Geometric Inverse Problems: With Emphasis on Two Dimensions

This book provides an introduction to geometric inverse problems, focusing on two-dimensional cases. It covers topics such as geodesic X-ray transforms, boundary rigidity, tensor tomography, attenuated X-ray transforms, and the Calderón problem. It includes exercises and examples for self-study and is suitable for a one-semester course or seminar.

Format: Hardback
Length: 350 pages
Publication date: 05 January 2023
Publisher: Cambridge University Press

This comprehensive and up-to-date treatment of recent advancements in geometric inverse problems offers a captivating introduction to graduate students and researchers in the field. Focusing on the two-dimensional case, the book delves into a wide range of topics, including geodesic X-ray transforms, boundary rigidity, tensor tomography, attenuated X-ray transforms, and the Calderón problem. The presentation is self-contained, beginning with motivating examples such as the Radon transform and radial sound speeds. Geometric background is developed in depth, specifically within the context of simple manifolds with boundary. An in-depth analysis of various geodesic X-ray transforms is conducted, along with related uniqueness, stability, reconstruction, and range characterization results. Highlights include a proof of boundary rigidity for simple surfaces and scattering rigidity for connections. The concluding chapter explores current open problems and related topics, making this book an invaluable resource for self-study or as a text for a one-semester course or seminar.

Geometric Inverse Problems: Recent Developments and Applications

In recent years, geometric inverse problems have gained significant attention in various scientific and engineering disciplines. These problems involve the reconstruction of shapes or properties from limited data, often in the presence of noise or uncertainty. The field has witnessed remarkable progress, with new techniques and algorithms emerging to tackle complex challenges.

One of the key areas of research in geometric inverse problems is the study of geodesic X-ray transforms. These transforms are used to reconstruct images from X-ray projections, which are commonly used in medical imaging. Geodesic X-ray transforms have applications in various fields, including computed tomography (CT), magnetic resonance imaging (MRI), and positron emission tomography (PET).

Boundary rigidity is another important topic in geometric inverse problems. It refers to the property of a shape or surface that remains unchanged when subjected to small perturbations. Boundary rigidity plays a crucial role in understanding the behavior of physical systems, such as fluids, elastic structures, and biological tissues. Researchers have developed algorithms to detect and analyze boundary rigidity, which has implications in fields such as medical imaging, computer graphics, and robotics.

Tensor tomography is a technique that reconstructs the internal structure of a three-dimensional object from multiple X-ray projections. It is particularly useful in medical imaging, where it can provide detailed images of organs and tissues. Tensor tomography has applications in fields such as nuclear medicine, computed tomography, and magnetic resonance imaging.

Attenuated X-ray transforms are used to reconstruct images from low-dose X-ray projections, which are less harmful to the patient. These transforms are particularly useful in medical imaging, where it is important to reduce radiation exposure. Attenuated X-ray transforms have applications in fields such as dental imaging, chest imaging, and mammography.

The Calderón problem is a fundamental problem in geometric inverse problems that involves the reconstruction of a smooth surface from a set of scattered data. It has applications in various fields, including physics, engineering, and computer graphics. The Calderón problem has been studied extensively, and researchers have developed algorithms to solve it efficiently.

In addition to these topics, geometric inverse problems have broader applications in areas such as computer vision, shape optimization, and image processing. Computer vision involves the extraction of meaningful information from images, while shape optimization involves the design of structures that minimize certain criteria. Image processing involves the manipulation and analysis of images for various purposes, such as medical imaging, surveillance, and computer graphics.

Overall, geometric inverse problems have significant potential to revolutionize various fields. With the continued development of new techniques and algorithms, we can expect to see more exciting applications in the years to come.

Weight: 704g
Dimension: 159 x 236 x 30 (mm)
ISBN-13: 9781316510872