We consider a general setting for dynamic tensor field tomography in an inhomogeneous refracting and absorbing medium as inverse source problem for the associated transport equation. Following Fermat's principle the Riemannian metric in the considered domain is generated by the refractive index of the medium. There is wealth of results for the inverse problem of recovering a tensor field from its longitudinal ray transform in a static euclidean setting, whereas there are only few inversion formulas and algorithms existing for general Riemannian metrics and time-dependent tensor fields. It is a well-known fact that tensor field tomography is equivalent to an inverse source problem for a transport equation where the ray transform serves as given boundary data. We prove that this result extends to the dynamic case. Interpreting dynamic tensor tomography as inverse source problem represents a holistic approach in this field. To guarantee that the forward mappings are well-defined, it is necessary to prove existence and uniqueness for the underlying transport equations. Unfortunately, the bilinear forms of the associated weak formulations do not satisfy the coercivity condition. To this end we transfer to viscosity solutions and prove their unique existence in appropriate Sobolev (static case) and Sobolev-Bochner (dynamic case) spaces under a certain assumption that allows only small variations of the refractive index. Numerical evidence is given that the viscosity solution solves the original transport equation if the viscosity term turns to zero.
翻译:我们认为,在一个不相容的折叠和吸收介质的不相容的复合场面成色器中,动态抗冲场断层是相关运输方程的反源问题的一般设置。 按照Fermat 原则, 被考虑域的里曼尼度是介质的折射指数产生的。 在静态的欧球底体环境下, 将抗冲场从长线变异中恢复过来的反向问题有丰富的结果, 而对于通用的里曼尼特度和基于时间的色素字段来说, 仅有很少的反向公式和算法存在。 不幸的是, 相联的直径方方度旋转法相当于一个运输方程的反向源问题, 光线变异性作为给定的边界数据。 我们证明,这一结果延伸到动态。 将动态色色谱成反源的动态场是这一领域的整体方法。 要保证远方图的正确定义, 只能证明基础运输方位空间和基于时间的色谱字段字段。 不幸的是, 相联的双向方形方形方形方形方形方形的方形方形方形方形的方形方形方形方程式相当于无法证明其真实性变变现的内, 直态性方形法则使得我们方形性方形性方形方形变变变的内, 直方形方形方形方形方形方形的方形方形方形的方形法的方形法的方形法使得的正正正正正正正正正正正正正的方形法使得的方形能性能能性能能能能能能能能能能能能能能能能能性能能能能能能能能能能让的方形变。