
This Article  
 
Share  
Bibliographic References  
Add to:  
Digg Furl Spurl Blink Simpy Del.icio.us Y!MyWeb  
Search  
 
ASCII Text  x  
T. Lindeberg, "ScaleSpace for Discrete Signals," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, no. 3, pp. 234254, March, 1990.  
BibTex  x  
@article{ 10.1109/34.49051, author = {T. Lindeberg}, title = {ScaleSpace for Discrete Signals}, journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence}, volume = {12}, number = {3}, issn = {01628828}, year = {1990}, pages = {234254}, doi = {http://doi.ieeecomputersociety.org/10.1109/34.49051}, 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  ScaleSpace for Discrete Signals IS  3 SN  01628828 SP234 EP254 EPD  234254 A1  T. Lindeberg, PY  1990 KW  signal processing; nonnegative kernels; unimodal kernels; spatial domain; discrete signals; discrete scalespace theory; diffusion equation; Gaussian kernel; discrete smoothing transformations; frequency domain; discrete systems; signal processing VL  12 JA  IEEE Transactions on Pattern Analysis and Machine Intelligence ER   
A basic and extensive treatment of discrete aspects of the scalespace theory is presented. A genuinely discrete scalespace theory is developed and its connection to the continuous scalespace theory is explained. Special attention is given to discretization effects, which occur when results from the continuous scalespace theory are to be implemented computationally. The 1D problem is solved completely in an axiomatic manner. For the 2D problem, the author discusses how the 2D discrete scale space should be constructed. The main results are as follows: the proper way to apply the scalespace theory to discrete signals and discrete images is by discretization of the diffusion equation, not the convolution integral; the discrete scale space obtained in this way can be described by convolution with the kernel, which is the discrete analog of the Gaussian kernel, a scalespace implementation based on the sampled Gaussian kernel might lead to undesirable effects and computational problems, especially at fine levels of scale; the 1D discrete smoothing transformations can be characterized exactly and a complete catalogue is given; all finite support 1D discrete smoothing transformations arise from repeated averaging over two adjacent elements (the limit case of such an averaging process is described); and the symmetric 1D discrete smoothing kernels are nonnegative and unimodal, in both the spatial and the frequency domain.
[1] M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions(Applied Mathematics Series, 55), Nat. Bureau Standards, 1964.
[2] T. Ando, "Totally positive matrices,"Linear Algebra Applicat., vol. 90, pp. 165219, 1987.
[3] 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.
[4] F. Bergholm, "Edge focusing,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI9, pp. 726741, 1987.
[5] G. Dahlquist,Å. Björk, and N. Anderson,Numerical Methods. London: PrenticeHall, 1974.
[6] R. M. Gray, "On the asymptotic eigenvalue distribution of Toeplitz matrices,"IEEE Trans. Inform. Theory, vol. IT18, pp. 725730, 1972.
[7] U. Grenander and G. Szegö,Toeplitz Forms and Their Applications. Los Angeles, CA: University of California Press, 1958.
[8] W. E. L. Grimson and E. C. Hildreth, "Comments on digital step edges from zero crossings of second directional derivatives,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI7, no. 1, pp. 121 127, 1985.
[9] E. Hille, and R. S. Phillips,Functional Analysis and SemiGroups, vol. XXXI, Providence, RI: Amer. Math. Soc. Colloquium Publ., 1957.
[10] I. I. Hirschmann and D. V. Widder,The Convolution Transform. Princeton, NJ: Princeton University Press, 1955.
[11] J. J. Koenderink and A. J. van Doorn, "The structure of images,"Biol. Cybern., vol. 50, pp. 363370, 1984.
[12] J. J. Koenderink and A. J. van Doorn, "Dynamic shape,"Biol. Cybern., vol. 53, pp. 383396, 1986.
[13] S. Karlin,Total Positivity, vol. I. Stanford, CA: Stanford University Press, 1968.
[14] L. M. Lifshitz and S. M. Pizer, "A multiresolution hierarchical approach to image segmentation based on intensity extrema," Dep. Comput. Sci. and Radiol., Univ. North Carolina, Chapel Hill, Internal Rep., 1987.
[15] T. P. Lindeberg, "On the construction of a scalespace for discrete images," Dep. Numer. Anal. Comput. Sci., Royal Inst. Technol., Stockholm, Sweden, Internal Rep. TRITANAP8808, 1988.
[16] T. P. Lindeberg, "Twodimensional discrete scalespace formulation," Dep. Numer. Anal. Comput. Sci., Royal Inst. Technol. Stockholm, Sweden, Internal Rep., 1989.
[17] D. Marr and E. Hildreth, "Theory of edge detection,"Proc. Roy. Soc. London, ser. B, vol. 207, pp. 187217, 1980.
[18] E. Norman, "A discrete analogue of the Weierstrass transform,"Proc. Amer. Math. Soc., vol. 11, pp. 596604, 1960.
[19] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,Numerical Recipes. London: Cambridge University Press, 1986.
[20] I. J. Schoenberg, "Über variationsvermindernde lineare Transformationen,"Mathematische Zeitschrift, vol. 32, pp. 321328, 1930.
[21] I. J. Schoenberg, "Some analytical aspects of the problem of smoothing," inStudies and Essays. New York: Courant Anniversary Volume, 1948, pp. 351370.
[22] I. J. Schoenberg, "On smoothing operations and their generating functions,"Bull. Amer. Math. Soc., vol. 59, pp. 199230, 1953.
[23] A. P. Witkin, "Scalespace filtering," inProc. 7th Int. Joint Conf. Artificial Intelligence, 1983, pp. 10191022.