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Generating Connected Textured Fractal Patterns Using Markov Random Fields
August 1991 (vol. 13 no. 8)
pp. 819-825

An algorithm that yields textured and connected binary fractals is presented. The texture is imposed by modelling the fractal as a Markov random field (MRF) at every resolution level. The model size and the parameters specify the texture. The generation starts at a coarser level and continues at finer levels. Connectivity, which is a global property, is maintained by restricting the flow of the sample generating Markov chain within a limited subset of all possible outcomes of the Markov random field. The texture is controlled by the parameters of the MRF model being used. Sample patterns are shown.

[1] J. E. Martin and K. D. Keefer, "Regular random fractals and then- parameter model,"J. Phys. A, vol. 18, pp. L625-L631, 1985.
[2] D. C. Hong, S. Havlin, and H. E. Stanley, "Family of growth fractals with continuously tunable chemical dimension,"J. Phys. A, vol. 18, pp. L1103-L1107, 1985.
[3] P. Meakin, "Diffusion-controlled cluster formation in 2-6 dimensional space,"Phys. Rev. A, vol. 27, no. 3, pp. 1495-1507, Mar. 1983.
[4] P. Meakin, "Diffusion-limited aggregation in three dimensions: results from a new cluster-cluster aggregation model,"J. Colloid Interface Sci., vol. 102, no. 2, pp. 491-504, Dec. 1984.
[5] P. Meakin, "The structure of two-dimensional Witten-Sander aggregates,"J. Phys. A, vol. 18, pp. L661-L666, 1985.
[6] P. Meakin and Z. B. Djordjevic´, "Cluster-cluster aggregation in two-monomer systems,"J. Phys. A, vol. 19, pp. 2137-2153, 1986.
[7] T. A. Witten and M. E. Cates, "Tenuous structures from disorderly growth processes,"Sci., vol. 232, pp. 1607-1612, June 27, 1986.
[8] T. A. Witten and L. M. Sander, "Diffusion-limited aggregation,"Phys. Rev. B, vol. 27, no. 9, pp. 5686-5697, May 1983.
[9] T. A. Witten, Jr. and L. M. Sander, "Diffusion-limited aggregation, a kinetic critical phenomenon,"Phys. Rev. Lett., vol. 47, no. 19, pp. 1400-1403, Nov. 1981.
[10] M. Kolb, "Reversible diffusion-limited cluster aggregation,"J. Phys. A, vol. 19, pp. L263-L268, 1986.
[11] R. Jullien, "A new model of cluster aggregation,"J. Phys. A, vol. 19, pp. 2129-2136, 1986.
[12] S. R. Forrest and T. A. Witten, Jr., "Long-range correlations in smoke-particle aggregates,"J. Phys. A, vol. 12, no. 5, pp. L109-L117, 1979.
[13] M. D. Normand and M. Peleg, "Determination of the fractal dimension of a particle silhouette using image-processing techniques,"Powder Technol., vol. 45, pp. 271-275, 1986.
[14] R. M. Malzbender, R. C. Mockler, and W. J. O'Sullivan, "Topological and geometrical properties of DLA clusters,"J. Phys. A, vol. 18, pp. L1143-L1147, 1985.
[15] A. Margolina, "The fractal dimension of cluster perimeters generated by a kinetic walk,"J. Phys. A, vol. 18, pp. L651-L656, 1985.
[16] R. Orbach, "Dynamics of fractal networks,"Sci., vol. 231, pp. 814-819, 21 Feb. 1986.
[17] P. Pfeifer and D. Avnir, "Chemistry in noninteger dimensions between two and three, 1. Fractal theory of heterogeneous surfaces,"J. Chem. Phys., vol. 79, no. 7, pp. 3558-3565, Oct. 1983.
[18] M. Lewis and D. C. Rees, "Fractal surfaces of proteins,"Sci., vol. 230, pp. 1163-1165, Dec. 1985.
[19] B. H. Kaye, "Fractal dimension and signature waveform characterization of fine particle shape,"Amer. Lab., pp. 55-63, Apr. 1985.
[20] B. H. Kaye, "Specification of the ruggedness and/or texture of a fine particle profile by its fractal dimension,"Powder Technol., vol. 21, pp. 1-16, 1978.
[21] N. N. Clark, "Three techniques for implementing digital fractal analysis of particle shape,"Powder Technol., vol. 46, pp. 45-52, 1986.
[22] A. P. Pentland, "Fractal based description of natural scenes,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-6, no. 6, pp. 661-674, Nov. 1984.
[23] B. B. Mandelbrott,Fractals: Form, Chance and Dimension. San Fransisco, CA: Freeman, 1977.
[24] B. B. Mandelbrott,The Fractal Geometry of Nature. San Fransisco, CA: Freeman, 1982.
[25] J. Besag, "Spatial interactions and the statistical analysis of lattice systems,"J. Roy. Stat. Soc. B, vol. 36, pp. 192-236, 1974.
[26] J. Moussouris, "Gibbs and Markov random systems with constraints,"J. Stat. Phys., vol. 10, no. 1, pp. 11-33, 1974.
[27] H. Derin, H. Elliott, R. Cristi, and D. Geman, "Bayes smoothing algorithms for segmentation of binary images modeled by Markov random fields,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-6, no. 6, pp. 707-720, Nov. 1984.
[28] G. R. Cross and A. K. Jain, "Markov random field texture models,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-5, no. 1, pp. 25-39, Jan. 1983.
[29] M. Hassner and J. Sklansky, "The use of Markov random fields as models of texture,"Comput. Graphics Image Processing, vol. 12, pp. 357-370, 1980.
[30] S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-6, no. 6, pp. 721-741, Nov. 1984.
[31] H. Derin and H. Elliott, "Modeling and segmentation of noisy and textured images using Gibbs random fields,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-9, pp. 39-55, Jan. 1987.
[32] B. D. Ripley, "Statistics, images, and pattern recognition,"Canadian J. Stat., vol. 14, no. 2, pp. 83-111, 1986.
[33] F. S. Cohen and D. B. Cooper, "Simple parallel hierarchical and relaxation algorithms for segmenting noncausal Markovian fields,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-9, pp. 195-219, March 1987.
[34] J. M. Hammersley and D. C. Handscomb,Monte Carlo Methods. London: Methuen, 1964.
[35] P. A. Flinn, "Monte Carlo calculation of phase separation in a two-dimensional Ising system,"J. Stat. Phys., vol. 10, no. 1, pp. 89-97, 1974.
[36] V. Isham, "An introduction to spatial point processes and Markov random fields,"Int. Stat. Rev., vol. 49, pp. 21-43, 1981.
[37] R. Kinderman and J. L. Snell,Markov Random Fields and Their Applications. Providence, RI: American Mathematical Society, 1980.
[38] Y. A. Rosanov,Markov Random Fields. New York: Springer-Verlag, 1982.

Index Terms:
picture processing; pattern recognition; statistical analysis; connected textured fractal patterns; Markov random fields; binary fractals; sample generating Markov chain; fractals; Markov processes; pattern recognition; picture processing; statistical analysis
L. Onural, "Generating Connected Textured Fractal Patterns Using Markov Random Fields," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 8, pp. 819-825, Aug. 1991, doi:10.1109/34.85673
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