Abstract
Stochastic representation of discrete images by partial differential equation operators is considered. It is shown that these representations can fit random images, with nonseparable, isotropic covariance functions, better than other common covariance models. Application of these models in image restoration, data compression, edge detection, image synthesis, etc., is possible.
Different representations based on classification of partial differential equations are considered. Examples on different images show the advantages of using these representations. The previously introduced notion of fast Karhunen-Loeve transform is extended to images with nonseparable or nearly isotropic covariance functions, or both.
Similar content being viewed by others
References
Nahi, N. E., andAssefi, T.,Bayesian Recursive Image Enhancement, IEEE Transactions on Computers, Vol. C-21, pp. 734–738, 1972.
Franks, L. E.,A Model for Random Video Processes, Bell Systems Technical Journal, Vol. 45, pp. 609–630, 1966.
Jain, A. K., andAngel, E.,Image Restoration, Modelling, and Reduction of Dimensionality, IEEE Transactions on Computers, Vol. C-23, pp. 470–476, 1974.
Jain, A. K.,Image Coding via a Nearest Neighbors Image Model, IEEE Transactions on Communications, Vol. COM-23, pp. 318–331, 1975.
Angel, E., andJain, A. K.,Filtering of Multidimensional Diffusion Processes, Sixth Asilomar Conference, Pacific Grove, Asilomar, California, 1972.
Jain, A. K.,A Semicausal Model for Recursive Filtering of Two Dimensional Images, IEEE Transactions on Computers, Vol. C-26, pp. 343–350, 1977.
Jain, A. K.,A Fast Karhunen-Loeve Transform for Recursive Filtering of Images Corrupted by White and Colored Noise, IEEE Transactions on Computers (to appear).
Jain, A. K.,Computer Program for Fast Karhunen-Loeve Transform Algorithm, SUNY Buffalo, Department of Electrical Engineering, Final Report on NASA Contract No. NAS8-31434, 1976.
Habibi, A.,Study of On-Board Compression of Earth Resources Data, TRW Systems Group, Redondo Beach, California, Final Report on NASA Contract No. NAS2-8394, 1975.
Davisson, L. D.,Rate Distortion Theory and Application, Proceedings of the IEEE, Vol. 60, pp. 800–808, 1972.
Garabedian, P. R.,Partial Differential Equations, John Wiley and Sons, New York, New York, 1964.
Woods, J. W.,Two Dimensional Discrete Markovian Fields, IEEE Transactions on Information Theory, Vol. IT-18, pp. 232–240, 1972.
Angel, E., andJain, A. K.,Initial-Value Transformations for Elliptic Boundary-Value Problems, Journal of Mathematical Analysis and Applications, Vol. 35, pp. 496–502, 1971.
Angel, E., Distefano, N., andJain, A. K.,Invariant Imbedding and Reduction of Boundary Value Problems of Thin Plate Theory to Cauchy Formulations, International Journal of Engineering Science, Vol. 9, pp. 933–945, 1971.
Angel, E., andJain, A. K.,Invariant Imbedding and Partial Differential Equations, Invariant Imbedding, Edited by R. Bellman and E. Denman, Springer-Verlag, New York, New York, 1971.
Angel, E., Jain, A. K., andKalaba, R.,Initial-Value Problems in Potential Theory, Journal of Optimization Theory and Applications, Vol. 11, pp. 274–283, 1973.
Distefano, N.,Nonlinear Processes in Engineering, Academic Press, New York, New York, 1974.
Angel, E., andBellman, R.,Dynamic Programming and Partial Differential Equations, Academic Press, New York, New York, 1972.
Jain, A. K.,A Fast Karhunen-Loeve Transform for a Class of Stochastic Processes, IEEE Transactions on Communications, Vol. COM-24, pp. 1023–1029, 1976.
Ahmed, N., Natarajan, T., andRao, K. R.,Discrete Cosine Transform, IEEE Transactions on Computers, Vol. C-23, pp. 90–93, 1974.
Jain, A. K.,Multidimensional Methods in Computer Image Processing (book, to appear).
Jain, A. K., Wang, S. H., andLiao, Y. Z.,Fast Karhunen Transform Data Compression Studies, National Telecommunications Conference, Dallas, Texas, 1976.
Bellman, R.,Introduction to Matrix Analysis, McGraw-Hill Book Company, New York, New York, 1970.
Jain, A. K.,A Fast Karhunen-Loeve Transform for Finite Discrete Images, Proceedings of the National Electronics Conference, Chicago, Illinois, 1974.
Jain, A. K., andJain, J. R.,Partial Differential Equations and Finite-Difference Methods in Image Processing, SUNY Buffalo, Department of Electrical Engineering, Technical Report No. AJ-76-003, 1976.
Author information
Authors and Affiliations
Additional information
Communicated by G. Leitmann
Rights and permissions
About this article
Cite this article
Jain, A.K. Partial differential equations and finite-difference methods in image processing, part 1: Image representation. J Optim Theory Appl 23, 65–91 (1977). https://doi.org/10.1007/BF00932298
Issue Date:
DOI: https://doi.org/10.1007/BF00932298