Jaakko Astola

Logic networks and automata are facets of digital systems. The change of the design of logic networks from skills and art into a scientific discipline was possible by the development of the underlying mathematical theory called the Switching Theory. The fundamentals of this theory come from the attempts towards an algebraic description of laws of thoughts presented in the works by George J. Boole and the works on logic by Augustus De Morgan. As often the case in engineering, when the importance of a problem and the need for solving it reach certain limits, the solutions are searched by many scholars in different parts of the word, simultaneously or at about the same time, however, quite independently and often unaware of the work by other scholars. The formulation and rise of Switching Theory is such an example. This book presents a brief account of the developments of Switching Theory and highlights some less known facts in the history of it. The readers will find the book a fresh look into the development of the field revealing how difficult it has been to arrive at many of the concepts that we now consider obvious . Researchers in the history or philosophy of computing will find this book a valuable source of information that complements the standard presentations of the topic.

Compared to binary switching functions, the multiple-valued functions (MV) offer more compact representations of the information content of signals modeled by logic functions and, therefore, their use fits very well in the general settings of data compression attempts and approaches. The first task in dealing with such signals is to provide mathematical methods for their representation in a way that will make their application in practice feasible. Representation of Multiple-Valued Logic Functions is aimed at providing an accessible introduction to these mathematical techniques that are necessary for application of related implementation methods and tools. This book presents in a uniform way different representations of multiple-valued logic functions, including functional expressions, spectral representations on finite Abelian groups, and their graphical counterparts (various related decision diagrams). Three-valued, or ternary functions, are traditionally used as the first extensionfrom the binary case. They have a good feature that the ratio between the number of bits and the number of different values that can be encoded with the specified number of bits is favourable for ternary functions. Four-valued functions, also called quaternary functions, are particularly attractive, since in practical realization within today prevalent binary circuits environment, they may be easy coded by binary values and realized with two-stable state circuits. At the same time, there is much more considerable advent in design of four-valued logic circuits than for other $p$-valued functions. Therefore, this book is written using a hands-on approach such that after introducing the general and necessarily abstract background theory, the presentation is based on a large number of examples for ternary and quaternary functions that should provide an intuitive understanding of various representation methods and the interconnections among them. Table of Contents: Multiple-Valued Logic Functions / Functional Expressions for Multiple-Valued Functions / Spectral Representations of Multiple-Valued Functions / Decision Diagrams for Multiple-Valued Functions / Fast Calculation Algorithms

Logic networks and automata are facets of digital systems. The change of the design of logic networks from skills and art into a scientific discipline was possible by the development of the underlying mathematical theory called the Switching Theory. The fundamentals of this theory come from the attempts towards an algebraic description of laws of thoughts presented in the works by George J. Boole and the works on logic by Augustus De Morgan. As often the case in engineering, when the importance of a problem and the need for solving it reach certain limits, the solutions are searched by many scholars in different parts of the word, simultaneously or at about the same time, however, quite independently and often unaware of the work by other scholars. The formulation and rise of Switching Theory is such an example. This book presents a brief account of the developments of Switching Theory and highlights some less known facts in the history of it. The readers will find the book a fresh look into the development of the field revealing how difficult it has been to arrive at many of the concepts that we now consider obvious . Researchers in the history or philosophy of computing will find this book a valuable source of information that complements the standard presentations of the topic.

Nonlinear Filters for Image Processing Editors: Edward R. Dougherty, Texas A&M University Jaakko T. Astola, Tampere University of Technology Part of the SPIE/IEEE Series on Imaging Science & Engineering This text covers key mathematical principles and algorithms for nonlinear filters used in image processing. Readers will gain an in-depth understanding of the underlying mathematical and filter design methodologies needed to construct and utilize nonlinear filters in a variety of applications. The 11 chapters, written by experts in the field, explore topics of contemporary interest as well as fundamentals drawn from nonlinear filtering’s historical roots in mathematical morphology and digital signal processing. Linear filtering has dominated image processing, partly because the mathematical analysis is much easier than for nonlinear operators. However, nonlinear filters often yield superior results. This book explains in depth various filter options and the types of applications for which they are best suited. The presentation is rigorous, yet accessible to engineers with a solid background in mathematics. Contents: Logical image operators (E. R. Dougherty, J. Barrera). Computational gray-scale operators (E. R. Dougherty, J. Barrera). Translation-invariant set operators (E. R. Dougherty). Granulometric filters (E. R. Dougherty, Y. Chen). Easy recipes for morphological filters (H. J. A. M. Heijmans). Introduction to connected operators (H. J. A. M. Heijmans). Representation and optimization of stack filters (J. T. Astola, P. Kuosmanen). Invariant signals of median and stack filters (J. T. Astola, P. Kuosmanen). Binary polynomial transforms and logical correlation (K. O. Egiazarian, J. T. Astola, S. S. Agaian). Applications of binary polynomial transforms (K. O. Egiazarian, J. T. Astola, S. S. Agaian, R. Öktem). Random sets in view of image filtering applications (I. S. Molchanov).

Discover applications of Fourier analysis on finite non-Abelian groups The majority of publications in spectral techniques consider Fourier transform on Abelian groups. However, non-Abelian groups provide notable advantages in efficient implementations of spectral methods. Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design examines aspects of Fourier analysis on finite non-Abelian groups and discusses different methods used to determine compact representations for discrete functions providing for their efficient realizations and related applications. Switching functions are included as an example of discrete functions in engineering practice. Additionally, consideration is given to the polynomial expressions and decision diagrams defined in terms of Fourier transform on finite non-Abelian groups. A solid foundation of this complex topic is provided by beginning with a review of signals and their mathematical models and Fourier analysis. Next, the book examines recent achievements and discoveries in: Matrix interpretation of the fast Fourier transformOptimization of decision diagramsFunctional expressions on quaternion groupsGibbs derivatives on finite groupsLinear systems on finite non-Abelian groupsHilbert transform on finite groups Among the highlights is an in-depth coverage of applications of abstract harmonic analysis on finite non-Abelian groups in compact representations of discrete functions and related tasks in signal processing and system design, including logic design. All chapters are self-contained, each with a list of references to facilitate the development of specialized courses or self-study. With nearly 100 illustrative figures and fifty tables, this is an excellent textbook for graduate-level students and researchers in signal processing, logic design, and system theory-as well as the more general topics of computer science and applied mathematics.

This book discusses in a uniform way binary, ternary, and quaternary bent functions, while most of the existing books on bent functions refer to just binary bent functions. The authors describe the differences between binary and multiple-valued cases and the construction methods for bent functions are focused on the application of two types of permutation matrices. These matrices are derived from a class of differential operators on finite groups and Fast Fourier transform algorithms, respectively. The approach presented is based on the observation that given certain bent functions, many other bent functions can be constructed by manipulating them. Permutations are possible manipulations that are easy to implement. These permutations perform spectral invariant operations which ensure that they preserve bentness.

This book discusses in a uniform way binary, ternary, and quaternary bent functions, while most of the existing books on bent functions refer to just binary bent functions. The authors describe the differences between binary and multiple-valued cases and the construction methods for bent functions are focused on the application of two types of permutation matrices. These matrices are derived from a class of differential operators on finite groups and Fast Fourier transform algorithms, respectively. The approach presented is based on the observation that given certain bent functions, many other bent functions can be constructed by manipulating them. Permutations are possible manipulations that are easy to implement. These permutations perform spectral invariant operations which ensure that they preserve bentness.

Fundamentals of Nonlinear Digital Filtering is the first book of its kind, presenting and evaluating current methods and applications in nonlinear digital filtering. Written for professors, researchers, and application engineers, as well as for serious students of signal processing, this is the only book available that functions as both a reference handbook and a textbook. Solid introductory material, balanced coverage of theoretical and practical aspects, and dozens of examples provide you with a self-contained, comprehensive information source on nonlinear filtering and its applications.

Universal codes efficiently compress sequences generated by stationary and ergodic sources with unknown statistics, and they were originally designed for lossless data compression. In the meantime, it was realized that they can be used for solving important problems of prediction and statistical analysis of time series, and this book describes recent results in this area. The first chapter introduces and describes the application of universal codes to prediction and the statistical analysis of time series; the second chapter describes applications of selected statistical methods to cryptography, including attacks on block ciphers; and the third chapter describes a homogeneity test used to determine authorship of literary texts. The book will be useful for researchers and advanced students in information theory, mathematical statistics, time-series analysis, and cryptography. It is assumed that the reader has some grounding in statistics and in information theory.

Universal codes efficiently compress sequences generated by stationary and ergodic sources with unknown statistics, and they were originally designed for lossless data compression. In the meantime, it was realized that they can be used for solving important problems of prediction and statistical analysis of time series, and this book describes recent results in this area. The first chapter introduces and describes the application of universal codes to prediction and the statistical analysis of time series; the second chapter describes applications of selected statistical methods to cryptography, including attacks on block ciphers; and the third chapter describes a homogeneity test used to determine authorship of literary texts. The book will be useful for researchers and advanced students in information theory, mathematical statistics, time-series analysis, and cryptography. It is assumed that the reader has some grounding in statistics and in information theory.

Switching theory and logic design provide mathematical foundations and tools for digital system design that is an essential part in the research and development in almost all areas of modern technology. The vast complexity of modern digital systems implies that they can only be handled by computer aided design tools that are built on sophisticated mathematical models. Fundamentals of Switching Theory and Logic Design is aimed at providing an accessible introduction to these mathematical techniques that underlie the design tools and that are necessary for understanding their capabilities and limitations. As is typical to many disciplines a high level of abstraction enables a unified treatment of many methodologies and techniques as well as provides a deep understanding of the subject in general. The drawback is that without a hands-on touch on the details it is difficult to develop an intuitive understanding of the techniques. We try to combine these views by providing hands-on examples on the techniques while binding these to the more general theory that is developed in parallel. For instance, the use of vector spaces and group theory unifies the spectral (Fourier-like) interpretation of polynomial, and graphic (decision diagrams) representations of logic functions, as well as provides new methods for optimization of logic functions. Consequently, Fundamentals of Switching Theory and Logic Design discusses the fundamentals of switching theory and logic design from a slightly alternative point of view and also presents links between switching theory and related areas of signal processing and system theory. It also covers the core topics recommended in IEEE/ACM curricula for teaching and study in this area. Further, it contains several elective sections discussing topics for further research work in this area

Switching theory and logic design provide mathematical foundations and tools for digital system design that is an essential part in the research and development in almost all areas of modern technology. The vast complexity of modern digital systems implies that they can only be handled by computer aided design tools that are built on sophisticated mathematical models. Fundamentals of Switching Theory and Logic Design is aimed at providing an accessible introduction to these mathematical techniques that underlie the design tools and that are necessary for understanding their capabilities and limitations. As is typical to many disciplines a high level of abstraction enables a unified treatment of many methodologies and techniques as well as provides a deep understanding of the subject in general. The drawback is that without a hands-on touch on the details it is difficult to develop an intuitive understanding of the techniques. We try to combine these views by providing hands-on examples on the techniques while binding these to the more general theory that is developed in parallel. For instance, the use of vector spaces and group theory unifies the spectral (Fourier-like) interpretation of polynomial, and graphic (decision diagrams) representations of logic functions, as well as provides new methods for optimization of logic functions. Consequently, Fundamentals of Switching Theory and Logic Design discusses the fundamentals of switching theory and logic design from a slightly alternative point of view and also presents links between switching theory and related areas of signal processing and system theory. It also covers the core topics recommended in IEEE/ACM curricula for teaching and study in this area. Further, it contains several elective sections discussing topics for further research work in this area

Microarray technology provides researchers in the life sciences with a revolutionary tool for measuring gene expression. However, this highly developed process involves multiple steps, from sample selection to data analysis, each susceptible to potentially costly errors. Without sound quality control, experimental microarrays may produce useless or, even worse, misleading results. Microarray Quality Control provides a comprehensive resource for ensuring quality control in every step of this complex process. From experimental design to data processing, analysis, and interpretation, the emphasis in this text remains on practical advice for each stage of planning and running a microarray study. Chapters cover: *Quality of biological samples *Quality of DNA *Hybridization protocols Scanning *Data acquisition *Image analysis *Data analysis Written for the broad group of workers–biologists, mathematicians, statisticians, engineers, physicians, and computational scientists–involved in microarray studies, Microarray Quality Control features a straightforward style easily accessed by various disciplines. Useful checklists and tips help ensure the integrity of results, and each chapter contains a thorough review of pertinent literature. The only complete, systematic treatment of the topic available, Microarray Quality Control offers students and practitioners an invaluable resource for improving experimental quality and efficiency.

Fundamentals of Nonlinear Digital Filtering is the first book of its kind, presenting and evaluating current methods and applications in nonlinear digital filtering. Written for professors, researchers, and application engineers, as well as for serious students of signal processing, this is the only book available that functions as both a reference handbook and a textbook. Solid introductory material, balanced coverage of theoretical and practical aspects, and dozens of examples provide you with a self-contained, comprehensive information source on nonlinear filtering and its applications.