Geometric Inverse Problems With Emphasis on Two Dimensions

Description

This up-to-date treatment of recent developments in geometric inverse problems introduces graduate students and researchers to an exciting area of research. With an emphasis on the two-dimensional case, topics covered include geodesic X-ray transforms, boundary rigidity, tensor tomography, attenuated X-ray transforms and the Calderón problem. The presentation is self-contained and begins with the Radon transform and radial sound speeds as motivating examples. The required geometric background is developed in detail in the context of simple manifolds with boundary. An in-depth analysis of various geodesic X-ray transforms is carried out together with related uniqueness, stability, reconstruction and range characterization results. Highlights include a proof of boundary rigidity for simple surfaces as well as scattering rigidity for connections. The concluding chapter discusses current open problems and related topics. The numerous exercises and examples make this book an excellent self-study resource or text for a one-semester course or seminar. 1 The Radon Transform in the Plane 1-24 2 Radial Sound Speed 25-51 3 Geometric Preliminaries 52-106 4 The Geodesic X-ray Transform 107-129 5 Regularity Results for the Transport Equation 130-141 6 Vertical Fourier Analysis 142-170 7 The X-ray Transform in Non-positive Curvature 171-188 8 Microlocal Aspects, Surjectivity of I0* 189-207 9 Inversion Formulas and Range 208-232 10 Tensor Tomography 233-240 11 Boundary Rigidity 241-268 12 The Attenuated Geodesic X-ray Transform 269-276 13 Non-Abelian X-ray Transforms 277-303 14 Non-Abelian X-ray Transforms II 304-325 15 Open Problems and Related Topics 326-331 References 332-341 Index 342-344