On the broken ray transform (bibtex)
by Joonas Ilmavirta
Abstract:
This PhD thesis studies the broken ray transform, a generalization of the geodesic X-ray transform where geodesics are replaced with broken rays that reflect on a part of the boundary. The fundamental question is whether this transform is injective. We employ four different methods to approach this question, and each of them gives interesting results. Direct calculation can be used in a ball, where the geometry is particularly simple. If the reflecting part of the boundary is (piecewise) flat, a reflection argument can be used to reduce the problem to the usual X-ray transform. In some geometries one can use broken rays near the boundary to determine the values of the unknown function at the reflector, and even construct its Taylor series. One can also use energy estimates -- which in this context are known as Pestov identities -- to show injectivity in the presence of one convex reflecting obstacle. Many of these methods work also on Riemannian manifolds. We also discuss the periodic broken ray transform, where the integrals are taken over periodic broken rays. The broken ray transform and its periodic version have applications in other inverse problems, including Calder\'on's problem and problems related to spectral geometry. (More detailed abstract in the PDF file. The PDF only contains the introductory part of the thesis.)
Reference:
On the broken ray transform (Joonas Ilmavirta), PhD thesis, University of Jyväskylä, Department of Mathematics and Statistics, Report 140, 2014. (advisor: Mikko Salo) [show abstract] [hide abstract] This PhD thesis studies the broken ray transform, a generalization of the geodesic X-ray transform where geodesics are replaced with broken rays that reflect on a part of the boundary. The fundamental question is whether this transform is injective. We employ four different methods to approach this question, and each of them gives interesting results. Direct calculation can be used in a ball, where the geometry is particularly simple. If the reflecting part of the boundary is (piecewise) flat, a reflection argument can be used to reduce the problem to the usual X-ray transform. In some geometries one can use broken rays near the boundary to determine the values of the unknown function at the reflector, and even construct its Taylor series. One can also use energy estimates – which in this context are known as Pestov identities – to show injectivity in the presence of one convex reflecting obstacle. Many of these methods work also on Riemannian manifolds. We also discuss the periodic broken ray transform, where the integrals are taken over periodic broken rays. The broken ray transform and its periodic version have applications in other inverse problems, including Calderón's problem and problems related to spectral geometry. (More detailed abstract in the PDF file. The PDF only contains the introductory part of the thesis.) [arXiv] [eprint]
Bibtex Entry:
```@phdthesis{phd-brt,
author    = "Joonas Ilmavirta",
title     = "On the broken ray transform",
school    = "University of Jyv\"askyl\"a, Department of Mathematics and Statistics, Report 140",
year      = "2014",
month     = aug,
arxiv = {1409.7500},
isbn = {978-951-39-5743-8},
issn = {1457-8905},
url = {http://users.jyu.fi/~jojapeil/thesis/brt_290814.pdf},
eprint = {http://urn.fi/URN:ISBN:978-951-39-5743-8},
gsid = {5359084840896215613},
abstract = {This PhD thesis studies the broken ray transform, a generalization of the geodesic X-ray transform where geodesics are replaced with broken rays that reflect on a part of the boundary.
The fundamental question is whether this transform is injective.
We employ four different methods to approach this question, and each of them gives interesting results.

Direct calculation can be used in a ball, where the geometry is particularly simple.
If the reflecting part of the boundary is (piecewise) flat, a reflection argument can be used to reduce the problem to the usual X-ray transform.
In some geometries one can use broken rays near the boundary to determine the values of the unknown function at the reflector, and even construct its Taylor series.
One can also use energy estimates -- which in this context are known as Pestov identities -- to show injectivity in the presence of one convex reflecting obstacle.
Many of these methods work also on Riemannian manifolds.

We also discuss the periodic broken ray transform, where the integrals are taken over periodic broken rays.
The broken ray transform and its periodic version have applications in other inverse problems, including Calder\'on's problem and problems related to spectral geometry.

(More detailed abstract in the PDF file. The PDF only contains the introductory part of the thesis.)}
}```