Abstract
The image algebra, developed by Ritter et al. at the University of Florida, is an algebraic structure for image processing. The three commonly used high-level image-template operations provided by the image algebra are the generalized convolution ⊕, the additive maximum or generalized lattice convolution {ie23-1}, and the multiplicative maximum {ie23-2}. These are used to realize various nonrecursive image transformations, including morphological transformations. Along with nonrecursive transformations, a class of recursive transformations, such as IIR filters, adaptive dithering, and predictive coding, are also widely used in signal and image processing. In this paper the notions of recursive templates and recursive template operations are introduced; these allow the image algebra to express a set of linear and nonlinear recursive transformations. Algebraic properties of these recursive operations are given, providing a mathematical basis for recursive template composition and decomposition. Finally, applications of recursive template operations in specifying some image processing algorithms are presented.
Similar content being viewed by others
References
T.S. Huang, ed., Two Dimensional Digital Signal Processing I—Linear Filters, Topics in Applied Physics, vol. 42, New York: Springer-Verlag, 1981.
A. Rosenfeld and A.C. Kak, Digital Picture Processing, 2nd ed., New York: Academic Press, 1982.
D.H. Ballard and C.M. Brown, Computer Vision, Englewood Cliffs, NJ: Prentice-Hall, 1982.
R. Ulichney, Digital Halftoning, Cambridge, MA: MIT Press, 1987.
A. Rosenfeld and J. Pfaltz, “Sequential Operation in Digital Picture Processing,” JACM, vol. 12(4), 1966, pp. 471–494.
S.Y. Kung, VLSI Array Processors, Englewood Cliffs, NJ: Prentice-Hall, 1988.
G.X. Ritter, J.N. Wilson, and J.L. Davidson, “Image Algebra: An Overview,” Comput. Vis., Graph., Image Process., vol. 49, 1990, pp. 297–331.
G.X. Ritter, M.A. Schrader-Frechette, and J.N. Wilson, “Image Algebra: A Rigorous and Translucent Way of Expressing All Image Processing Operations,” in Proceedings of the 1987 SPIE Technical Symposium Southeast on Optics, Electro-Optics, and Sensors, Orlando, FL, May 1987.
G.X. Ritter and P.D. Gader, “Image Algebra Techniques for Parallel Image Processing,” J. Parallel Distr. Comput., vol. 4(5), 1987, pp. 7–44.
P.D. Gader, “Image Algebra Techniques for Parallel Computation of Discrete Fourier Transforms and General Linear Transforms,” Ph.D. thesis, University of Florida, Gainesville, FL, 1986.
G.X. Ritter, J.L. Davidson, and J.N. Wilson, “Beyond Mathematical Morphology,” in Proceedings of the SPIE Conference-Visual Communication and Image Processing II, Cambridge, MA, October 1987, pp. 260–269.
J.L. Davidson, “Lattice Structures in the Image Algebra and Their Applications to Image Processing,” Ph.D. thesis, University of Florida, Gainesville, FL, 1989.
R.A. Cuninghame-Green, Minimax Algebra, Lecture Notes in Economics and Mathematical Systems, vol. 166, New York: Springer-Verlag, 1979.
I. Gohberg, P. Lancaster, and L. Rodman, Invariant Subspaces of Matrices with Applications, New York: John Wiley & Sons, 1986.
J.G. Wade, Signal Coding and Processing—An Introduction Based on Video Systems, London: Ellis Horwood, 1983.
G. Borgefors, “Distance Transformations in Digital Images,” Comput. Vis. Graph., Image Process., vol. 34, 1986, pp. 344–371.
A. Rosenfeld and J. Pfaltz, “Distance Function on Digital Pictures,” Patt. Recog., vol. 1(1), 1968, pp. 33–61.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Li, D. Recursive operations in image algebra. J Math Imaging Vis 1, 23–42 (1992). https://doi.org/10.1007/BF00135223
Issue Date:
DOI: https://doi.org/10.1007/BF00135223