On the Statistical Analysis of Dirty Pictures

may 7th, 1986, Professor A. F. M. Smith in the Chair] SUMMARY A continuous two-dimensional region is partitioned into a fine rectangular array of sites or "pixels", each pixel having a particular "colour" belonging to a prescribed finite set. The true colouring of the region is unknown but, associated with each pixel, there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model. The aim is to reconstruct the true scene, with the additional knowledge that pixels close together tend to have the same or similar colours. In this paper, it is assumed that the local characteristics of the true scene can be represented by a nondegenerate Markov random field. Such information can be combined with the records by Bayes' theorem and the true scene can be estimated according to standard criteria. However, the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field. Thus, a simple, iterative method of reconstruction is proposed, which does not depend on these large-scale characteristics. The method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field. Some complications, including parameter estimation, are discussed. Potential applications are mentioned briefly.

[1]  D. R. Fulkerson,et al.  Maximal Flow Through a Network , 1956 .

[2]  Laveen N. Kanal,et al.  Classification of binary random patterns , 1965, IEEE Trans. Inf. Theory.

[3]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[4]  L. Baum,et al.  A Maximization Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains , 1970 .

[5]  C. Preston Gibbs States on Countable Sets , 1974 .

[6]  Tzay Y. Young,et al.  Classification, Estimation and Pattern Recognition , 1974 .

[7]  J. Besag Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[8]  J. Besag Statistical Analysis of Non-Lattice Data , 1975 .

[9]  T. Bush,et al.  Fading Characteristics of Panchromatic Radar Backscatter from Selected Agricultural Targets , 1973, IEEE Transactions on Geoscience Electronics.

[10]  H. D. Ratliff,et al.  Minimum cuts and related problems , 1975, Networks.

[11]  R. Galbraith,et al.  ON A TWO-DIMENSIONAL BINARY PROCESS , 1976 .

[12]  Azriel Rosenfeld,et al.  Scene Labeling by Relaxation Operations , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[13]  D. J. Strauss Clustering on coloured lattices , 1977, Journal of Applied Probability.

[14]  J. Besag On Spatial-Temporal Models and Markov Fields , 1977 .

[15]  A. M. W. Verhagen,et al.  A three parameter isotropic distribution of atoms and the hard‐core square lattice gas , 1977 .

[16]  D. K. Pickard A curious binary lattice process , 1977, Journal of Applied Probability.

[17]  J. Besag Efficiency of pseudolikelihood estimation for simple Gaussian fields , 1977 .

[18]  S. Silvey,et al.  An algorithm for optimal designs on a design space , 1978 .

[19]  Casimir A. Kulikowski,et al.  A Model-Based Method for Computer-Aided Medical Decision-Making , 1978, Artif. Intell..

[20]  S. Gull,et al.  Image reconstruction from incomplete and noisy data , 1978, Nature.

[21]  H. Atkins,et al.  Emission tomography. , 1979, New York state journal of medicine.

[22]  Paul Switzer,et al.  Extensions of linear discriminant analysis for statistical classification of remotely sensed satellite imagery , 1980 .

[23]  J. Laurie Snell,et al.  Markov Random Fields and Their Applications , 1980 .

[24]  Laveen N. Kanal,et al.  Markov mesh models , 1980 .

[25]  D. K. Pickard,et al.  Unilateral Markov fields , 1980, Advances in Applied Probability.

[26]  David J. Hand,et al.  Discrimination and Classification , 1982 .

[27]  Steven W. Zucker,et al.  Continuous Relaxation and Local Maxima Selection: Conditions for Equivalence , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[28]  Ulf Grenander Metric Pattern Theory , 1981 .

[29]  Azriel Rosenfeld,et al.  Digital Picture Processing, Volume 1 , 1982 .

[30]  L. Shepp,et al.  Maximum Likelihood Reconstruction for Emission Tomography , 1983, IEEE Transactions on Medical Imaging.

[31]  F. R. Hansen,et al.  Image segmentation using simple markov field models , 1982, Computer Graphics and Image Processing.

[32]  Josef Kittler,et al.  Pattern recognition : a statistical approach , 1982 .

[33]  B. Torsney A Moment Inequality and Monotonicity of an Algorithm , 1983 .

[34]  N. Wermuth,et al.  Graphical and recursive models for contingency tables , 1983 .

[35]  Steven W. Zucker,et al.  On the Foundations of Relaxation Labeling Processes , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[36]  Anil K. Jain,et al.  Markov Random Field Texture Models , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[37]  John Skilling,et al.  Image restoration by a powerful maximum entropy method , 1982, Comput. Vis. Graph. Image Process..

[38]  King-Sun Fu,et al.  Recursive contextual classification using a spatial stochastic model , 1983, Pattern Recognit..

[39]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[40]  A. Owen A neighbourhood-based classifier for LANDSAT data , 1984 .

[41]  Josef Kittler,et al.  Contextual classification of multispectral pixel data , 1984, Image Vis. Comput..

[42]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[43]  David A. Landgrebe,et al.  Adaptive Relaxation Labeling , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[44]  D. Titterington,et al.  The maximum entropy method for data analysis , 1984, Nature.

[45]  D. M. Titterington,et al.  Comments on "Application of the Conditional Population-Mixture Model to Image Segmentation" , 1984, IEEE Trans. Pattern Anal. Mach. Intell..

[46]  John Skilling,et al.  Data analysis: The maximum entropy method , 1984, Nature.

[47]  Donald Geman,et al.  Bayes Smoothing Algorithms for Segmentation of Binary Images Modeled by Markov Random Fields , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[48]  T. Speed,et al.  Recursive causal models , 1984, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[49]  John C. Russ,et al.  Image processing in a general purpose microcomputer , 1984 .

[50]  On the covariance of two noncentral F random variables and the variance of the estimated liner discriminant function , 1984 .

[51]  W. Nagel,et al.  The comparison by simulation of solutions of Wicksell's corpuscle problem , 1984 .

[52]  J. Mazziotta,et al.  Positron emission tomography and autoradiography , 1985 .

[53]  D. Titterington Common structure of smoothing techniques in statistics , 1985 .

[54]  B. Silverman,et al.  Some Aspects of the Spline Smoothing Approach to Non‐Parametric Regression Curve Fitting , 1985 .

[55]  David J. Spiegelhalter,et al.  Probabilistic Reasoning in Predictive Expert Systems , 1985, Conference on Uncertainty in Artificial Intelligence.

[56]  John Haslett,et al.  Maximum likelihood discriminant analysis on the plane using a Markovian model of spatial context , 1985, Pattern Recognit..

[57]  L. Shepp,et al.  A Statistical Model for Positron Emission Tomography , 1985 .

[58]  Judea Pearl,et al.  A Constraint-Propagation Approach to Probabilistic Reasoning , 1985, UAI.

[59]  A. Seheult,et al.  Analysis of Field Experiments by Least Squares Smoothing , 1985 .

[60]  Josef Kittler,et al.  Contextual Pattern Recognition Applied to Cloud Detection and Identification , 1985, IEEE Transactions on Geoscience and Remote Sensing.

[61]  B. Gidas Global optimization via the Langevin equation , 1985, 1985 24th IEEE Conference on Decision and Control.

[62]  S. Geman,et al.  Diffusions for global optimizations , 1986 .

[63]  D. M. Titterington,et al.  Image labelling and the statistical analysis of incomplete data , 1986 .

[64]  D. Spiegelhalter A statistical view of uncertainty in expert systems , 1986 .


[66]  Donald Geman,et al.  Bayesian Image Analysis , 1986 .

[67]  J. F. Boyce,et al.  Relaxation labelling and the entropy of neighbourhood information , 1987, Pattern Recognit. Lett..

[68]  Rolf Sundberg,et al.  Confidence and Conflict in Multivariate Calibration , 1987 .