Proper analysis of image and multimedia data requires efficient extraction and segmentation techniques. Among the many computational intelligence approaches, the soft computing paradigm is best equipped with several tools and techniques that incorporate intelligent concepts and principles. This book is dedicated to object extraction, image segmentation, and edge detection using soft computing techniques with extensive real-life application to image and multimedia data. The authors start with a comprehensive tutorial on the basics of brain structure and learning, and then the key soft computing techniques, including evolutionary computation, neural networks, fuzzy sets and fuzzy logic, and rough sets. They then present seven chapters that detail the application of representative techniques to complex image processing tasks such as image recognition, lighting control, target tracking, object extraction, and edge detection. These chapters follow a structured approach with detailed explanations of the problems, solutions, results, and conclusions. This is both a standalone textbook for graduates in computer science, electrical engineering, system science, and information technology, and a reference for researchers and engineers engaged with pattern recognition, image processing, and soft computing.

An Image Processing Tour of College Mathematics aims to provide meaningful context for reviewing key topics of the college mathematics curriculum, to help students gain confidence in using concepts and techniques of applied mathematics, to increase student awareness of recent developments in mathematical sciences, and to help students prepare for graduate studies. The topics covered include a library of elementary functions, basic concepts of descriptive statistics, probability distributions of functions of random variables, definitions and concepts behind first- and second-order derivatives, most concepts and techniques of traditional linear algebra courses, an introduction to Fourier analysis, and a variety of discrete wavelet transforms - all of that in the context of digital image processing. Features Pre-calculus material and basic concepts of descriptive statistics are reviewed in the context of image processing in the spatial domain. Key concepts of linear algebra are reviewed both in the context of fundamental operations with digital images and in the more advanced context of discrete wavelet transforms. Some of the key concepts of probability theory are reviewed in the context of image equalization and histogram matching. The convolution operation is introduced painlessly and naturally in the context of naive filtering for denoising and is subsequently used for edge detection and image restoration. An accessible elementary introduction to Fourier analysis is provided in the context of image restoration. Discrete wavelet transforms are introduced in the context of image compression, and the readers become more aware of some of the recent developments in applied mathematics. This text helps students of mathematics ease their way into mastering the basics of scientific computer programming.

In view of better results expected from examination of medical datasets (images) with hybrid (integration of thresholding and segmentation) image processing methods, this work focuses on implementation of possible hybrid image examination techniques for medical images. It describes various image thresholding and segmentation methods which are essential for the development of such a hybrid processing tool. Further, this book presents the essential details, such as test image preparation, implementation of a chosen thresholding operation, evaluation of threshold image, and implementation of segmentation procedure and its evaluation, supported by pertinent case studies. Aimed at researchers/graduate students in the medical image processing domain, image processing, and computer engineering, this book: Provides broad background on various image thresholding and segmentation techniques Discusses information on various assessment metrics and the confusion matrix Proposes integration of the thresholding technique with the bio-inspired algorithms Explores case studies including MRI, CT, dermoscopy, and ultrasound images Includes separate chapters on machine learning and deep learning for medical image processing

The aim of this book is to outline the concept of entropy, various types of entropies and their implementation to evaluate a variety of biomedical signals/images. The book emphasizes various entropy-based image pre-processing methods which are essential for the development of suitable computerized examination systems. The recent research works on biomedical signal evaluation confirms that signal analysis provides vital information regarding the physiological condition of the patient, and the efficient evaluation of these signals can help to diagnose the nature and the severity of the disease. This book emphasizes various entropy-based image pre-processing methods which are essential for the development of suitable computerized examination systems for the analysis of biomedical images recorded with a variety of modalities. The work discusses the image pro-processing methods with the Entropies, such as Kapur, Tsallis, Shannon and Fuzzy on a class of RGB-scaled and gray-scaled medical pictures. The performance of the proposed technique is justified with the help of suitable case studies, which involves x-ray image analysis, MRI analysis and CT analysis. This book is intended for medical signal/image analysts, undergraduate and postgraduate students, researchers, and medical scientists interested in biomedical data evaluation.

The combination of high-resolution visual and depth sensing, supported by machine learning, opens up new opportunities to solve real-world problems in computer vision.

This authoritative text/reference presents an interdisciplinary selection of important, cutting-edge research on RGB-D based computer vision. Divided into four sections, the book opens with a detailed survey of the field, followed by a focused examination of RGB-D based 3D reconstruction, mapping and synthesis. The work continues with a section devoted to novel techniques that employ depth data for object detection, segmentation and tracking, and concludes with examples of accurate human action interpretation aided by depth sensors.

Topics and features: discusses the calibration of color and depth cameras, the reduction of noise on depth maps, and methods for capturing human performance in 3D; reviews a selection of applications which use RGB-D information to reconstruct human figures, evaluate energy consumption, and obtain accurate action classification; presents an innovative approach for 3D object retrieval, and for the reconstruction of gas flow from multiple Kinect cameras; describes an RGB-D computer vision system designed to assist the visually impaired, and another for smart-environment sensing to assist elderly and disabled people; examines the effective features that characterize static hand poses, and introduces a unified framework to enforce both temporal and spatial constraints for hand parsing; proposes a new classifier architecture for real-time hand pose recognition, and a novel hand segmentation and gesture recognition system.

Researchers and practitioners working in computer vision, HCI and machine learning will find this to be a must-read text. The book also serves as a useful reference for graduate students studying computer vision, pattern recognition or multimedia.

Long used in undergraduate and introductory graduate courses, Astrophysical Techniques, Seventh Edition provides an accessible yet comprehensive account of the innovate instruments, detectors, and techniques employed in astronomy and astrophysics. Emphasizing the underlying unity of all astronomical observations, this popular textbook provides a coherent state-of-the-art account of the instruments and techniques used in current astronomy and astrophysics. Fully updated throughout, this seventh edition builds upon the sixth edition, covering improved techniques and cutting-edge methods in the field, as well as other exciting new developments in gravitational waves, dark matter and energy, the use of photonics, and astronomy education and outreach, in addition to further detailed discussions on the latest scientific instruments and individual detectors. The book is written in a very accessible manner, and most of the mathematics is accessible to those who have attended a mathematics course in their final years at school. Nevertheless, the treatment of the topics in general is at a sufficiently high level to be of use to those professionals seeking technical information in areas of astronomy with which they might not be completely familiar. Key Features: Details the instrumentation and theory of astronomical observations, including radio waves, gamma rays, cosmic rays, neutrinos, gravitational waves and dark matter and energy and more Presents the background theory and operating practice of state-of-the-art detectors and instruments Fully updated to contain the latest technology and research developments

With the ubiquitous use of digital imaging, a new profession has emerged: imaging engineering. Designed for newcomers to imaging science and engineering, Theoretical Foundations of Digital Imaging Using MATLAB (R) treats the theory of digital imaging as a specific branch of science. It covers the subject in its entirety, from image formation to image perfecting. Based on the author's 50 years of working and teaching in the field, the text first addresses the problem of converting images into digital signals that can be stored, transmitted, and processed on digital computers. It then explains how to adequately represent image transformations on computers. After presenting several examples of computational imaging, including numerical reconstruction of holograms and virtual image formation through computer-generated display holograms, the author introduces methods for image perfect resampling and building continuous image models. He also examines the fundamental problem of the optimal estimation of image parameters, such as how to localize targets in images. The book concludes with a comprehensive discussion of linear and nonlinear filtering methods for image perfecting and enhancement. Helping you master digital imaging, this book presents a unified theoretical basis for understanding and designing methods of imaging and image processing. To facilitate a deeper understanding of the major results, it offers a number of exercises supported by MATLAB programs, with the code available at www.crcpress.com.

Principles of Synthetic Aperture Radar Imaging: A System Simulation Approach demonstrates the use of image simulation for SAR. It covers the various applications of SAR (including feature extraction, target classification, and change detection), provides a complete understanding of SAR principles, and illustrates the complete chain of a SAR operation. The book places special emphasis on a ground-based SAR, but also explains space and air-borne systems. It contains chapters on signal speckle, radar-signal models, sensor-trajectory models, SAR-image focusing, platform-motion compensation, and microwave-scattering from random media. While discussing SAR image focusing and motion compensation, it presents processing algorithms and applications that feature extraction, target classification, and change detection. It also provides samples of simulation on various scenarios, and includes simulation flowcharts and results that are detailed throughout the book. Introducing SAR imaging from a systems point of view, the author: Considers the recent development of MIMO SAR technology Includes selected GPU implementation Provides a numerical analysis of system parameters (including platforms, sensor, and image focusing, and their influence) Explores wave-target interactions, signal transmission and reception, image formation, motion compensation Covers all platform motion compensation and error analysis, and their impact on final image radiometric and geometric quality Describes a ground-based SFMCW system Principles of Synthetic Aperture Radar Imaging: A System Simulation Approach is dedicated to the use, study, and development of SAR systems. The book focuses on image formation or focusing, treats platform motion and image focusing, and is suitable for students, radar engineers, and micr

This book intends to provide a comprehensive overview on different aspects of mechanisms and techniques for information security. It is written for students, researchers, and professionals studying in the field of multimedia security and steganography. Multimedia security and steganography is especially relevant due to the global scale of digital multimedia and the rapid growth of the Internet. Digital watermarking technology can be used to guarantee authenticity and can be applied as proof that the content has not been altered since insertion. Updated techniques and advances in watermarking are explored in this new edition. The combinational spatial and frequency domains watermarking technique provides a new concept of enlarging the embedding capacity of watermarks. The genetic algorithm (GA) based watermarking technique solves the rounding error problem and provide an efficient embedding approach. Each chapter provides the reader with a fundamental, theoretical framework, while developing the extensive advanced techniques and considering the essential principles of the digital watermarking and steganographic systems. Several robust algorithms that are presented throughout illustrate the framework and provide assistance and tools in understanding and implementing the fundamental principles.

Inspection is crucial to the management of ageing infrastructure. Visual information on structures is regularly collected but very little work exists on its organised and quantitative analysis, even though image processing can significantly enhance these inspection processes and transfer real financial and safety benefits to the managers, owners and users. Additionally, new opportunities exist in the fast evolving sectors of wind and wave energy to add value to image-based inspection techniques. This book is a first for structural engineers and inspectors who wish to harness the full potential of cameras as an inspection tool. It is particularly directed to the inspection of offshore and marine structures and the application of image-based methods in underwater inspections. It outlines a set of best practice guidelines for obtaining imagery, then the fundamentals of image processing are covered along with several image processing techniques which can be used to assess multiple damage forms: crack detection, corrosion detection, and depth analysis of marine growth on offshore structures. The book provides benchmark performance measures for these techniques under various visibility conditions using an image repository which will help inspectors to envisage the effectiveness of the techniques when applied. MATLAB (R) scripts and access to the underwater image repository are included so readers can run these techniques themselves. Practising engineers and managers of infrastructure assets are guided in image processing based inspection. Researchers can use this book as a primer, and it also suits advanced graduate courses in infrastructure management or on applied image processing.

Long used in undergraduate and introductory graduate courses, Astrophysical Techniques, Seventh Edition provides an accessible yet comprehensive account of the innovate instruments, detectors, and techniques employed in astronomy and astrophysics. Emphasizing the underlying unity of all astronomical observations, this popular textbook provides a coherent state-of-the-art account of the instruments and techniques used in current astronomy and astrophysics. Fully updated throughout, this seventh edition builds upon the sixth edition, covering improved techniques and cutting-edge methods in the field, as well as other exciting new developments in gravitational waves, dark matter and energy, the use of photonics, and astronomy education and outreach, in addition to further detailed discussions on the latest scientific instruments and individual detectors. The book is written in a very accessible manner, and most of the mathematics is accessible to those who have attended a mathematics course in their final years at school. Nevertheless, the treatment of the topics in general is at a sufficiently high level to be of use to those professionals seeking technical information in areas of astronomy with which they might not be completely familiar. Key Features: Details the instrumentation and theory of astronomical observations, including radio waves, gamma rays, cosmic rays, neutrinos, gravitational waves and dark matter and energy and more Presents the background theory and operating practice of state-of-the-art detectors and instruments Fully updated to contain the latest technology and research developments

In today's world, deep learning source codes and a plethora of open access geospatial images are readily available and easily accessible. However, most people are missing the educational tools to make use of this resource. Deep Learning for Remote Sensing Images with Open Source Software is the first practical book to introduce deep learning techniques using free open source tools for processing real world remote sensing images. The approaches detailed in this book are generic and can be adapted to suit many different applications for remote sensing image processing, including landcover mapping, forestry, urban studies, disaster mapping, image restoration, etc. Written with practitioners and students in mind, this book helps link together the theory and practical use of existing tools and data to apply deep learning techniques on remote sensing images and data. Specific Features of this Book: The first book that explains how to apply deep learning techniques to public, free available data (Spot-7 and Sentinel-2 images, OpenStreetMap vector data), using open source software (QGIS, Orfeo ToolBox, TensorFlow) Presents approaches suited for real world images and data targeting large scale processing and GIS applications Introduces state of the art deep learning architecture families that can be applied to remote sensing world, mainly for landcover mapping, but also for generic approaches (e.g. image restoration) Suited for deep learning beginners and readers with some GIS knowledge. No coding knowledge is required to learn practical skills. Includes deep learning techniques through many step by step remote sensing data processing exercises.

Find the right algorithm for your image processing application Exploring the recent achievements that have occurred since the mid-1990s, Circular and Linear Regression: Fitting Circles and Lines by Least Squares explains how to use modern algorithms to fit geometric contours (circles and circular arcs) to observed data in image processing and computer vision. The author covers all facets-geometric, statistical, and computational-of the methods. He looks at how the numerical algorithms relate to one another through underlying ideas, compares the strengths and weaknesses of each algorithm, and illustrates how to combine the algorithms to achieve the best performance. After introducing errors-in-variables (EIV) regression analysis and its history, the book summarizes the solution of the linear EIV problem and highlights its main geometric and statistical properties. It next describes the theory of fitting circles by least squares, before focusing on practical geometric and algebraic circle fitting methods. The text then covers the statistical analysis of curve and circle fitting methods. The last chapter presents a sample of "exotic" circle fits, including some mathematically sophisticated procedures that use complex numbers and conformal mappings of the complex plane. Essential for understanding the advantages and limitations of the practical schemes, this book thoroughly addresses the theoretical aspects of the fitting problem. It also identifies obscure issues that may be relevant in future research.

Image Processing and Acquisition using Python provides readers with a sound foundation in both image acquisition and image processing-one of the first books to integrate these topics together. By improving readers' knowledge of image acquisition techniques and corresponding image processing, the book will help them perform experiments more effectively and cost efficiently as well as analyze and measure more accurately. Long recognized as one of the easiest languages for non-programmers to learn, Python is used in a variety of practical examples. A refresher for more experienced readers, the first part of the book presents an introduction to Python, Python modules, reading and writing images using Python, and an introduction to images. The second part discusses the basics of image processing, including pre/post processing using filters, segmentation, morphological operations, and measurements. The second part describes image acquisition using various modalities, such as x-ray, CT, MRI, light microscopy, and electron microscopy. These modalities encompass most of the common image acquisition methods currently used by researchers in academia and industry. Features Covers both the physical methods of obtaining images and the analytical processing methods required to understand the science behind the images. Contains many examples, detailed derivations, and working Python examples of the techniques. Offers practical tips on image acquisition and processing. Includes numerous exercises to test the reader's skills in Python programming and image processing, with solutions to selected problems, example programs, and images available on the book's web page. New to this edition Machine learning has become an indispensable part of image processing and computer vision, so in this new edition two new chapters are included: one on neural networks and the other on convolutional neural networks. A new chapter on affine transform and many new algorithms. Updated Python code aligned to the latest version of modules.

Over the past decade, the field of image processing has made tremendous advances. One type of image processing that is currently of particular interest is "tomographic imaging," a technique for computing the density function of a body, or discontinuity surfaces of this function. Today, tomography is widely used, and has applications in such fields as medicine, engineering, physics, geophysics, and security. The Radon Transform and Local Tomography clearly explains the theoretical, computational, and practical aspects of applied tomography. It includes sufficient background information to make it essentially self-contained for most readers.

Hyperspectral Imaging: Techniques for Spectral Detection and Classification is an outgrowth of the research conducted over the years in the Remote Sensing Signal and Image Processing Laboratory (RSSIPL) at the University of Maryland, Baltimore County. It explores applications of statistical signal processing to hyperspectral imaging and further develops non-literal (spectral) techniques for subpixel detection and mixed pixel classification. This text is the first of its kind on the topic and can be considered a recipe book offering various techniques for hyperspectral data exploitation. In particular, some known techniques, such as OSP (Orthogonal Subspace Projection) and CEM (Constrained Energy Minimization) that were previously developed in the RSSIPL, are discussed in great detail. This book is self-contained and can serve as a valuable and useful reference for researchers in academia and practitioners in government and industry.

This much-needed text brings the treatment of optical pattern recognition up-to-date in one comprehensive resource. Optical pattern recognition, one of the first implementations of Fourier Optics, is now widely used, and this text provides an accessible introduction for readers who wish to get to grips with how holography is applied in a practical context. A wide range of devices are addressed from a user perspective and are accompanied with detailed tables enabling performance comparison, in addition to chapters exploring computer-generated holograms, optical correlator systems, and pattern matching algorithms. This book will appeal to both lecturers and research scientists in the field of electro-optic devices and systems. Features: Covers a range of new developments, including computer-generated holography and 3D image recognition Accessible without a range of prior knowledge, providing a clear exposition of technically difficult concepts Contains extensive examples throughout to reinforce learning

Automatic biometrics recognition techniques are increasingly important in corporate and public security systems and have increased in methods due to rapid field development. This book discusses classic behavioral biometrics as well as collects the latest advances in techniques, theoretical approaches, and dynamic applications. This future-looking book is an important reference tool for researchers, practitioners, academicians, and technologists. While there are existing books that focus on physiological biometrics or algorithmic approaches deployed in biometrics, this book addresses a gap in the existing literature for a text that is solely dedicated to the topic of behavioral biometrics.

Handbook of Robust Low-Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing shows you how robust subspace learning and tracking by decomposition into low-rank and sparse matrices provide a suitable framework for computer vision applications. Incorporating both existing and new ideas, the book conveniently gives you one-stop access to a number of different decompositions, algorithms, implementations, and benchmarking techniques. Divided into five parts, the book begins with an overall introduction to robust principal component analysis (PCA) via decomposition into low-rank and sparse matrices. The second part addresses robust matrix factorization/completion problems while the third part focuses on robust online subspace estimation, learning, and tracking. Covering applications in image and video processing, the fourth part discusses image analysis, image denoising, motion saliency detection, video coding, key frame extraction, and hyperspectral video processing. The final part presents resources and applications in background/foreground separation for video surveillance. With contributions from leading teams around the world, this handbook provides a complete overview of the concepts, theories, algorithms, and applications related to robust low-rank and sparse matrix decompositions. It is designed for researchers, developers, and graduate students in computer vision, image and video processing, real-time architecture, machine learning, and data mining.

Remote sensing of impervious surfaces has matured using advances in geospatial technology so recent that its applications have received only sporadic coverage in remote sensing literature. Remote Sensing of Impervious Surfaces is the first to focus entirely on this developing field. It provides detailed coverage of mapping, data extraction, and modeling techniques specific to analyzing impervious surfaces, such as roads and buildings. Written by renowned experts in the field, this book reviews the major approaches that apply to this emerging field as well as current challenges, developments, and trends. The authors introduce remote sensing digital image processing techniques for estimating and mapping impervious surfaces in urban and rural areas. Presenting the latest modeling tools and algorithms for data extraction and analysis, the book explains how to differentiate roads, roofs, and other manmade structures from remotely sensed images for individual analysis. The final chapters examine how to use impervious surface data for predicting the flow of storm- or floodwater and studying trends in population, land use, resource distribution, and other real-world applications in environmental, urban, and regional planning. Each chapter offers a consistent format including a concise review of basic concepts and methodologies, timely case studies, and guidance for solving problems and analyzing data using the techniques presented.

This volume gathers papers presented at the Workshop on Computational Diffusion MRI (CDMRI'18), which was held under the auspices of the International Conference on Medical Image Computing and Computer Assisted Intervention in Granada, Spain on September 20, 2018. It presents the latest developments in the highly active and rapidly growing field of diffusion MRI. The reader will find papers on a broad range of topics, from the mathematical foundations of the diffusion process and signal generation, to new computational methods and estimation techniques for the in-vivo recovery of microstructural and connectivity features, as well as harmonisation and frontline applications in research and clinical practice. The respective papers constitute invited works from high-profile researchers with a specific focus on three topics that are now gaining momentum within the diffusion MRI community: i) machine learning for diffusion MRI; ii) diffusion MRI outside the brain (e.g. in the placenta); and iii) diffusion MRI for multimodal imaging. The book shares new perspectives on the latest research challenges for those currently working in the field, but also offers a valuable starting point for anyone interested in learning computational techniques in diffusion MRI. It includes rigorous mathematical derivations, a wealth of full-colour visualisations, and clinically relevant results. As such, it will be of interest to researchers and practitioners in the fields of computer science, MRI physics and applied mathematics alike.

Medical image analysis using advanced fuzzy set theoretic techniques is an exciting and dynamic branch of image processing. Since the introduction of fuzzy set theory, there has been an explosion of interest in advanced fuzzy set theories-such as intuitionistic fuzzy and Type II fuzzy set-that represent uncertainty in a better way. Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques deals with the application of intuitionistic fuzzy and Type II fuzzy set theories for medical image analysis. Designed for graduate and doctorate students, this higher-level text: Provides a brief introduction to advanced fuzzy set theory, fuzzy/intuitionistic fuzzy aggregation operators, and distance/similarity measures Covers medical image enhancement using advanced fuzzy sets, including MATLAB (R)-based examples to increase contrast of the images Describes intuitionistic fuzzy and Type II fuzzy thresholding techniques that separate different regions/leukocyte types/abnormal lesions Demonstrates the clustering of unwanted lesions/regions even in the presence of noise by applying intuitionistic fuzzy clustering Highlights the edges of poorly illuminated images and uses intuitionistic fuzzy edge detection to find the edges of different regions Defines fuzzy mathematical morphology and explores its application using the Lukasiewicz operator, t-norms, and t-conorms Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques is useful not only for students, but also for teachers, engineers, scientists, and those interested in the field of medical image analysis. A basic knowledge of fuzzy set is required, along with a solid understanding of mathematics and image processing.

This book provides insights into the benefits of using remote sensing data from a geoscientist's perspective, by integrating the data with the understanding of Earth's surface and subsurface. In 3 sections, the book takes a detailed look at what data explorationists use when they explore for hydrocarbon resources, assess different terrain types for planning and hazards and extract present-day geologic analogs for subsurface geologic settings. The book presents the usage of remote sensing data in exploration in a structured way by detecting individual geologic features as building blocks for complex geologic systems. This concept enables readers to build their own workflows for the assessment of complex geologic systems using various combinations of remote sensing data. Section 1 introduces readers to the foundations of remote sensing for exploration, covers various methods of image processing and studies different digital elevation and bathymetry models. Section 2 presents the concept of geomorphology as a means to integrate surface and subsurface data. Different aspects of rendering in 2D and 3D are explained and used for the interpretation and extraction of geologic features that are used in exploration. Section 3 addresses remote sensing for hydrocarbon exploration in detail, from geophysical data acquisition to development and infrastructure planning. The organization of this chapter follows an exploration workflow from regional to local modeling studying basin and petroleum system modeling as well as logistics planning of seismic surveys and near-surface modeling. Aspects of field development and infrastructure planning comprise multi-temporal and dynamic modeling. The section closes with a structured approach to extracting geologic analogs from interpreted remote sensing data. The book will be of interest to professionals and students working in exploration for hydrocarbons and water resources, as well as geoscientists and engineers using remote sensing for infrastructure planning, hazard assessment and dynamic environmental studies.

This book lays out all the latest research in the area of multimedia data hiding. The book introduces multimedia signal processing and information hiding techniques. It includes multimedia representation, digital watermarking fundamentals and requirements of watermarking. It moves on to cover the recent advances in multimedia signal processing, before presenting information hiding techniques including steganography, secret sharing and watermarking. The final part of this book includes practical applications of intelligent multimedia signal processing and data hiding systems.

Goals of the Book Overthelast thirty yearsthere has been arevolutionindiagnostic radiology as a result oftheemergenceofcomputerized tomography (CT), which is the process of obtaining the density distribution within the human body from multiple x-ray projections. Since an enormous variety of possible density values may occur in the body, a large number of projections are necessary to ensure the accurate reconstruction oftheir distribution. There are other situations in which we desire to reconstruct an object from its projections, but in which we know that the object to be recon structed has only a small number of possible values. For example, a large fraction of objects scanned in industrial CT (for the purpose of nonde structive testing or reverse engineering) are made of a single material and so the ideal reconstruction should contain only two values: zero for air and the value associated with the material composing the object. Similar as sumptions may even be made for some specific medical applications; for example, in angiography ofthe heart chambers the value is either zero (in dicating the absence of dye) or the value associated with the dye in the chamber. Another example arises in the electron microscopy of biological macromolecules, where we may assume that the object to be reconstructed is composed of ice, protein, and RNA. One can also apply electron mi croscopy to determine the presenceor absence ofatoms in crystallinestruc tures, which is again a two-valued situation."