Preface

Imaging techniques in science, engineering and medicine have evolved to expand our ability to visualize the internal information in an object such as the human body. Examples may include X-ray computed tomography (CT), magnetic resonance imaging (MRI), ultrasound imaging and positron emission tomography (PET). They provide cross-sectional images of the human body, which are solutions of corresponding inverse problems. Information embedded in such an image depends on the underlying physical principle, which is described in its forward problem. Since each imaging modality has limited viewing capability, there have been numerous research efforts to develop new techniques producing additional contrast information not available from existing methods.

There are such imaging techniques of practical significance, which can be formulated as nonlinear inverse problems. Electrical impedance tomography (EIT), magnetic induction tomography (MIT), diffuse optical tomography (DOT), magnetic resonance electrical impedance tomography (MREIT), magnetic resonance electrical property tomography (MREPT), magnetic resonance elastography (MRE), electrical source imaging and others have been developed and adopted in application areas where new contrast information is in demand. Unlike X-ray CT, MRI and PET, they manifest some nonlinearity, which result in their image reconstruction processes being represented by nonlinear inverse problems.

Visualizing new contrast information on the electrical, optical and mechanical properties of materials inside an object will widen the applications of imaging methods in medicine, biotechnology, non-destructive testing, geophysical exploration, monitoring of industrial processes and other areas. Some are advantageous in terms of non-invasiveness, portability, convenience of use, high temporal resolution, choice of dimensional scale and total cost. Others may offer a higher spatial resolution, sacrificing some of these merits.

Owing primarily to nonlinearity and low sensitivity, in addition to the lack of sufficient information to solve an inverse problem in general, these nonlinear inverse problems share the technical difficulties of ill-posedness, which may result in images with a low spatial resolution. Deep understanding of the underlying physical phenomena as well as the implementation details of image reconstruction algorithms are prerequisites for finding solutions with practical significance and value.

Research outcomes during the past three decades have accumulated enough knowledge and experience that we can deal with these topics in graduate programs of applied mathematics and engineering. This book covers nonlinear inverse problems associated with some of these imaging modalities. It focuses on methods rather than applications. The methods mainly comprise mathematical and numerical tools to solve the problems. Instrumentation will be treated only in enough detail to describe practical limitations imposed by measurement methods.

Readers will acquire the diverse knowledge and skills needed to deal effectively with nonlinear inverse problems in imaging by following the steps below.

1. Understand the underlying physical phenomena and the constraints imposed on the problem, which may enable solutions of nonlinear inverse problems to be improved. Physics, chemistry and also biology play crucial roles here. No attempt is made to be comprehensive in terms of physics, chemistry and biology.
2. Understand forward problems, which usually are the processes of information loss. They provide strategic insights into seeking solutions of nonlinear inverse problems. The underlying principles are described here so that readers can understand their mathematical formulations.
3. Formulate forward problems in such a way that they can be dealt with systematically and quantitatively.
4. Understand how to probe the imaging object and what is measurable using available engineering techniques. Practical limitations associated with the measurement sensitivity and specificity, such as noise, artifacts, interface between target object and instrument, data acquisition time and so on, must be properly understood and analyzed.
5. Understand what is feasible in a specific nonlinear inverse problem.
6. Formulate proper nonlinear inverse problems by defining the image contrast associated with physical quantities. Mathematical formulations should include any interrelation between those qualities and measurable data.
7. Construct inversion methods to produce images of contrast information.
8. Develop computer programs and properly address critical issues of numerical analysis.
9. Customize the inversion process by including a priori information.
10. Validate results by simulations and experiments.

This book is for advanced graduate courses in applied mathematics and engineering. Prerequisites for students with a mathematical background are vector calculus, linear algebra, partial differential equations and numerical analysis. For students with an engineering background, we recommend taking linear algebra, numerical analysis, electromagnetism, signal and system and also preferably instrumentation.

Lecture notes, sample codes, experimental data and other teaching material are available at http://mimaging.yonsei.ac.kr/NIPI.

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