolic.

[1] V. I. Arnold,Singularity Theory. Cambridge, MA: Cambridge University Press, 1981.
[2] J. Babaud, A. P. Witkin, M. Baudin, and R. O. Duda, "Uniqueness of the Gaussian kernel for scale-space filtering,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 26-33, Jan. 1986.
[3] J. F. Canny, "A computational approach to edge detection,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 679-697, 1986.
[4] J. J. Clark, "Authenticating edges produced by zero crossing algorithms," Harvard Robotics Laboratory Tech. Rep., 1986; alsoIEEE Trans. Pattern Anal. Machine Intell., to be published.
[5] J. J. Clark, "Singularities of contrast functions in scale space," inProc. 1st Int. Conf. Computer Vision, London, England, 1987, pp. 491- 496.
[6] R. Gilmore,Catastrophe Theory for Scientists and Engineers. New York: Wiley-Interscience, 1981.
[7] V. Guillemin and A. Pollack,Differential Topology. Englewood Cliffs, NJ: Prentice-Hall, 1974.
[8] R. M. Haralick, "Digital step edges from zero crossings of second directional derivatives,"IEEE Trans. Pattern Anal. Machine Intell., vol. 6, no. 1, pp. 58-68, 1984.
[9] R. A. Hummel and B. C. Gidas, "Zero crossings and the heat equation," Courant Inst. Math. Sci., Comput. Sci. Division, New York Univ., Tech. Rep. 111, 1984.
[10] J. J. Koenderink, "The structure of images,"Biol. Cybern., vol. 50, pp. 363-370, 1984.
[11] D. Marr,Vision. San Francisco, CA: W. H. Freeman, 1982.
[12] D. Marr and E. C. Hildreth, "Theory of edge detection,"Proc. Roy. Soc. London, series B, vol. 207, pp. 187-217, 1980.
[13] D. Marr and T. Poggio, "A computational theory of human stereo vision,"Proc. Roy. Soc. London, series B, vol. 207, pp. 187-217, 1980.
[14] J. Milnor,Morse Theory, Annals of Mathematics Studies Number 51. Princeton, NJ: Princeton University, 1963.
[15] T. Poston and I. Stewart,Catastrophe Theory and its Applications. London: Pitman, 1978.
[16] J. Richter and S. Ullman, "Non-linearities in cortical simple cells and the possible detection of zero-crossings,"Biol. Cybern., vol. 53, pp. 195-202, 1986.
[17] A. Rosenfeld and M. Thurston, "Edge and curve detection for visual scene analysis,"IEEE Trans. Comput., vol. 20, pp. 562-569, 1971.
[18] M. Shah, A. Sood, and R. Jain, "Pulse and staircase edge models",Comput. Vision Graphics Image Processing, vol. 34, pp. 321-343, 1986.
[19] J. L. Stansfield, "Conclusions from the commodity expert project," Massachusetts Inst. Technol., AI Lab. Memo 601, 1980.
[20] R. Thom,Structural Stability and Morphogenesis(translated D. H. Fowler). New York: Benjamin-Addison Wesley, 1975.
[21] V. Torre and T. A. Poggio, "On edge detection,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 147-163, Mar. 1986.
[22] A. Witkin, "Scale space filtering," inProc. 8th Int. Joint Conf. Artificial Intelligence, Karlsruhe, 1983, pp. 1019-1022.
[23] A. Yuille and T. Poggio, "Scaling theorems for zero crossings,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, no. 1, pp. 15-25, 1986.

Index Terms:
1D signal; picture processing; edge detection; pattern recognition; singularity theory; phantom edges; scale space; contrast functions; filter scale; catastrophy theory; scale map contours; catastrophe theory; filtering and prediction theory; pattern recognition; picture processing; signal processing
Citation:
J.J. Clark, "Singularity Theory and Phantom Edges in Scale Space," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 10, no. 5, pp. 720-727, Sept. 1988, doi:10.1109/34.6782
Usage of this product signifies your acceptance of the Terms of Use.