Geometric Inverse Problems: With Emphasis on Two Dimensions

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.

這本關於幾何反問題最新發展的書籍，向研究生和研究人員介紹了一個令人興奮的研究領域。重點放在二維情況下，涵蓋的主題包括測地線X射線轉換、邊界剛性、張量斷層攝影、衰減X射線轉換和卡爾德隆問題。本書的介紹是自成一體的，並以Radon轉換和徑向聲速作為引例。所需的幾何背景在具有邊界的簡單流形的背景下詳細發展。對各種測地線X射線轉換進行了深入分析，並提供了相關的唯一性、穩定性、重建和範圍特徵結果。亮點包括對簡單曲面的邊界剛性證明以及對連接的散射剛性。結尾章節討論了當前的開放問題和相關主題。大量的練習和例子使本書成為一個優秀的自學資源，也適合作為一學期課程或研討會的教材。