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J.J. Clark, "Singularity Theory and Phantom Edges in Scale Space," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 10, no. 5, pp. 720727, September, 1988.  
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@article{ 10.1109/34.6782, author = {J.J. Clark}, title = {Singularity Theory and Phantom Edges in Scale Space}, journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence}, volume = {10}, number = {5}, issn = {01628828}, year = {1988}, pages = {720727}, doi = {http://doi.ieeecomputersociety.org/10.1109/34.6782}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Pattern Analysis and Machine Intelligence TI  Singularity Theory and Phantom Edges in Scale Space IS  5 SN  01628828 SP720 EP727 EPD  720727 A1  J.J. Clark, PY  1988 KW  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 VL  10 JA  IEEE Transactions on Pattern Analysis and Machine Intelligence ER   
The process of detecting edges in a onedimensional 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 scalespace filtering,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI8, pp. 2633, Jan. 1986.
[3] J. F. Canny, "A computational approach to edge detection,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI8, pp. 679697, 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: WileyInterscience, 1981.
[7] V. Guillemin and A. Pollack,Differential Topology. Englewood Cliffs, NJ: PrenticeHall, 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. 5868, 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. 363370, 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. 187217, 1980.
[13] D. Marr and T. Poggio, "A computational theory of human stereo vision,"Proc. Roy. Soc. London, series B, vol. 207, pp. 187217, 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, "Nonlinearities in cortical simple cells and the possible detection of zerocrossings,"Biol. Cybern., vol. 53, pp. 195202, 1986.
[17] A. Rosenfeld and M. Thurston, "Edge and curve detection for visual scene analysis,"IEEE Trans. Comput., vol. 20, pp. 562569, 1971.
[18] M. Shah, A. Sood, and R. Jain, "Pulse and staircase edge models",Comput. Vision Graphics Image Processing, vol. 34, pp. 321343, 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: BenjaminAddison Wesley, 1975.
[21] V. Torre and T. A. Poggio, "On edge detection,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI8, pp. 147163, Mar. 1986.
[22] A. Witkin, "Scale space filtering," inProc. 8th Int. Joint Conf. Artificial Intelligence, Karlsruhe, 1983, pp. 10191022.
[23] A. Yuille and T. Poggio, "Scaling theorems for zero crossings,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI8, no. 1, pp. 1525, 1986.