Markov Random Fields and Stochastic Image Models

In this paper, I will buildup the basic framework of Markov Chains over finite state spaces using analytical techniques. In particular, we will see how the study of Markov Chains over finite state spaces reduces to the study of powers of matrices. Using this framework, I will prove that under mild restrictions, Markov Chains converge to a unique stationary distribution. Finally, I will discuss some interesting connections between Markov Chains and Linear Algebra.

[1]  Max Born,et al.  Statistical theory of adsorption with interaction between the adsorbed atoms , 1936, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  R. Peierls.  On Ising's model of ferromagnetism , 1936, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  L. Onsager.  Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[4]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[5]  I. G. BONNER CLAPPISON.  Editor , 1960, The Electric Power Engineering Handbook - Five Volume Set.

[6]  L. Baum,et al.  Statistical Inference for Probabilistic Functions of Finite State Markov Chains , 1966 .

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

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

[9]  J. Besag.  Nearest‐Neighbour Systems and the Auto‐Logistic Model for Binary Data , 1972 .

[10]  P. Peskun,et al.  Optimum Monte-Carlo sampling using Markov chains , 1973 .

[11]  B. R. Hunt,et al.  The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer , 1973, IEEE Transactions on Computers.

[12]  H. Akaike.  A new look at the statistical model identification , 1974 .

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

[14]  J. Besag,et al.  On the estimation and testing of spatial interaction in Gaussian lattice processes , 1975 .

[15]  Bobby R. Hunt,et al.  Bayesian Methods in Nonlinear Digital Image Restoration , 1977, IEEE Transactions on Computers.

[16]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[17]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

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

[19]  David B. Cooper,et al.  Maximum Likelihood Estimation of Markov-Process Blob Boundaries in Noisy Images , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[20]  D. K. Pickard.  Asymptotic inference for an Ising lattice III. Non-zero field and ferromagnetic states , 1979 .

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

[22]  H. Elliott,et al.  Stochastic boundary estimation and object recognition , 1980 .

[23]  A.K. Jain,et al.  Advances in mathematical models for image processing , 1981, Proceedings of the IEEE.

[24]  David B. Cooper,et al.  Implementation, Interpretation, and Analysis of a Suboptimal Boundary Finding Algorithm , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[25]  Rama Chellappa,et al.  Texture classification using features derived from random field models , 1982, Pattern Recognit. Lett..

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

[27]  Charles W. Therrien,et al.  An estimation-theoretic approach to terrain image segmentation , 1983, Comput. Vis. Graph. Image Process..

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

[29]  New York Dover,et al.  ON THE CONVERGENCE PROPERTIES OF THE EM ALGORITHM , 1983 .

[30]  J. Rissanen.  A UNIVERSAL PRIOR FOR INTEGERS AND ESTIMATION BY MINIMUM DESCRIPTION LENGTH , 1983 .

[31]  W. Rey.  Introduction to Robust and Quasi-Robust Statistical Methods , 1983 .

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

[33]  G. W. Wecksung,et al.  Bayesian approach to limited-angle reconstruction in computed tomography , 1983 .

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

[35]  R. Redner,et al.  Mixture densities, maximum likelihood, and the EM algorithm , 1984 .

[36]  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.

[37]  D. Rubin,et al.  Estimation and Hypothesis Testing in Finite Mixture Models , 1985 .

[38]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[39]  Rama Chellappa,et al.  Classification of textures using Gaussian Markov random fields , 1985, IEEE Trans. Acoust. Speech Signal Process..

[40]  Editors , 1986, Brain Research Bulletin.

[41]  Demetri Terzopoulos,et al.  Image Analysis Using Multigrid Relaxation Methods , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[42]  J. Besag.  On the Statistical Analysis of Dirty Pictures , 1986 .

[43]  T.F. Quatieri,et al.  Statistical model-based algorithms for image analysis , 1986, Proceedings of the IEEE.

[44]  H. Derin,et al.  Segmentation of textured images using Gibbs random fields , 1986 .

[45]  Andrew Blake,et al.  Visual Reconstruction , 1987, Deep Learning for EEG-Based Brain–Computer Interfaces.

[46]  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.

[47]  Chee Sun Won,et al.  A parallel image segmentation algorithm using relaxation with varying neighborhoods and its mapping to array processors , 1987, Computer Vision Graphics and Image Processing.

[48]  Stuart Geman,et al.  Statistical methods for tomographic image reconstruction , 1987 .

[49]  Haluk Derin,et al.  Modeling and Segmentation of Noisy and Textured Images Using Gibbs Random Fields , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[50]  John W. Woods,et al.  Image Estimation Using Doubly Stochastic Gaussian Random Field Models , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[51]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[52]  Tomaso Poggio,et al.  Probabilistic Solution of Ill-Posed Problems in Computational Vision , 1987 .

[53]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

[54]  Haluk Derin,et al.  Segmentation of noisy textured images using simulated annealing , 1987, ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[55]  D. K. Pickard.  Inference for Discrete Markov Fields: The Simplest Nontrivial Case , 1987 .

[56]  Thrasyvoulos N. Pappas,et al.  An Adaptive Clustering Algorithm For Image Segmentation , 1988, [1988 Proceedings] Second International Conference on Computer Vision.

[57]  Charles A. Bouman,et al.  Segmentation of textured images using a multiple resolution approach , 1988, ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing.

[58]  Rangasami L. Kashyap,et al.  Robust image modeling techniques with an image restoration application , 1988, IEEE Trans. Acoust. Speech Signal Process..

[59]  Demetri Terzopoulos,et al.  The Computation of Visible-Surface Representations , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[60]  Charles A. Bouman,et al.  A Multiple Resolution Approach To Regularization , 1988, Other Conferences.

[61]  Jin Luo,et al.  Computing motion using analog and binary resistive networks , 1988, Computer.

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

[63]  Patrick A. Kelly,et al.  Adaptive segmentation of speckled images using a hierarchical random field model , 1988, IEEE Trans. Acoust. Speech Signal Process..

[64]  Andrew Blake,et al.  Comparison of the Efficiency of Deterministic and Stochastic Algorithms for Visual Reconstruction , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[65]  Jun Zhang,et al.  A Markov random field model-based approach to image interpretation , 1989, Proceedings CVPR '89: IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[66]  H. Derin,et al.  Discrete-index Markov-type random processes , 1989, Proc. IEEE.

[67]  Anil K. Jain,et al.  Random field models in image analysis , 1989 .

[68]  Basilis Gidas,et al.  A Renormalization Group Approach to Image Processing Problems , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[69]  Edward J. Delp,et al.  A comparative cost function approach to edge detection , 1989, IEEE Trans. Syst. Man Cybern..

[70]  T. Hebert,et al.  A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors. , 1989, IEEE transactions on medical imaging.

[71]  Lawrence R. Rabiner,et al.  A tutorial on hidden Markov models and selected applications in speech recognition , 1989, Proc. IEEE.

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

[73]  C. W. Therrien,et al.  Decision, Estimation and Classification: An Introduction to Pattern Recognition and Related Topics , 1989 .

[74]  Anil K. Jain,et al.  Segmentation of Document Images , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[75]  Reginald L. Lagendijk,et al.  Identification and restoration of noisy blurred images using the expectation-maximization algorithm , 1990, IEEE Trans. Acoust. Speech Signal Process..

[76]  Kenneth Lange,et al.  Overview of Bayesian methods in image reconstruction , 1990, Optics & Photonics.

[77]  R. Cristi,et al.  Markov and recursive least squares methods for the estimation of data with discontinuities , 1990, IEEE Trans. Acoust. Speech Signal Process..

[78]  Aggelos K. Katsaggelos,et al.  Image identification and restoration based on the expectation-maximization algorithm , 1990 .

[79]  R. Mersereau,et al.  Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration , 1990 .

[80]  Robert M. Haralick,et al.  Multispectral image context classification using stochastic relaxation , 1990, IEEE Trans. Syst. Man Cybern..

[81]  A. Benveniste,et al.  Multiscale system theory , 1990, 29th IEEE Conference on Decision and Control.

[82]  Y. Ogata.  A Monte Carlo method for an objective Bayesian procedure , 1990 .

[83]  P. Green.  Bayesian reconstructions from emission tomography data using a modified EM algorithm. , 1990, IEEE transactions on medical imaging.

[84]  Donald Geman,et al.  Boundary Detection by Constrained Optimization , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[85]  S. Mitter,et al.  On sampling methods and annealing algorithms , 1990 .

[86]  Anil K. Jain,et al.  MRF model-based algorithms for image segmentation , 1990, [1990] Proceedings. 10th International Conference on Pattern Recognition.

[87]  Rama Chellappa,et al.  Stochastic and deterministic networks for texture segmentation , 1990, IEEE Trans. Acoust. Speech Signal Process..

[88]  Jun Zhang,et al.  A Model-Fitting Approach to Cluster Validation with Application to Stochastic Model-Based Image Segmentation , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[89]  A. Murat Tekalp,et al.  Maximum likelihood image and blur identification: a unifying , 1990 .

[90]  Rama Chellappa,et al.  Relaxation algorithms for MAP estimation of gray-level images with multiplicative noise , 1990, IEEE Trans. Inf. Theory.

[91]  K. Lange.  Convergence of EM image reconstruction algorithms with Gibbs smoothing. , 1990, IEEE transactions on medical imaging.

[92]  Levent Onural,et al.  Generating Connected Textured Fractal Patterns Using Markov Random Fields , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[93]  D. M. Titterington,et al.  A Study of Methods of Choosing the Smoothing Parameter in Image Restoration by Regularization , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

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

[95]  John K. Goutsias.  Unilateral approximation of Gibbs random field images , 1991, CVGIP Graph. Model. Image Process..

[96]  P. Doerschuk.  Bayesian Signal Reconstruction, Markov Random Fields, and X-Ray Crystallography , 1991 .

[97]  S. Mitter,et al.  Recursive stochastic algorithms for global optimization in R d , 1991 .

[98]  F. S. Cohen,et al.  Classification of Rotated and Scaled Textured Images Using Gaussian Markov Random Field Models , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[99]  Alan S. Willsky,et al.  Modeling and estimation of multiscale stochastic processes , 1991, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing.

[100]  John W. Woods,et al.  Compound Gauss-Markov random fields for image estimation , 1991, IEEE Trans. Signal Process..

[101]  Abhir Bhalerao,et al.  Multiresolution image segmentation , 1991 .

[102]  Michèle Basseville,et al.  Modeling and estimation of multiresolution stochastic processes , 1992, IEEE Trans. Inf. Theory.

[103]  Anil K. Jain,et al.  Texture classification and segmentation using multiresolution simultaneous autoregressive models , 1992, Pattern Recognit..

[104]  Eric Dubois,et al.  Bayesian Estimation of Motion Vector Fields , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[105]  Russell M. Mersereau,et al.  Blur identification by the method of generalized cross-validation , 1992, IEEE Trans. Image Process..

[106]  D.A. Landgrebe,et al.  Classification with spatio-temporal interpixel class dependency contexts , 1992, IEEE Trans. Geosci. Remote. Sens..

[107]  Zhigang Fan,et al.  Maximum likelihood unsupervised textured image segmentation , 1992, CVGIP Graph. Model. Image Process..

[108]  Alvaro R. De Pierro,et al.  On methods for maximum a posteriori image reconstruction with a normal prior , 1992, J. Vis. Commun. Image Represent..

[109]  F. Heitz,et al.  Multiscale Markov random fields and constrained relaxation in low level image analysis , 1992, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[110]  Rama Chellappa,et al.  Segmentation of polarimetric synthetic aperture radar data , 1992, IEEE Trans. Image Process..

[111]  Ken D. Sauer,et al.  Bayesian estimation of transmission tomograms using segmentation based optimization , 1992 .

[112]  P. Green,et al.  Metropolis Methods, Gaussian Proposals and Antithetic Variables , 1992 .

[113]  Nikolas P. Galatsanos,et al.  Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation , 1992, IEEE Trans. Image Process..

[114]  Arnoldo Frigessi,et al.  Stochastic models, statistical methods, and algorithms in image analysis : proceedings of the special year on image analysis held in Rome, Italy, 1990 , 1992 .

[115]  Donald Geman,et al.  Constrained Restoration and the Recovery of Discontinuities , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[116]  A. Benveniste,et al.  Multiscale autoregressive processes. I. Schur-Levinson parametrizations , 1992, IEEE Trans. Signal Process..

[117]  Jun Zhang.  The mean field theory in EM procedures for Markov random fields , 1992, IEEE Trans. Signal Process..

[118]  Chee Sun Won,et al.  Unsupervised segmentation of noisy and textured images using Markov random fields , 1992, CVGIP Graph. Model. Image Process..

[119]  Paul Cohen,et al.  Gibbs Random Fields, Fuzzy Clustering, and the Unsupervised Segmentation of Textured Images , 1993, CVGIP Graph. Model. Image Process..

[120]  Patrick Bouthemy,et al.  Multimodal Estimation of Discontinuous Optical Flow using Markov Random Fields , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[121]  Richard M. Leahy,et al.  An approximate method of evaluating the joint likelihood for first-order GMRFs , 1993, IEEE Trans. Image Process..

[122]  Rama Chellappa,et al.  Mean field annealing using compound Gauss-Markov random fields for edge detection and image estimation , 1993, IEEE Trans. Neural Networks.

[123]  Josiane Zerubia,et al.  Parallel image classification using multiscale Markov random fields , 1993, 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[124]  Ken D. Sauer,et al.  A local update strategy for iterative reconstruction from projections , 1993, IEEE Trans. Signal Process..

[125]  Jun Zhang,et al.  The mean field theory in EM procedures for blind Markov random field image restoration , 1993, IEEE Trans. Image Process..

[126]  Brian D. Jeffs,et al.  Restoration of blurred star field images by maximally sparse optimization , 1993, IEEE Trans. Image Process..

[127]  Ken D. Sauer,et al.  A generalized Gaussian image model for edge-preserving MAP estimation , 1993, IEEE Trans. Image Process..

[128]  S. Mitter,et al.  Metropolis-type annealing algorithms for global optimization in R d , 1993 .

[129]  W. Clem Karl,et al.  Multiscale representations of Markov random fields , 1993, 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[130]  Julian Besag,et al.  Towards Bayesian image analysis , 1993 .

[131]  Sridhar Lakshmanan,et al.  Valid parameter space for 2-D Gaussian Markov random fields , 1993, IEEE Trans. Inf. Theory.

[132]  Simon R. Cherry,et al.  Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images , 1994, IEEE Trans. Medical Imaging.

[133]  Alfred O. Hero,et al.  Space-alternating generalized expectation-maximization algorithm , 1994, IEEE Trans. Signal Process..

[134]  Ken D. Sauer,et al.  Maximum likelihood scale estimation for a class of Markov random fields , 1994, Proceedings of ICASSP '94. IEEE International Conference on Acoustics, Speech and Signal Processing.

[135]  Charles A. Bouman,et al.  A multiscale random field model for Bayesian image segmentation , 1994, IEEE Trans. Image Process..

[136]  Michel Barlaud,et al.  Motion estimation based on Markov random fields , 1994, Proceedings of 1st International Conference on Image Processing.

[137]  Ibrahim M. Elfadel,et al.  Gibbs Random Fields, Cooccurrences, and Texture Modeling , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[138]  Edward J. Delp,et al.  Parameter estimation and segmentation of noisy or textured images using the EM algorithm and MPM estimation , 1994, Proceedings of 1st International Conference on Image Processing.

[139]  Levent Onural,et al.  On a Parameter Estimation Method for Gibbs-Markov Random Fields , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[140]  Andrew Lumsdaine,et al.  Maximum likelihood parameter estimation for non-Gaussian prior signal models , 1994, Proceedings of 1st International Conference on Image Processing.

[141]  Michel Barlaud,et al.  Two deterministic half-quadratic regularization algorithms for computed imaging , 1994, Proceedings of 1st International Conference on Image Processing.

[142]  Stefano Alliney,et al.  An algorithm for the minimization of mixed l1 and l2 norms with application to Bayesian estimation , 1994, IEEE Trans. Signal Process..

[143]  Martin G. Bello,et al.  A combined Markov random field and wave-packet transform-based approach for image segmentation , 1994, IEEE Trans. Image Process..

[144]  Robert L. Stevenson,et al.  A Bayesian approach to image expansion for improved definitio , 1994, IEEE Trans. Image Process..

[145]  Jun Zhang,et al.  Maximum-likelihood parameter estimation for unsupervised stochastic model-based image segmentation , 1994, IEEE Trans. Image Process..

[146]  W. Clem Karl,et al.  Efficient multiscale regularization with applications to the computation of optical flow , 1994, IEEE Trans. Image Process..

[147]  D. M. Titterington,et al.  An Empirical Study of the Simulation of Various Models used for Images , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[148]  B. Claus,et al.  Multiscale signal processing: isotropic random fields on homogeneous trees , 1994 .

[149]  Ravi Mazumdar,et al.  Wavelet representations of stochastic processes and multiresolution stochastic models , 1994, IEEE Trans. Signal Process..

[150]  Levent Onural,et al.  Gibbs Random Field Model Based Weight Selection for the 2-D Adaptive Weighted Median Filter , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[151]  Robert L. Stevenson,et al.  Stochastic modeling and estimation of multispectral image data , 1994, Proceedings of ICASSP '94. IEEE International Conference on Acoustics, Speech and Signal Processing.

[152]  D. Politis.  Markov Chains in Many Dimensions , 1994, Advances in Applied Probability.

[153]  Yee-Hong Yang,et al.  Multiresolution Color Image Segmentation , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[154]  Thomas J. Hebert,et al.  Expectation-maximization algorithms, null spaces, and MAP image restoration , 1995, IEEE Trans. Image Process..

[155]  Rangasami L. Kashyap,et al.  Bayesian decision feedback for segmentation of binary images , 1995, 1995 International Conference on Acoustics, Speech, and Signal Processing.

[156]  Shape parameter estimation for generalized Gaussian Markov random field models used in MAP image restoration , 1995, Conference Record of The Twenty-Ninth Asilomar Conference on Signals, Systems and Computers.

[157]  Anthony A. Maciejewski,et al.  A multiscale stochastic image model for automated inspection , 1995, IEEE Transactions on Image Processing.

[158]  Peter C. Doerschuk,et al.  Cluster Expansions for the Deterministic Computation of Bayesian Estimators Based on Markov Random Fields , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[159]  Alan S. Willsky,et al.  Likelihood calculation for a class of multiscale stochastic models, with application to texture discrimination , 1995, IEEE Trans. Image Process..

[160]  Peter C. Doerschuk,et al.  Tree Approximations to Markov Random Fields , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[161]  Jan P. Allebach,et al.  Fast image search using a multiscale stochastic model , 1995, Proceedings., International Conference on Image Processing.

[162]  Jun Zhang,et al.  The application of mean field theory to image motion estimation , 1995, IEEE Trans. Image Process..

[163]  Stan Z. Li,et al.  Markov Random Field Modeling in Computer Vision , 1995, Computer Science Workbench.

[164]  Ken D. Sauer,et al.  A unified approach to statistical tomography using coordinate descent optimization , 1996, IEEE Trans. Image Process..

[165]  Ken D. Sauer,et al.  Efficient ML estimation of the shape parameter for generalized Gaussian MRFs , 1996, 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Conference Proceedings.

[166]  Benveniste,et al.  Multiscale Autoregressive Processes , Part II : Lattice Structures for Whitening and Modeling-? '-koI-6 , 2022 .