*A Mathematical Introduction*

**Author**: Gerhard Winkler

**Publisher:** Springer Science & Business Media

**ISBN:** 3642557600

**Category:** Mathematics

**Page:** 387

**View:** 7642

"This book is concerned with a probabilistic approach for image analysis, mostly from the Bayesian point of view, and the important Markov chain Monte Carlo methods commonly used....This book will be useful, especially to researchers with a strong background in probability and an interest in image analysis. The author has presented the theory with rigor...he doesn’t neglect applications, providing numerous examples of applications to illustrate the theory." -- MATHEMATICAL REVIEWS

This text is concerned with a probabilistic approach to image analysis as initiated by U. GRENANDER, D. and S. GEMAN, B.R. HUNT and many others, and developed and popularized by D. and S. GEMAN in a paper from 1984. It formally adopts the Bayesian paradigm and therefore is referred to as 'Bayesian Image Analysis'. There has been considerable and still growing interest in prior models and, in particular, in discrete Markov random field methods. Whereas image analysis is replete with ad hoc techniques, Bayesian image analysis provides a general framework encompassing various problems from imaging. Among those are such 'classical' applications like restoration, edge detection, texture discrimination, motion analysis and tomographic reconstruction. The subject is rapidly developing and in the near future is likely to deal with high-level applications like object recognition. Fascinating experiments by Y. CHOW, U. GRENANDER and D.M. KEENAN (1987), (1990) strongly support this belief.

Image processing is concerned with the analysis and manipulation of images by computer. Providing a thorough treatment of image processing with an emphasis on those aspects most used in computer graphics, the authors concentrate on describing and analyzing the underlying concepts rather than on presenting algorithms or pseudocode. As befits a modern introduction to this topic, a good balance is struck between discussing the underlying mathematics and the main topics: signal processing, data discretization, the theory of colour and different colour systems, operations in images, dithering and half-toning, warping and morphing and image processing. This second edition reflects recent trends in science andtechnology that exploit image processing in computer graphics and vision applications. Stochastic image models and statistical methods for image processing are covered as are: A modern approach and new developments in the area, Probability theory for image processing, Applications in image analysis and computer vision.

In various scientific and industrial fields, stochastic simulations are taking on a new importance. This is due to the increasing power of computers and practitioners’ aim to simulate more and more complex systems, and thus use random parameters as well as random noises to model the parametric uncertainties and the lack of knowledge on the physics of these systems. The error analysis of these computations is a highly complex mathematical undertaking. Approaching these issues, the authors present stochastic numerical methods and prove accurate convergence rate estimates in terms of their numerical parameters (number of simulations, time discretization steps). As a result, the book is a self-contained and rigorous study of the numerical methods within a theoretical framework. After briefly reviewing the basics, the authors first introduce fundamental notions in stochastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and stochastic differential equations. In particular, they review the essential properties of Itô integrals and prove fundamental results on the probabilistic analysis of parabolic partial differential equations. These results in turn provide the basis for developing stochastic numerical methods, both from an algorithmic and theoretical point of view. The book combines advanced mathematical tools, theoretical analysis of stochastic numerical methods, and practical issues at a high level, so as to provide optimal results on the accuracy of Monte Carlo simulations of stochastic processes. It is intended for master and Ph.D. students in the field of stochastic processes and their numerical applications, as well as for physicists, biologists, economists and other professionals working with stochastic simulations, who will benefit from the ability to reliably estimate and control the accuracy of their simulations.

Primarily an introduction to the theory of stochastic processes at the undergraduate or beginning graduate level, the primary objective of this book is to initiate students in the art of stochastic modelling. However it is motivated by significant applications and progressively brings the student to the borders of contemporary research. Examples are from a wide range of domains, including operations research and electrical engineering. Researchers and students in these areas as well as in physics, biology and the social sciences will find this book of interest.

Mathematical Methods for Signal and Image Analysis and Representation presents the mathematical methodology for generic image analysis tasks. In the context of this book an image may be any m-dimensional empirical signal living on an n-dimensional smooth manifold (typically, but not necessarily, a subset of spacetime). The existing literature on image methodology is rather scattered and often limited to either a deterministic or a statistical point of view. In contrast, this book brings together these seemingly different points of view in order to stress their conceptual relations and formal analogies. Furthermore, it does not focus on specific applications, although some are detailed for the sake of illustration, but on the methodological frameworks on which such applications are built, making it an ideal companion for those seeking a rigorous methodological basis for specific algorithms as well as for those interested in the fundamental methodology per se. Covering many topics at the forefront of current research, including anisotropic diffusion filtering of tensor fields, this book will be of particular interest to graduate and postgraduate students and researchers in the fields of computer vision, medical imaging and visual perception.

The first text to bridge the gap between image processing and jump regression analysis Recent statistical tools developed to estimate jump curves and surfaces have broad applications, specifically in the area of image processing. Often, significant differences in technical terminologies make communication between the disciplines of image processing and jump regression analysis difficult. In easy-to-understand language, Image Processing and Jump Regression Analysis builds a bridge between the worlds of computer graphics and statistics by addressing both the connections and the differences between these two disciplines. The author provides a systematic analysis of the methodology behind nonparametric jump regression analysis by outlining procedures that are easy to use, simple to compute, and have proven statistical theory behind them. Key topics include: Conventional smoothing procedures Estimation of jump regression curves Estimation of jump location curves of regression surfaces Jump-preserving surface reconstruction based on local smoothing Edge detection in image processing Edge-preserving image restoration With mathematical proofs kept to a minimum, this book is uniquely accessible to a broad readership. It may be used as a primary text in nonparametric regression analysis and image processing as well as a reference guide for academicians and industry professionals focused on image processing or curve/surface estimation.

In a family study of breast cancer, epidemiologists in Southern California increase the power for detecting a gene-environment interaction. In Gambia, a study helps a vaccination program reduce the incidence of Hepatitis B carriage. Archaeologists in Austria place a Bronze Age site in its true temporal location on the calendar scale. And in France, researchers map a rare disease with relatively little variation. Each of these studies applied Markov chain Monte Carlo methods to produce more accurate and inclusive results. General state-space Markov chain theory has seen several developments that have made it both more accessible and more powerful to the general statistician. Markov Chain Monte Carlo in Practice introduces MCMC methods and their applications, providing some theoretical background as well. The authors are researchers who have made key contributions in the recent development of MCMC methodology and its application. Considering the broad audience, the editors emphasize practice rather than theory, keeping the technical content to a minimum. The examples range from the simplest application, Gibbs sampling, to more complex applications. The first chapter contains enough information to allow the reader to start applying MCMC in a basic way. The following chapters cover main issues, important concepts and results, techniques for implementing MCMC, improving its performance, assessing model adequacy, choosing between models, and applications and their domains. Markov Chain Monte Carlo in Practice is a thorough, clear introduction to the methodology and applications of this simple idea with enormous potential. It shows the importance of MCMC in real applications, such as archaeology, astronomy, biostatistics, genetics, epidemiology, and image analysis, and provides an excellent base for MCMC to be applied to other fields as well.

From the reviews: "Paul Glasserman has written an astonishingly good book that bridges financial engineering and the Monte Carlo method. The book will appeal to graduate students, researchers, and most of all, practicing financial engineers [...] So often, financial engineering texts are very theoretical. This book is not." --Glyn Holton, Contingency Analysis

Simulation and Monte Carlo is aimed at students studying for degrees in Mathematics, Statistics, Financial Mathematics, Operational Research, Computer Science, and allied subjects, who wish an up-to-date account of the theory and practice of Simulation. Its distinguishing features are in-depth accounts of the theory of Simulation, including the important topic of variance reduction techniques, together with illustrative applications in Financial Mathematics, Markov chain Monte Carlo, and Discrete Event Simulation. Each chapter contains a good selection of exercises and solutions with an accompanying appendix comprising a Maple worksheet containing simulation procedures. The worksheets can also be downloaded from the web site supporting the book. This encourages readers to adopt a hands-on approach in the effective design of simulation experiments. Arising from a course taught at Edinburgh University over several years, the book will also appeal to practitioners working in the finance industry, statistics and operations research.

The theory of spatial models over lattices, or random fields as they are known, has developed significantly over recent years. This book provides a graduate-level introduction to the subject which assumes only a basic knowledge of probability and statistics, finite Markov chains, and the spectral theory of second-order processes. A particular strength of this book is its emphasis on examples - both to motivate the theory which is being developed, and to demonstrate the applications which range from statistical mechanics to image analysis and from statistics to stochastic algorithms.

Stochastic geometry involves the study of random geometric structures, and blends geometric, probabilistic, and statistical methods to provide powerful techniques for modeling and analysis. Recent developments in computational statistical analysis, particularly Markov chain Monte Carlo, have enormously extended the range of feasible applications. Stochastic Geometry: Likelihood and Computation provides a coordinated collection of chapters on important aspects of the rapidly developing field of stochastic geometry, including: o a "crash-course" introduction to key stochastic geometry themes o considerations of geometric sampling bias issues o tesselations o shape o random sets o image analysis o spectacular advances in likelihood-based inference now available to stochastic geometry through the techniques of Markov chain Monte Carlo

"[This third edition] reflects the latest developments in the field and presents a fully updates and comprehensive account of state-of-the art theory, methods, and applications that have emerged in Monte Carlo simulation since the publication of the classic first edition over more than a quarter of a century ago. While maintaining its accessible and intuitive approach, this revised edition features a wealth of up-to-date information facilitating a deeper understanding of problem solving across a wide array of subject areas, such as engineering, statistics, computer science, mathematics, and the physical and life sciences. The book begins with a modernized introduction addressing the basic concepts of probability, Markov processes, and convex optimization. Subsequent chapters discuss dramatic changes that have occurred in the field of the Monte Carlo method, with coverage of many modern topics including : Markov chain Monte Carlo, variance reduction techniques such a importance (re)sampling and the transform likelihood ratio method, score function method for sensitivity analysis, stochastic approximation method and stochastic counter-part method for Monte Carlo optimization, cross-entropy method for rare events estimation and combinatorial optimization, and application of Monte Carlo techniques for counting problems. An extensive range of exercises is provided at the end of each chapter, as well as a generous sampling of applied examples." (source : 4ème de couverture).

This book is an introduction to the modern theory of Markov chains, whose goal is to determine the rate of convergence to the stationary distribution, as a function of state space size and geometry. This topic has important connections to combinatorics, statistical physics, and theoretical computer science. Many of the techniques presented originate in these disciplines. The central tools for estimating convergence times, including coupling, strong stationary times, and spectral methods, are developed. The authors discuss many examples, including card shuffling and the Ising model, from statistical mechanics, and present the connection of random walks to electrical networks and apply it to estimate hitting and cover times. The first edition has been used in courses in mathematics and computer science departments of numerous universities. The second edition features three new chapters (on monotone chains, the exclusion process, and stationary times) and also includes smaller additions and corrections throughout. Updated notes at the end of each chapter inform the reader of recent research developments.

This sequel to volume 19 of Handbook on Statistics on Stochastic Processes: Modelling and Simulation is concerned mainly with the theme of reviewing and, in some cases, unifying with new ideas the different lines of research and developments in stochastic processes of applied flavour. This volume consists of 23 chapters addressing various topics in stochastic processes. These include, among others, those on manufacturing systems, random graphs, reliability, epidemic modelling, self-similar processes, empirical processes, time series models, extreme value therapy, applications of Markov chains, modelling with Monte Carlo techniques, and stochastic processes in subjects such as engineering, telecommunications, biology, astronomy and chemistry. particular with modelling, simulation techniques and numerical methods concerned with stochastic processes. The scope of the project involving this volume as well as volume 19 is already clarified in the preface of volume 19. The present volume completes the aim of the project and should serve as an aid to students, teachers, researchers and practitioners interested in applied stochastic processes.

Stochastic Modeling of Scientific Data combines stochastic modeling and statistical inference in a variety of standard and less common models, such as point processes, Markov random fields and hidden Markov models in a clear, thoughtful and succinct manner. The distinguishing feature of this work is that, in addition to probability theory, it contains statistical aspects of model fitting and a variety of data sets that are either analyzed in the text or used as exercises. Markov chain Monte Carlo methods are introduced for evaluating likelihoods in complicated models and the forward backward algorithm for analyzing hidden Markov models is presented. The strength of this text lies in the use of informal language that makes the topic more accessible to non-mathematicians. The combinations of hard science topics with stochastic processes and their statistical inference puts it in a new category of probability textbooks. The numerous examples and exercises are drawn from astronomy, geology, genetics, hydrology, neurophysiology and physics.

Markov processes are used to model systems with limited memory. They are used in many areas including communications systems, transportation networks, image segmentation and analysis, biological systems and DNA sequence analysis, random atomic motion and diffusion in physics, social mobility, population studies, epidemiology, animal and insect migration, queueing systems, resource management, dams, financial engineering, actuarial science, and decision systems. This book, which is written for upper level undergraduate and graduate students, and researchers, presents a unified presentation of Markov processes. In addition to traditional topics such as Markovian queueing system, the book discusses such topics as continuous-time random walk,correlated random walk, Brownian motion, diffusion processes, hidden Markov models, Markov random fields, Markov point processes and Markov chain Monte Carlo. Continuous-time random walk is currently used in econophysics to model the financial market, which has traditionally been modelled as a Brownian motion. Correlated random walk is popularly used in ecological studies to model animal and insect movement. Hidden Markov models are used in speech analysis and DNA sequence analysis while Markov random fields and Markov point processes are used in image analysis. Thus, the book is designed to have a very broad appeal. - Provides the practical, current applications of Markov processes - Coverage of HMM, Point processes, and Monte Carlo - Includes enough theory to help students gain throrough understanding of the subject - Principles can be immediately applied in many specific research projects, saving researchers time - End of chapter exercises provide reinforcement, practice and increased understanding to the student

Monte Carlo methods are among the most used and useful computational tools available today, providing efficient and practical algorithims to solve a wide range of scientific and engineering problems. Applications covered in this book include optimization, finance, statistical mechanics, birth and death processes, and gambling systems. Explorations in Monte Carlo Methods provides a hands-on approach to learning this subject. Each new idea is carefully motivated by a realistic problem, thus leading from questions to theory via examples and numerical simulations. Programming exercises are integrated throughout the text as the primary vehicle for learning the material. Each chapter ends with a large collection of problems illustrating and directing the material. This book is suitable as a textbook for students of engineering and the sciences, as well as mathematics.

This updated new edition introduces the reader to the fundamentals of queueing theory, including Markov processes and random walks. It contains an extended treatment of queueing networks and matrix analytic methods as well as additional topics like Poisson's equation, Palm theory and heavy tails.

This book provides an undergraduate introduction to discrete and continuous-time Markov chains and their applications. A large focus is placed on the first step analysis technique and its applications to average hitting times and ruin probabilities. Classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes, are also covered. Two major examples (gambling processes and random walks) are treated in detail from the beginning, before the general theory itself is presented in the subsequent chapters. An introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times is also provided, and the book includes a chapter on spatial Poisson processes with some recent results on moment identities and deviation inequalities for Poisson stochastic integrals. The concepts presented are illustrated by examples and by 72 exercises and their complete solutions.