This Article 
 Bibliographic References 
 Add to: 
Boundary Detection by Constrained Optimization
July 1990 (vol. 12 no. 7)
pp. 609-628

A statistical framework is used for finding boundaries and for partitioning scenes into homogeneous regions. The model is a joint probability distribution for the array of pixel gray levels and an array of labels. In boundary finding, the labels are binary, zero, or one, representing the absence or presence of boundary elements. In partitioning, the label values are generic: two labels are the same when the corresponding scene locations are considered to belong to the same region. The distribution incorporates a measure of disparity between certain spatial features of block pairs of pixel gray levels, using the Kolmogorov-Smirnov nonparametric measures of difference between the distributions of these features. The number of model parameters is minimized by forbidding label configurations, which are assigned probability zero. The maximum a posteriori estimator of boundary placements and partitionings is examined. The forbidden states introduce constraints into the calculation of these configurations. Stochastic relaxation methods are extended to accommodate constrained optimization.

[1] E. Arts and P. van Laarhoven, "Simulated annealing: A pedestrian review of the theory and some applications," inNATO Advanced Study Institute on Pattern Recognition: Theory and Applications. Spa, Belgium. 1986.
[2] J. Besag, "On the statistical analysis of dirty pictures" (with discussion),J. Roy. Statist. Soc., series B, vol. 48, pp. 259-302, 1986.
[3] A. Blake, "The least disturbance principle and weak constraints,"Pattern Recog. Lett., vol. 1, pp. 393-399, 1983.
[4] E. Bonomi and J.-L. Lutton, "Then-city travelling salesman problem: Statistical mechanics and Metropolis algorithm,"SIAM Rev., vol. 26, pp. 551-568, 1984.
[5] A. Brandt, "Multi-level approaches to large scale problems," inProc. Int. Congr. Mathematicians 1986, A. M. Gleason, Ed., Amer. Math. Soc., Providence, RI, 1987.
[6] P. Brodatz,Texture: A Photographic Album for Artists and Designers. New York: Dover, 1966.
[7] T. M. Cannon, H. J. Trussell, and B. R. Hunt. "Comparison of image restoration methods,"Appl. Opt., vol. 17, pp. 3385-3390, 1978.
[8] V. Cerny, "A thermodynamical approach to the travelling salesman problem: An efficient simulation algorithm," Inst. Phys. Biophys., Comenius Univ., Bratislava, Czechoslovakia, 1982.
[9] B. Chalmond, "Image restoration using an estimated Markov model," Dep. Math., Univ. Paris, Orsay, France, 1987.
[10] T.-S. Chiang and Y. Chow, "On eigenvalues and optimal annealing rate," Inst. Math., Academia Sinica, Taipei, Taiwan, preprint 1987.
[11] 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.
[12] H. Derin and W. S. Cole, "Segmentation of textured images using Gibbs random fields,"Comput. Vision, Graphics, Image Processing, vol. 35, pp. 72-98, 1986.
[13] 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.
[14] H. Derin and C. Won, "A parallel image segmentation algorithm using relaxation with varying neighborhoods and its mapping to array processors," Dep. Elec. Comput. Eng., Univ. Massachusetts, Tech. Rep., 1986.
[15] P. A. Devijver and M. M. Dekesel, "Learning the parameters of a hidden Markov random field image model: A simple example," inPattern Recognition Theory and Applications, P. A. Devijver and J. Kittler. Eds. Heidelberg: Springer-Verlag, 1987, pp. 141-163.
[16] A. Gagalowicz and Song De Ma, "Sequential synthesis of natural textures,"Comput. Vision, Graphics, Image Processing, vol. 30, pp. 289-315. 1985.
[17] D. Geman, "A stochastic model for boundary detection,"Image Vision Comput., May 1987.
[18] D. Geman and S. Geman, "Relaxation and annealing with constraints," Division Appl. Math., Brown Univ., Complex Systems Tech. Rep. 35, 1987.
[19] D. Geman, S. Geman, and C. Graffigne, "Locating texture and object boundaries," inPattern Recognition Theory and Applications, P. A. Devijver and J. Kittler, Eds. Heidelberg: Springer-Verlag, 1987.
[20] S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 721-741, 1984.
[21] S. Geman and C. Graffigne, "Markov random field image models and their applications to computer vision," inProc. Int. Congr. Mathematicians, 1986, A. M. Gleason, Ed., Amer. Math. Soc., Providence, RI, 1987.
[22] S. Geman and D. E. McClure, "Statistical methods for tomographic image reconstruction," inBull. ISI (Proc. 46th Session Int. Statistical Institute), vol. 52, 1987.
[23] B. Gidas, "Nonstationary Markov chains and convergence of the annealing algorithm,"J. Statist. Phys., vol. 39, pp. 73-131, 1985.
[24] B. Gidas, "A renormalization group approach to image processing problems,"IEEE Trans. Pattern Anal. Machine Intell., vol. 11, pp. 164- 180, Feb. 1989.
[25] C. Graffigne, "Experiments in texture analysis and segmentation," Ph.D. dissertation, Division Appli. Math., Brown Univ., 1987.
[26] P. J. Green, "Discussion on "On the statistical analysis of dirty pictures," J. Besag,J. Roy, Statist. Soc., vol. B-48, pp. 259-302. 1986.
[27] D. M. Greig, B. T. Porteous, and A. H. Seheult, Discussion on "On the statistical analysis of dirty pictures," by J. Besag.J. Roy. Statist. Soc., vol. B-48, pp. 259-302, 1986.
[28] U. Grenander, "Tutorial in pattern theory," Division Appl. Math., Brown Univ., Lecture Notes Volume, 1984.
[29] W. E. L. Grimson and T. Pavlidis, "Discontinuity detection for visual surface reconstruction,"Comput. Vision, Graphics, Image Processing, vol. 30, pp. 316-330, 1985.
[30] B. Hajek, "A tutorial survey of theory and applications of simulated annealing," inProc. 24th IEEE Conf. Decision and Control, 1985, pp. 755-760.
[31] R. M. Haralick, K. Shanmugam, and I. Denstein, "Textural features for image classification,"IEEE Trans. Syst., Man, Cybern., vol. 6, pp. 610-621, 1973.
[32] R. Holley and D. Stroock, "Simulated annealing via Sobolev inequalities," preprint, 1987.
[33] B. K. P. Horn and M. J. Brooks, "The variational approach to shape from shading,"Comp. vision, Graphics, and Image Processing, vol. 33, no. 2, pp. 174-208, Feb. 1986.
[34] C.R. Hwang and S.-J. Sheu, "Large time behaviors of perturbed diffusion Markov processes with applications," I, II, and III, Inst. Math., Academic Sinica, Taipei, Taiwan. preprint 1987.
[35] A. Kashko and B. F. Buxton, "Markov random fantasies," preprint, 1986.
[36] A. Kashko, H. Buxton, and B. F. Buxton, "Parallel stochastic optimization in computer vision," preprint, 1987.
[37] R. L. Kashyap and K. Eom, "Texture boundary detection based on the long correlation model,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-2, pp. 58-67, 1989.
[38] R. L. Kashyap and A. Khotanzad, "A Model-Based Method for Rotation Invariant Texture Classification,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 472-481, July 1986.
[39] S. Kirkpatrick, C. D. Gellatt, Jr., and M. P. Vecchi, "Optimization by simulated annealing,"Science, vol. 220, pp. 671-680, 1983.
[40] K. Laws, "Textured image segmentation," Ph.D. dissertation, Univ. Southern California, Los Angeles, USCIPI Rep. 940, 1980.
[41] R. Linsker, "An iterative-improvement penalty-function-driven wire routing system,"IBM J. Res. Develop., vol. 28, pp. 613-624, 1984.
[42] C. von der Malsburg and E. Bienenstock, "Statistical coding and short-term synaptic plasticity: A scheme for knowledge representation in the brain," inDisordered Systems and Biological Organization(NATO ASI Series), E. Bienenstock, F. Fajelman Soulié, and G. Weisbuch, Eds. Berlin: Springer-Verlag, 1986.
[43] J. L. Marroquin, "Surface reconstruction preserving discontinuities," Massachusetts Inst. Technol., Cambridge, A.I. Memo 792, 1984.
[44] J. L. Marroquin, S. Mitter, and T. Poggio, "Probabilistic solution of ill-posed problems in computational vision,"J. Amer. Statist. Assoc., vol. 82, pp. 76-89, 1987.
[45] J. W. Modestino, R. W. Fries, and A. L. Vickers, "Texture discrimination based upon an assumed stochastic texture model,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-3, pp. 557-580, 1981.
[46] J. Moussouris, "Gibbs and Markov random systems with constraints,"J. Statist. Phys., vol. 10, pp. 11-33, 1974.
[47] D. W. Murray and B. F. Buxton, "Scene segmentation from visual motion using global optimization,"IEEE Trans. Patt. Analy. Machine Intell., vol. PAMI-9, pp. 161-180, 1987.
[48] T. Pavlidis, "A critical survey of image analysis methods," Expanded version of paper delivered to 8th Int. Conf. Pattern Recognition, Paris, France, Oct. 1986.
[49] T. Poggio, V. Torre, and C. Koch, "Computational vision and regularization theory,"Nature, vol. 317, pp. 314-319, 1985.
[50] B. D. Ripley, "Statistics, images, and pattern recognition,"Canadian J. Statist., vol. 14, pp. 83-111, 1986.
[51] A. Rosenfeld and L. S. Davis. "Image segmentation and image models,"Proc. IEEE, vol. 67, pp. 764-772, 1979.
[52] B. W. Silverman, Discussion on "On the statistical analysis of dirty pictures," by J. Besag,J. Roy. Statist. Soc., vol. B-48, pp. 259-302, 1986.
[53] T. Simchony and R. Chellappa, "Stochastic and deterministic algorithms for texture segmentation," Dep. EE--Syst., Univ. Southern California, 1988.
[54] R. H. Swendsen and J.-S. Wang, "Nonuniversal critical dynamics in Monte Carto simulations,"Phys. Rev. Lett., vol. 58, pp. 86-88, 1987.
[55] H. Szu, "Non-convex optimization," inProc. SPIE Conf. Real Time Signal Processing IX, vol. 698, 1986.
[56] D. Terzopoulos, "Regularization of inverse visual problems involving discontinuities,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, no. 4, pp. 413-424, July 1986.
[57] W. B. Thompson, "Textural boundary analysis,"IEEE Trans. Comput., vol. C-26, pp. 272-276, 1977.
[58] E. E. Triendl, "Texturerkennung and texturreproduktion,"Kybernetik, vol. 13. pp. 1-5, 1973.
[59] E. Triendl and T. Henderson, "A model for texture edges," inProc. 5th Int. Conf. Pattern Recognition, Miami Beach, FL, Dec. 1-4, 1980, pp. 1100-1102.
[60] H. L. Voorhees, Jr., "Finding texture boundaries in images," Master's thesis, Dep. Elec. Eng. Comput. Sci., Massachusetts Inst. Technol., 1987.
[61] W. A. Yasnoff, J. K. Mui, and J. W. Bacus, "Error measures for scene segmentation,"Pattern Recog., vol. 9, pp. 217-231, 1977.
[62] A. I. Zobrist and W. B. Thompson, "Building a distance function for Gestalt grouping,"IEEE Trans. Comput., vol. C-24, pp. 718-728, 1975.

Index Terms:
scenes partitioning; boundary detection; stochastic relaxation; constrained optimization; probability distribution; pixel gray levels; scene locations; Kolmogorov-Smirnov; forbidding label configurations; boundary placements; optimisation; pattern recognition; picture processing; statistical analysis
D. Geman, S. Geman, C. Graffigne, P. Dong, "Boundary Detection by Constrained Optimization," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 609-628, July 1990, doi:10.1109/34.56204
Usage of this product signifies your acceptance of the Terms of Use.