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Nonlinear Inverse Problems in Imaging

ISBN: 9780470669426 | 047066942X

Edition: 1st
Format: Hardcover
Publisher: Wiley
Pub. Date: 1/4/2013

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Summary Table of Contents

This book provides researchers and engineers in the imaging field with the skills they need to effectively deal with nonlinear inverse problems associated with different imaging modalities, including impedance imaging, optical tomography, elastography, and electrical source imaging. Focusing on numerically implementable methods, the book bridges the gap between theory and applications, helping readers tackle problems in applied mathematics and engineering. Complete, self-contained coverage includes basic concepts, models, computational methods,... MORE numerical simulations, examples, and case studies. Provides a step-by-step progressive treatment of topics for ease of understanding. Discusses the underlying physical phenomena as well as implementation details of image reconstruction algorithms as prerequisites for finding solutions to non linear inverse problems with practical significance and value. Includes end of chapter problems, case studies and examples with solutions throughout the book. Companion website will provide further examples and solutions, experimental data sets, open problems, teaching material such as PowerPoint slides and software including MATLAB m files. Essential reading for Graduate students and researchers in imaging science working across the areas of applied mathematics, biomedical engineering, and electrical engineering and specifically those involved in nonlinear imaging techniques, impedance imaging, optical tomography, elastography, and electrical source imaging

Imaging techniques in science, engineering and medicine have evolved to expand our ability to visualize the electrical and mechanical properties of materials inside an object in the form of a cross-sectional image. There have been numerous research efforts to develop new techniques which produce additional contrast information unavailable from existing methods. This book discusses some of these latest imaging modalities: electrical impedance tomography (EIT), magnetic resonance electrical impedance tomography (MREIT) and magnetic resonance elastography (MRE). These imaging techniques all manifest some nonlinearity and this book focuses on their associated nonlinear inverse problems. The authors explain the mathematical and numerical tools needed to solve these problems in order to better visualize the electrical and mechanical properties inside a given object through these imaging techniques. Key features: Provides an understanding of the underlying physical phenomena and implementation details of image reconstruction algorithms as prerequisites for finding solutions with practical significance and value. Offers a self-contained treatment of the topic taking the reader from basic concepts, the mathematical background, computational methods and numerical simulations through to solved examples, practical limitations and case studies. Describes forward problems and their mathematical formulations as they are often the processes of information loss and provide strategic insights in seeking solutions to nonlinear inverse problems. Focuses on methods rather than applications. Includes an accompanying website hosted by the authors containing lecture notes, computer programs and experimental data.

Preface

List of Abbreviations

1 Introduction

1.1 Forward Problem

1.2 Inverse Problem

1.3 Issues in Inverse Problem Solving

1.4 Linear, Nonlinear and Linearized Problems

2 Signal and System as Vectors

2.1 Vector Space

2.1.1 Vector Space and Subspace

2.1.2 Basis, Norm and Inner Product

2.1.3 Hilbert Space

2.2 Vector Calculus

2.2.1 Gradient

2.2.2 Divergence

2.2.3 Curl

2.2.4 Curve

2.2.5 Curvature

2.3 Taylor’s Expansion

... MORE

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