Methods for numerical integration of high-dimensional posterior densities with application to statistical image models

Numerical computation with Bayesian posterior densities has recently received much attention both in the applied statistics and image processing communities. This paper surveys previous literature and presents efficient methods for computing marginal density values for image models that have been widely considered in computer vision and image processing. The particular models chosen are a Markov random field (MRF) formulation, implicit polynomial surface models, and parametric polynomial surface models. The computations can be used to make a variety of statistically based decisions, such as assessing region homogeneity for segmentation or performing model selection. Detailed descriptions of the methods are provided, along with demonstrative experiments on real imagery.

[1]  Mehmet Celenk,et al.  A color clustering technique for image segmentation , 1990, Comput. Vis. Graph. Image Process..

[2]  J. Alison Noble,et al.  Toward template-based tolerancing from a Bayesian viewpoint , 1993, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[3]  Zane W. Bell.  A Bayesian/Monte Carlo Segmentation Method for Images Dominated by Gaussian Noise , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[4]  David B. Cooper,et al.  Bayesian Clustering for Unsupervised Estimation of Surface and Texture Models , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[5]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[6]  G. Casella,et al.  Explaining the Gibbs Sampler , 1992 .

[7]  Jean Ponce,et al.  On using geometric distance fits to estimate 3D object shape, pose, and deformation from range, CT, and video images , 1993, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[8]  J. Besag,et al.  Spatial Statistics and Bayesian Computation , 1993 .

[9]  P.K Sahoo,et al.  A survey of thresholding techniques , 1988, Comput. Vis. Graph. Image Process..

[10]  R. Kass,et al.  Approximate Bayes Factors and Orthogonal Parameters, with Application to Testing Equality of Two Binomial Proportions , 1992 .

[11]  M. Aitkin.  Posterior Bayes Factors , 1991 .

[12]  J. Berger,et al.  Testing Precise Hypotheses , 1987 .

[13]  Gabriel Taubin,et al.  Estimation of Planar Curves, Surfaces, and Nonplanar Space Curves Defined by Implicit Equations with Applications to Edge and Range Image Segmentation , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[14]  Bradley P. Carlin,et al.  Predicting Working Memory Failure: A Subjective Bayesian Approach to Model Selection , 1992 .

[15]  B. Ripley,et al.  Pattern Recognition , 1968, Nature.

[16]  Ramesh C. Jain,et al.  Segmentation through Variable-Order Surface Fitting , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[17]  Jake K. Aggarwal,et al.  Segmentation of 3D range images using pyramidal data structures , 1990, [1990] Proceedings Third International Conference on Computer Vision.

[18]  David B. Cooper,et al.  Simple Parallel Hierarchical and Relaxation Algorithms for Segmenting Noncausal Markovian Random Fields , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[19]  Stamatis Cambanis,et al.  Trapezoidal Monte Carlo integration , 1990 .

[20]  Charles R. Dyer,et al.  Model-based recognition in robot vision , 1986, CSUR.

[21]  Steven M. LaValle,et al.  A Bayesian Segmentation Methodology for Parametric Image Models , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[22]  Glenn Healey,et al.  Unsupervised segmentation of textured color images using Markov random field models , 1993, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[23]  Michael L. Baird.  Structural Pattern Recognition , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[24]  Ian W. Evett,et al.  Statistical Inference in Crime Investigations Using Deoxyribonucleic Acid Profiling , 1992 .

[25]  Anil K. Jain,et al.  Segmentation and Classification of Range Images , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[26]  Anil K. Jain,et al.  Algorithms for Clustering Data , 1988 .

[27]  D. Lindley.  A STATISTICAL PARADOX , 1957 .

[28]  I. P. Mysovskih.  Approximate Calculation of Integrals , 1969 .

[29]  Charles A. Bouman,et al.  Multiple Resolution Segmentation of Textured Images , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[30]  Ian W. Evett.  A Quantitative Theory for Interpreting Transfer Evidence in Criminal Cases , 1984 .

[31]  Sridhar Lakshmanan,et al.  Simultaneous Parameter Estimation and Segmentation of Gibbs Random Fields Using Simulated Annealing , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[32]  Steven M. LaValle,et al.  A Framework for Constructing Probability Distributions on the Space of Image Segmentations , 1995, Comput. Vis. Image Underst..

[33]  Ruzena Bajcsy,et al.  Segmentation as the search for the best description of the image in terms of primitives , 1990, [1990] Proceedings Third International Conference on Computer Vision.

[34]  Rama Chellappa,et al.  Estimation and choice of neighbors in spatial-interaction models of images , 1983, IEEE Trans. Inf. Theory.

[35]  R. Haralick,et al.  A facet model for image data , 1981 .

[36]  David J. Kriegman,et al.  On Recognizing and Positioning Curved 3-D Objects from Image Contours , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[37]  Jerrold H. Zar.  Approximations for the Percentage Points of the Chi‐Squared Distribution , 1978 .

[38]  D. Spiegelhalter,et al.  Bayes Factors and Choice Criteria for Linear Models , 1980 .

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

[40]  David B. Cooper,et al.  On Optimally Combining Pieces of Information, with Application to Estimating 3-D Complex-Object Position from Range Data , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[41]  Adrian F. M. Smith,et al.  Bayesian computation via the gibbs sampler and related markov chain monte carlo methods (with discus , 1993 .

[42]  RESEARCH AT THE UNIVERSITY OF ILLINOIS. , 1942, Science.

[43]  Baba C. Vemuri,et al.  On Three-Dimensional Surface Reconstruction Methods , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[44]  Peter E. Hart,et al.  Pattern classification and scene analysis , 1974, A Wiley-Interscience publication.

[45]  D. Kriegman,et al.  On recognizing and positioning curved 3D objects from image contours , 1989, [1989] Proceedings. Workshop on Interpretation of 3D Scenes.

[46]  Ian W. Evett,et al.  Evaluation of the likelihood ratio for fibre transfer evidence in criminal cases , 1987 .

[47]  Richard O. Duda,et al.  Pattern classification and scene analysis , 1974, A Wiley-Interscience publication.

[48]  Kim L. Boyer,et al.  An information theoretic robust sequential procedure for surface model order selection in noisy range data , 1992, Proceedings 1992 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[49]  Colin Aitken,et al.  Estimation of the bayes' factor in a forensic science problem , 1989 .

[50]  S. Yakowitz,et al.  Weighted Monte Carlo Integration , 1978 .

[51]  David M. Bloom,et al.  Linear Algebra and Geometry , 1979 .

[52]  M. Hebert,et al.  The Representation, Recognition, and Locating of 3-D Objects , 1986 .

[53]  Kannan,et al.  ON IMAGE SEGMENTATION TECHNIQUES , 2022 .