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Geodesic ray transforms and the transport equation

测地射线变换和输运方程

基本信息

批准号：
EP/M023842/1
负责人：
Gabriel Paternain
金额：
$ 24.01万
依托单位：
University of Cambridge
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2015
资助国家：
英国
起止时间：
2015 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FM023842%2F1
关键词：
Geodesic ray transforms transport equation

项目摘要

A basic problem in geophysics is to reconstruct the interior structure of the Earth from measurements carried out at the surface. A basic problem in medicine is to obtain structural information about tumours inside the body without invasive surgery. Both are examples of inverse problems in three dimensions. Over the last five years, many-decade old problems concerning inverse problems in two dimensions have been resolved by the PI and his collaborators. This proposal addresses the key remaining questions, with a view of making progress in the higher dimensional setting which are central to potential applications and which formed the original motivation for this branch of Analysis.The linearization of many of these important inverse problems takes naturally to the geodesic ray transform where one integrates a function or a tensor field along geodesics of a Riemannian metric.The standard X-ray transform, where one integrates a function along straight lines, corresponds to the case of the Euclidean metric and is the basis of medical imaging techniques such as CT and PET. The case of integration along more general geodesics arises in geophysical imaging in determining the inner structure of the Earth since the speed of elastic waves generally increases with depth, thus curving the rays back to the Earth's surface. It also arises in ultrasound imaging, where the Riemannian metric models the anisotropic index of refraction. In tensor tomography problems one would like to determine a symmetric tensor field up to natural obstruction from its integrals over geodesics.The proposal aims to get further insight into the injectivity property of the geodesic ray transform acting on symmetric tensors by relating it to the existence of special solutions to the transport equation. This relationship has been crucial for recent successes in solving geometric inverse problems in two dimensions.

地球物理学的一个基本问题是根据在地表进行的测量重建地球的内部结构。医学的一个基本问题是在不进行侵入性手术的情况下获取体内肿瘤的结构信息。两者都是三维逆问题的例子。在过去的五年里，PI 和他的合作者解决了几十年来有关二维反问题的老问题。该提案解决了剩下的关键问题，着眼于在高维环境中取得进展，这些问题是潜在应用的核心，也是这一分析分支的最初动机。许多重要逆问题的线性化自然需要测地射线变换，其中沿黎曼度量的测地线积分函数或张量场。标准 X 射线变换，沿直线积分函数，对应于以下情况欧几里得度量，是 CT 和 PET 等医学成像技术的基础。在确定地球内部结构的地球物理成像中，会出现沿着更一般的测地线积分的情况，因为弹性波的速度通常随着深度的增加而增加，从而将射线弯曲回到地球表面。它也出现在超声成像中，黎曼度量对各向异性折射率进行建模。在张量断层扫描问题中，人们希望通过测地线上的积分来确定一个对称张量场，直至自然阻碍。该提案旨在通过将其与输运方程特殊解的存在性联系起来，进一步深入了解作用于对称张量的测地线射线变换的单射性。这种关系对于最近解决二维几何逆问题的成功至关重要。