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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.
The Radon Transform in the Plane
Radial Sound Speed
Geometric Preliminaries
The Geodesic X-ray Transform
Regularity Results for the Transport Equation
Vertical Fourier Analysis
The X-ray Transform in Non-positive Curvature
Microlocal Aspects, Surjectivity of I0*
Inversion Formulas and Range 2
Tensor Tomography
Boundary Rigidity
The Attenuated Geodesic X-ray Transform
Non-Abelian X-ray Transforms
Non-Abelian X-ray Transforms II
Open Problems and Related Topics
References
Index