CRC Press, 2015. — 242 p.The objective of this book is to present an overview of the challenging problem of determining information about an object from measurements of the field scattered from that object. This problem is a very old one, since, in a fundamental sense, most of what we perceive and learn about objects around us is a result of electromagnetic or acoustic waves impinging on, interacting with and scattering from those objects. The theoretical formalism of a scattering problem is increasingly complex, as the extent of the interactions increase between the fields with the object. The forward or direct problem generally demands a good model for the anticipated response of the object. Deducing information about the object generally demands knowledge of that model or that (acceptable) approximations can be made to simplify matters. Theoretical approaches to solving inverse problems have been widely studied and as a broad class of problems are known to suffer from concerns over lack of uniqueness and solution stability (ill-conditioning) but, despite modeling a physically well-defined problem, could also be formulated in a way that the very existence of a solution is questionable. In the specific context of inverse scattering theories and algorithms, we present in this text an overview of some of the more widely used approaches to recover information about objects. We consider both the assumptions made a priori about the object as well as the consequences of having to recover object information from limited numbers of noisy measurements of the scattered fields. There is a wealth of literature dealing with scattering and inverse scattering methods for relatively simple structures embedded in a homogeneous background. We introduce the terminology and concepts early in the text and review some important inverse methods. When the scattering is assumed to be “weak,” which we define in the text, inversion methods allow more straightforward inverse algorithms to be exploited. We highlight the consequences of the widespread practice of adopting such methods when they are not justified while recognizing their attractiveness from a practical implementation point of view. Assuming weak scattering allows many well-established techniques developed in Fourier-based signal and image processing to be incorporated. The weak scattering models facilitate a simple mapping of scattered field data onto a locus of points in the Fourier domain of the object of interest. More rigorous scattering methods that rely on iterative techniques or strong prior knowledge of the forward scattering model are often slow to implement and may not yield reliable information. We have organized this book with graduate students and those practicing imaging from scattered fields in mind. This includes, for example, those working with medical, geophysical, defense, and industrial inspection inverse problems. It will be helpful for readers to have an understanding of basic electromagnetic principles, some background in calculus and Fourier analysis, and preferably familiarity with MATLAB (and possibly COMSOL) in order to take advantage of the source code provided. The text is self-contained and gives the required background theory to be able to design improved experiments and process measured data more effectively, to recover for a strongly scattering object an estimate that is not perfect, but probably the best that one can hope for from limited scattered field data.Section I Fundamentals Introduction to Inverse Scattering Electromagnetic Waves Scattering Fundamentals Inverse Scattering Fundamentals Section II Inversion Methods Data Processing Born Approximation Observations Alternate Inverse Methods Homomorphic (Cepstral) Filtering Section III Applications Applications to Real Measured Data Advanced Cepstral Filtering Advanced Topics in Inverse Imaging Section IV Appendices A: Review of Fourier Analysis Appendix B: The Phase Retrieval Problem C: Prior Discrete Fourier Transform D: The Poynting Vector E: Resolution and Degrees of Freedom . F: MATLAB Exercises with COMSOL Data
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