Applied Graph Theory in Computer Vision and Pattern Recognition

The theory of modular forms and especially the so-called 'Ramanujan Conjectures' have been applied to resolve problems in combinatorics, computer science, analysis and number theory. This tract, based on the Wittemore Lectures given at Yale University, is concerned with describing some of these applications. In order to keep the presentation reasonably self-contained, Professor Sarnak begins by developing the necessary background material in modular forms. He then considers the solution of three problems: the Ruziewicz problem concerning finitely additive rotationally invariant measures on the sphere; the explicit construction of highly connected but sparse graphs: 'expander graphs' and 'Ramanujan graphs'; and the Linnik problem concerning the distribution of integers that represent a given large integer as a sum of three squares. These applications are carried out in detail. The book therefore should be accessible to a wide audience of graduate students and researchers in mathematics

Bipartite graphs are perhaps the most basic of objects in graph theory, both from a theoretical and practical point of view. However, sometimes they have been considered only as a special class in some wider context. This book deals solely with bipartite graphs. Together with traditional material, the reader will also find many unusual results. Essentially all proofs are given in full; many of these have been streamlined specifically for this text. Numerous exercises of all standards have also been included. The theory is illustrated with many applications especially to problems in timetabling, chemistry, communication networks and computer science. For the most part the material is accessible to any reader with a graduate understanding of mathematics. However, the book contains advanced sections requiring much more specialized knowledge, which will be of interest to specialists in combinatorics and graph theory.

Bayesian probability theory has emerged not only as a powerful tool for building computational theories of vision, but also as a general paradigm for studying human visual perception. This 1996 book provides an introduction to and critical analysis of the Bayesian paradigm. Leading researchers in computer vision and experimental vision science describe general theoretical frameworks for modelling vision, detailed applications to specific problems and implications for experimental studies of human perception. The book provides a dialogue between different perspectives both within chapters, which draw on insights from experimental and computational work, and between chapters, through commentaries written by the contributors on each others' work. Students and researchers in cognitive and visual science will find much to interest them in this thought-provoking collection.

Inferring the precise locations and splicing patterns of genes in DNA is a difficult but important task, with broad applications to biomedicine. The mathematical and statistical techniques that have been applied to this problem are surveyed and organized into a logical framework based on the theory of parsing. Both established approaches and methods at the forefront of current research are discussed. Numerous case studies of existing software systems are provided, in addition to detailed examples that work through the actual implementation of effective gene-predictors using hidden Markov models and other machine-learning techniques. Background material on probability theory, discrete mathematics, computer science, and molecular biology is provided, making the book accessible to students and researchers from across the life and computational sciences. This book is ideal for use in a first course in bioinformatics at graduate or advanced undergraduate level, and for anyone wanting to kee

Inferring the precise locations and splicing patterns of genes in DNA is a difficult but important task, with broad applications to biomedicine. The mathematical and statistical techniques that have been applied to this problem are surveyed and organized into a logical framework based on the theory of parsing. Both established approaches and methods at the forefront of current research are discussed. Numerous case studies of existing software systems are provided, in addition to detailed examples that work through the actual implementation of effective gene-predictors using hidden Markov models and other machine-learning techniques. Background material on probability theory, discrete mathematics, computer science, and molecular biology is provided, making the book accessible to students and researchers from across the life and computational sciences. This book is ideal for use in a first course in bioinformatics at graduate or advanced undergraduate level, and for anyone wanting to kee

This 2007 volume contains survey articles based on the invited lectures given at the Twenty-first British Combinatorial Conference, held in July 2007 at the University of Reading. This biennial conference is a well-established international event and the articles are of the high quality that befits the event. By its nature this volume provides an overview of current research activity in several areas of combinatorics, ranging from graph theory to current applications of combinatorial mathematics, including efficient approximability of NP-hard optimization problems and cryptographic key management. The authors are some of the world's foremost researchers in their fields, and here they summarize existing results, and give a unique preview of work currently being written up. The book provides a valuable survey of the state of knowledge in combinatorics. It will be useful to research workers and advanced graduate students, primarily in mathematics but also in computer science, statistics a

A human observer can effortlessly identify visible portions of geometric objects present in the environment. However, computations of visible portions of objects from a viewpoint involving thousands of objects is a time consuming task even for high speed computers. To solve such visibility problems, efficient algorithms have been designed. This book presents some of these visibility algorithms in two dimensions. Specifically, basic algorithms for point visibility, weak visibility, shortest paths, visibility graphs, link paths and visibility queries are all discussed. Several geometric properties are also established through lemmas and theorems. With over 300 figures and hundreds of exercises, this book is ideal for graduate students and researchers in the field of computational geometry. It will also be useful as a reference for researchers working in algorithms, robotics, computer graphics and geometric graph theory, and some algorithms from the book can be used in a first course in

SuperFractals, first published in 2006, is the successor to Fractals Everywhere, in which the power and beauty of Iterated Function Systems were introduced and applied to producing startling and original images that reflect complex structures found for example in nature. This provoked the question of whether there is a deeper connection between topology, geometry, IFS and codes on the one hand and biology, DNA and protein development on the other. Now, 20 years later, Barnsley explains how IFS have developed in order to address this issue. Ideas such as fractal tops and superIFS are introduced, and the classical deterministic approach is combined with probabilistic ideas to produce new mathematics and algorithms that open a whole theory that could have applications in computer graphics, bioinformatics, economics, signal processing and beyond. For the first time these ideas are explained in book form, and illustrated with breathtaking pictures.

Percolation theory was initiated some fifty years ago as a mathematical framework for the study of random physical processes such as flow through a disordered porous medium. It has proved to be a remarkably rich theory, with applications beyond natural phenomena to topics such as network modelling. The aims of this book, first published in 2006, are twofold. First to present classical results in a way that is accessible to non-specialists. Second, to describe results of Smirnov in conformal invariance, and outline the proof that the critical probability for random Voronoi percolation in the plane is 1/2. Throughout, the presentation is streamlined, with elegant and straightforward proofs requiring minimal background in probability and graph theory. Numerous examples illustrate the important concepts and enrich the arguments. All-in-all, it will be an essential purchase for mathematicians, physicists, electrical engineers and computer scientists working in this exciting area.

Oriented matroids play the role of matrices in discrete geometry, when metrical properties, such as angles or distances, are neither required nor available. Thus they are of great use in such areas as graph theory, combinatorial optimization and convex geometry. The variety of applications corresponds to the variety of ways they can be defined. Each of these definitions corresponds to a differing data structure for an oriented matroid, and handling them requires computational support, best realised through a functional language. Haskell is used here, and, for the benefit of readers, the book includes a primer on it. The combination of concrete applications and computation, the profusion of illustrations, many in colour, and the large number of examples and exercises make this an ideal introductory text on the subject. It will also be valuable for self-study for mathematicians and computer scientists working in discrete and computational geometry.

This tract has two purposes: to show what is known about the n-dimensional unit cubes and to demonstrate how Analysis, Algebra, Combinatorics, Graph Theory, Hyperbolic Geometry, Number Theory, can be applied to the study of them. The unit cubes, from any point of view, are among the most important and fascinating objects in an n-dimensional Euclidean space. However, our knowledge about them is still quite limited and many basic problems remain unsolved. In this Tract eight topics about the unit cubes are introduced: cross sections, projections, inscribed simplices, triangulations, 0/1 polytopes, Minkowski's conjecture, Furtwangler's conjecture, and Keller's conjecture. In particular the author demonstrates how deep analysis like log concave measure and the Brascamp-Lieb inequality can deal with the cross section problem, how Hyperbolic Geometry helps with the triangulation problem, how group rings can deal with Minkowski's conjecture and Furtwangler's conjecture, and how Graph Theory h

Computer vision is a rapidly growing field which aims to make computers 'see' as effectively as humans. In this book Dr Shapiro presents a new computer vision framework for interpreting time-varying imagery. This is an important task, since movement reveals valuable information about the environment. The fully-automated system operates on long, monocular image sequences containing multiple, independently-moving objects, and demonstrates the practical feasibility of recovering scene structure and motion in a bottom-up fashion. Real and synthetic examples are given throughout, with particular emphasis on image coding applications. Novel theory is derived in the context of the affine camera, a generalisation of the familiar scaled orthographic model. Analysis proceeds by tracking 'corner features' through successive frames and grouping the resulting trajectories into rigid objects using new clustering and outlier rejection techniques. The three-dimensional motion parameters are then compu

This book gives applications of the theory of process algebra, or Algebra of Communicating Processes (ACP), that is the study of concurrent or communicating processes studied using an algebraic framework. The approach is axiomatic; the authors consider structures that are some set of mostly equational axioms, which are equipped with several operators. Thus the term 'algebra' is used in the model-theoretic sense. The axiomatic approach enables one to organize the field of process theories. The theory is applied systematically to a number of situations, including systolic algorithms, semantics of an object-oriented language, and protocols. It will be welcomed by computer scientists working in parallel programming.

Now available in the Cambridge Mathematical Library, the classic work from Luis Santaló. Integral geometry originated with problems on geometrical probability and convex bodies. Its later developments, however, have proved to be useful in several fields ranging from pure mathematics (measure theory, continuous groups) to technical and applied disciplines (pattern recognition, stereology). The book is a systematic exposition of the theory and a compilation of the main results in the field. The volume can be used to complement courses on differential geometry, Lie groups or probability or differential geometry. It is ideal both as a reference and for those wishing to enter the field.

Cowan and Steel's Manual has for many years had an essential role in every laboratory of microbiology and bacteriology. This substantially revised new edition, which is modelled on the successful pattern established in the previous two editions, has been fully updated and is suitable for all bacteriological laboratories using traditional diagnostic methods. It is essentially a practical manual with up-to-date contributions and key references by experts to the diagnostic characteristics of the bacteria likely to be encountered in public health and hospital microbiology laboratories, as well as in medical and veterinary practice. This edition contains new sections on rapid and mechanised test methods, and on the laboratory applications of computer theory and practice to the identification of bacteria. As in previous editions, the importance of laboratory quality control and proficiency procedures are emphasised. The Appendices give details of the laboratory methods and media for all the

A new text/reference covering all apsects of modern combinatorial design theory. Graduates and professionals in computer science, applied math, combinatorics, and applied statistics will find the book

An understanding of random processes is crucial to many engineering fields-including communication theory, computer vision, and digital signal processing in electrical and computer engineering, and vi

A systematic and comprehensive introduction to the study of nonlinear dynamical systems, in both discrete and continuous time, for nonmathematical students and researchers working in applied fields. An understanding of linear systems and the classical theory of stability are essential although basic reviews of the relevant material are provided. Further chapters are devoted to the stability of invariant sets, bifurcation theory, chaotic dynamics and the transition to chaos. In the final two chapters the authors approach the subject from a measure-theoretical point of view and compare results to those given for the geometrical or topological approach of the first eight chapters. Includes about one hundred exercises. A Windows-compatible software programme called DMC, provided free of charge through a website dedicated to the book, allows readers to perform numerical and graphical analysis of dynamical systems. Also available on the website are computer exercises and solutions to selecte

A systematic and comprehensive introduction to the study of nonlinear dynamical systems, in both discrete and continuous time, for nonmathematical students and researchers working in applied fields. An understanding of linear systems and the classical theory of stability are essential although basic reviews of the relevant material are provided. Further chapters are devoted to the stability of invariant sets, bifurcation theory, chaotic dynamics and the transition to chaos. In the final two chapters the authors approach the subject from a measure-theoretical point of view and compare results to those given for the geometrical or topological approach of the first eight chapters. Includes about one hundred exercises. A Windows-compatible software programme called DMC, provided free of charge through a website dedicated to the book, allows readers to perform numerical and graphical analysis of dynamical systems. Also available on the website are computer exercises and solutions to selecte

In this second edition of the now classic text, the already extensive treatment given in the first edition has been heavily revised by the author. The addition of two new sections, numerous new results and 150 references means that this represents a comprehensive account of random graph theory. The theory (founded by Erdös and Rényi in the late fifties) aims to estimate the number of graphs of a given degree that exhibit certain properties. It not only has numerous combinatorial applications, but also serves as a model for the probabilistic treatment of more complicated random structures. This book, written by an acknowledged expert in the field, can be used by mathematicians, computer scientists and electrical engineers, as well as people working in biomathematics. It is self-contained, and with numerous exercises in each chapter, is ideal for advanced courses or self study.

In this second edition of the now classic text, the already extensive treatment given in the first edition has been heavily revised by the author. The addition of two new sections, numerous new results and 150 references means that this represents a comprehensive account of random graph theory. The theory (founded by Erdös and Rényi in the late fifties) aims to estimate the number of graphs of a given degree that exhibit certain properties. It not only has numerous combinatorial applications, but also serves as a model for the probabilistic treatment of more complicated random structures. This book, written by an acknowledged expert in the field, can be used by mathematicians, computer scientists and electrical engineers, as well as people working in biomathematics. It is self-contained, and with numerous exercises in each chapter, is ideal for advanced courses or self study.

Now thoroughly revised and up-dated, this book describes techniques for handling and analysing data obtained from high-energy and nuclear physics experiments. The observation of particle interactions involves the analysis of large and complex data samples. Beginning with a chapter on real-time data triggering and filtering, the book describes methods of selecting the relevant events from a sometimes huge background. The use of pattern recognition techniques to group the huge number of measurements into physically meaningful objects like particle tracks or showers is then examined and the track and vertex fitting methods necessary to extract the maximum amount of information from the available measurements are explained. The final chapter describes tools and methods which are useful to the experimenter in the physical interpretation and in the presentation of the results. This indispensable guide will appeal to graduate students, researchers and computer and electronic engineers involve

Singularity theory is a broad subject with vague boundaries. It draws on many other areas of mathematics, and in turn has contributed to many areas both within and outside mathematics, in particular differential and algebraic geometry, knot theory, differential equations, bifurcation theory, Hamiltonian mechanics, optics, robotics and computer vision. This volume consists of two dozen articles from some of the best known figures in singularity theory, and it presents an up-to-date survey of research in this area.

This book gives a friendly introduction to Fourier analysis on finite groups, both commutative and non-commutative. Aimed at students in mathematics, engineering and the physical sciences, it examines the theory of finite groups in a manner that is both accessible to the beginner and suitable for graduate research. With applications in chemistry, error-correcting codes, data analysis, graph theory, number theory and probability, the book presents a concrete approach to abstract group theory through applied examples, pictures and computer experiments. In the first part, the author parallels the development of Fourier analysis on the real line and the circle, and then moves on to analogues of higher dimensional Euclidean space. The second part emphasizes matrix groups such as the Heisenberg group of upper triangular 2x2 matrices. The book concludes with an introduction to zeta functions on finite graphs via the trace formula.

This new edition of Professor Sethian's successful text provides an introduction to level set methods and fast marching methods, which are powerful numerical techniques for analyzing and computing interface motion in a host of settings. They rely on a fundamental shift in how one views moving boundaries; rethinking the natural geometric Lagrangian perspective and exchanging it for an Eulerian, initial value partial differential equation perspective. For this edition, the collection of applications provided in the text has been expanded, including examples from physics, chemistry, fluid mechanics, combustion, image processing, material science, fabrication of microelectronic components, computer vision, computer-aided design, and optimal control theory. This book will be a useful resource for mathematicians, applied scientists, practising engineers, computer graphic artists, and anyone interested in the evolution of boundaries and interfaces.

The Subjects of Art History provides an introduction to the historiography and theory of the history of art. Examining a variety of theoretical approaches, the editors and contributors to this volume provide interpretations of the history and contemporary relevance of such important methodologies as semiotics, phenomenology, feminism, gay and lesbian studies, museology, and computer applications, among other topics. Each essay, specially commissioned for this volume, gives a fresh perspective on the topic by demonstrating how a particular approach can be applied to the understanding and interpretation of specific works of art. This volume will be a timely contribution to the current debate on the theory and practice of art history.

Bipartite graphs are perhaps the most basic of objects in graph theory, both from a theoretical and practical point of view. However, sometimes they have been considered only as a special class in some wider context. This book deals solely with bipartite graphs. Together with traditional material, the reader will also find many unusual results. Essentially all proofs are given in full; many of these have been streamlined specifically for this text. Numerous exercises of all standards have also been included. The theory is illustrated with many applications especially to problems in timetabling, chemistry, communication networks and computer science. For the most part the material is accessible to any reader with a graduate understanding of mathematics. However, the book contains advanced sections requiring much more specialized knowledge, which will be of interest to specialists in combinatorics and graph theory.

Combinatorics on words, or finite sequences, is a field which grew simultaneously within disparate branches of mathematics such as group theory and probability. It has grown into an independent theory finding substantial applications in computer science automata theory and linguistics. This volume is the first to attempt to present a thorough treatment of this theory. All of the main results and techniques are covered. The presentation is accessible to undergraduate and graduate level students in mathematics and computer science as well as to specialists in all branches of applied mathematics. Topics discussed include: Thue's square free words, Van der Waerden's theorem, Ramsey's theorem; factorization of free monoids including application to free Lie algebras; equations in words, etc.

This book explores the theory and methods of systems analysis and computer modeling as applied to problems in ecology and natural resource management. It reflects the problems and conflicts between co

Bayesian probability theory has emerged not only as a powerful tool for building computational theories of vision, but also as a general paradigm for studying human visual perception. This 1996 book provides an introduction to and critical analysis of the Bayesian paradigm. Leading researchers in computer vision and experimental vision science describe general theoretical frameworks for modelling vision, detailed applications to specific problems and implications for experimental studies of human perception. The book provides a dialogue between different perspectives both within chapters, which draw on insights from experimental and computational work, and between chapters, through commentaries written by the contributors on each others' work. Students and researchers in cognitive and visual science will find much to interest them in this thought-provoking collection.

These notes are based on a series of lectures given at the Advanced Research Institute of Discrete Applied Mathematics held at Rutgers University. Their aim is to link together algorithmic problems arising in knot theory, statistical physics and classical combinatorics. Apart from the theory of computational complexity concerned with enumeration problems, introductions are given to several of the topics treated, such as combinatorial knot theory, randomised approximation algorithms, percolation and random cluster models. To researchers in discrete mathematics, computer science and statistical physics, this book will be of great interest, but any non-expert should find it an appealing guide to a very active area of research.

This book, the first in the Cambridge Nonlinear Science Series, presents the fundamentals of chaos theory in conservative systems, providing a systematic study of the theory of transitional states of physical systems which lie between deterministic and chaotic behaviour. The authors' treatment of transitions to chaos, the theory of stochastic layers and webs, and the numerous applications of this theory, particularly to pattern symmetry, will make the book of importance to scientists from many disciplines.The authors have been meticulous in providing a detailed presentation of the material, enabling the reader to learn the necessary computational methods and to apply them in other problems. The inclusion of a significant amount of computer graphics is also an important aid to understanding. The final section of the book contains a fascinating collection of patterns in art and living nature. The book will be of interest to graduate students and researchers in physics and mathematics who

In this 1991 book, Hilbers discusses the general case of how to use processors simultaneously in order to solve a single problem rather than any specific application, and develops a theory independent of particular architectures. He starts by introducing distributed computing with graph theory, and considers processor networks and their price/performance ratios. He goes on to look at obtaining homogeneous distributions of work over networks and considers examples. Finally he discusses message routing within a processor network. This is intended to be a fundamental treatment of the relevant subjects and is aimed at computer scientists and graduate students in computer science who have experience with parallel processing: it will also be useful to others interested in processor networks.

Although graph theory, design theory, and coding theory had their origins in various areas of applied mathematics, today they are to be found under the umbrella of discrete mathematics. Here the authors have considerably reworked and expanded their earlier successful books on graphs, codes and designs, into an invaluable textbook. They do not seek to consider each of these three topics individually, but rather to stress the many and varied connections between them. The discrete mathematics needed is developed in the text, making this book accessible to any student with a background of undergraduate algebra. Many exercises and useful hints are included througout, and a large number of references are given.

Although graph theory, design theory, and coding theory had their origins in various areas of applied mathematics, today they are to be found under the umbrella of discrete mathematics. Here the authors have considerably reworked and expanded their earlier successful books on graphs, codes and designs, into an invaluable textbook. They do not seek to consider each of these three topics individually, but rather to stress the many and varied connections between them. The discrete mathematics needed is developed in the text, making this book accessible to any student with a background of undergraduate algebra. Many exercises and useful hints are included througout, and a large number of references are given.

The study of spatial processes and their applications is an important topic in statistics and finds wide application particularly in computer vision and image processing. This book is devoted to statistical inference in spatial statistics and is intended for specialists needing an introduction to the subject and to its applications. One of the themes of the book is the demonstration of how these techniques give new insights into classical procedures (including new examples in likelihood theory) and newer statistical paradigms such as Monte-Carlo inference and pseudo-likelihood. Professor Ripley also stresses the importance of edge effects and of lack of a unique asymptotic setting in spatial problems. Throughout, the author discusses the foundational issues posed and the difficulties, both computational and philosophical, which arise. The final chapters consider image restoration and segmentation methods and the averaging and summarising of images. Thus, the book will find wide appeal

The application of time series techniques in economics has become increasingly important, both for forecasting purposes and in the empirical analysis of time series in general. In this book, Terence Mills not only brings together recent research at the frontiers of the subject, but also analyses the areas of most importance to applied economics. It is an up-to-date text which extends the basic techniques of analysis to cover the development of methods that can be used to analyse a wide range of economic problems. The book analyses three basic areas of time series analysis: univariate models, multivariate models, and non-linear models. In each case the basic theory is outlined and then extended to cover recent developments. Particular emphasis is placed on applications of the theory to important areas of applied economics and on the computer software and programs needed to implement the techniques. This book clearly distinguishes itself from its competitors by emphasising the techniques

This book, the first in the Cambridge Nonlinear Science Series, presents the fundamentals of chaos theory in conservative systems, providing a systematic study of the theory of transitional states of physical systems which lie between deterministic and chaotic behaviour. The authors' treatment of transitions to chaos, the theory of stochastic layers and webs, and the numerous applications of this theory, particularly to pattern symmetry, will make the book of importance to scientists from many disciplines.The authors have been meticulous in providing a detailed presentation of the material, enabling the reader to learn the necessary computational methods and to apply them in other problems. The inclusion of a significant amount of computer graphics is also an important aid to understanding. The final section of the book contains a fascinating collection of patterns in art and living nature. The book will be of interest to graduate students and researchers in physics and mathematics who

Written in an appealing and informal style, this text is a self-contained second course on mathematical methods dealing with topics in linear algebra and multivariate calculus that can be applied to statistics, operations research, computer science, econometrics and mathematical economics. The prerequisites are elementary courses in linear algebra and calculus, but the author has maintained a balance between a rigorous theoretical and a cookbook approach, giving concrete and geometric explanations, so that the material will be accessible to students who have not studied mathematics in depth. Indeed, as much of the material is normally available only in technical textbooks, this book will have wide appeal to students whose interest is in application rather than theory. The book is amply supplied with examples and exercises: complete solutions to a large proportion of these are provided.

Written in an appealing and informal style, this text is a self-contained second course on mathematical methods dealing with topics in linear algebra and multivariate calculus that can be applied to statistics, operations research, computer science, econometrics and mathematical economics. The prerequisites are elementary courses in linear algebra and calculus, but the author has maintained a balance between a rigorous theoretical and a cookbook approach, giving concrete and geometric explanations, so that the material will be accessible to students who have not studied mathematics in depth. Indeed, as much of the material is normally available only in technical textbooks, this book will have wide appeal to students whose interest is in application rather than theory. The book is amply supplied with examples and exercises: complete solutions to a large proportion of these are provided.