5 The weighted X-ray transform and applications
-
Hanming Zhou
Abstract
In this article, we survey recent developments of weighted geodesic X-ray transforms. A special case of weighted X-ray transform is the attenuated X-ray transform. We review both attenuated X-ray transforms and X-ray transforms with general weights, in particular the matrix version, with emphasis on the approaches using microlocal analysis. We also discuss applications of weighted X-ray transforms to nonlinear inverse problems, such as the non-Abelian X-ray transform and the lens rigidity problem.
Abstract
In this article, we survey recent developments of weighted geodesic X-ray transforms. A special case of weighted X-ray transform is the attenuated X-ray transform. We review both attenuated X-ray transforms and X-ray transforms with general weights, in particular the matrix version, with emphasis on the approaches using microlocal analysis. We also discuss applications of weighted X-ray transforms to nonlinear inverse problems, such as the non-Abelian X-ray transform and the lens rigidity problem.
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- 1 Generalized Radon transforms 1
- 2 Nonstandard Sobolev scales and the mapping properties of the X-ray transform on manifolds with strictly convex boundary 35
- 3 On geometric inverse problems and microlocal analysis 77
- 4 Inverse problems in cosmological X-ray tomography 139
- 5 The weighted X-ray transform and applications 167
- Index 187
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- 1 Generalized Radon transforms 1
- 2 Nonstandard Sobolev scales and the mapping properties of the X-ray transform on manifolds with strictly convex boundary 35
- 3 On geometric inverse problems and microlocal analysis 77
- 4 Inverse problems in cosmological X-ray tomography 139
- 5 The weighted X-ray transform and applications 167
- Index 187
