Higher-order dynamic ray tracing in ray-centred coordinates

A reflection approach to the broken ray transform

We reduce the broken ray transform on some Riemannian manifolds (with corners) to the geodesic ray transform on another manifold, which is obtained from the original one by reflection. We give examples of this idea and present injectivity results for the broken ray transform using corresponding earlier results for the geodesic ray transform. Examples of manifolds where the broken ray transform is injective include Euclidean cones and parts of the spheres $S^n$. In addition, we introduce the periodic broken ray transform and use the reflection argument to produce examples of manifolds where it is injective. We also give counterexamples to both periodic and nonperiodic cases. The broken ray t…

Recovery of time-dependent coefficients from boundary data for hyperbolic equations

We study uniqueness of the recovery of a time-dependent magnetic vector-valued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet to Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold.

Urysohn's metrization theorem for higher cardinals

In this paper a generalization of Urysohn's metrization theorem is given for higher cardinals. Namely, it is shown that a topological space with a basis of cardinality at most $|\omega_\mu|$ or smaller is $\omega_\mu$-metrizable if and only if it is $\omega_\mu$-additive and regular, or, equivalently, $\omega_\mu$-additive, zero-dimensional, and T\textsubscript{0}. Furthermore, all such spaces are shown to be embeddable in a suitable generalization of Hilbert's cube.

Neutrino transport in coherent quasiparticle approximation

We introduce the coherent quasiparticle approximation (cQPA), a model in thermal quantum field theory which describes various effects of temporally varying thermal medium on particle propagation. We present the cQPA Feynman rules and develop related calculational tools. Using these methods we calculate neutrino self energies in the Standard Model and derive an equation of motion for neutrino propagation in a very general framework. Some immediate implications of this equation are discussed. (More detailed abstract in the PDF file.)

Partial Data Problems and Unique Continuation in Scalar and Vector Field Tomography

AbstractWe prove that if P(D) is some constant coefficient partial differential operator and f is a scalar field such that P(D)f vanishes in a given open set, then the integrals of f over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields.

Pestov identities and X-ray tomography on manifolds of low regularity

We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds $(M,g)$ with $g \in C^{1,1}$. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This $C^{1,1}$-regularity is optimal on the H\"older scale. The bulk of the article is devoted to setting up a calculus of differential and curvature operators on the unit sphere bundle atop this non-smooth structure.

Unique continuation of the normal operator of the x-ray transform and applications in geophysics

We show that the normal operator of the X-ray transform in $\mathbb{R}^d$, $d\geq 2$, has a unique continuation property in the class of compactly supported distributions. This immediately implies uniqueness for the X-ray tomography problem with partial data and generalizes some earlier results to higher dimensions. Our proof also gives a unique continuation property for certain Riesz potentials in the space of rapidly decreasing distributions. We present applications to local and global seismology. These include linearized travel time tomography with half-local data and global tomography based on shear wave splitting in a weakly anisotropic elastic medium.

Torus computed tomography

We present a new computed tomography (CT) method for inverting the Radon transform in 2D. The idea relies on the geometry of the flat torus, hence we call the new method Torus CT. We prove new inversion formulas for integrable functions, solve a minimization problem associated to Tikhonov regularization in Sobolev spaces and prove that the solution operator provides an admissible regularization strategy with a quantitative stability estimate. This regularization is a simple post-processing low-pass filter for the Fourier series of a phantom. We also study the adjoint and the normal operator of the X-ray transform on the flat torus. The X-ray transform is unitary on the flat torus. We have i…

Spherically symmetric terrestrial planets with discontinuities are spectrally rigid

We establish spectral rigidity for spherically symmetric manifolds with boundary and interior interfaces determined by discontinuities in the metric under certain conditions. Rather than a single metric, we allow two distinct metrics in between the interfaces enabling the consideration of two wave types, like P- and S-polarized waves in isotropic elastic solids. Terrestrial planets in our solar system are approximately spherically symmetric and support toroidal and spheroidal modes. Discontinuities typically correspond with phase transitions in their interiors. Our rigidity result applies to such planets as we ensure that our conditions are satisfied in generally accepted models in the pres…

Tensor tomography in periodic slabs

The X-ray transform on the periodic slab $[0,1]\times\mathbb T^n$, $n\geq0$, has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless $n=0$. We characterize the kernel of the geodesic X-ray transform for $L^2$-regular $m$-tensors for any $m\geq0$. The characterization extends to more general manifolds, twisted slabs, including the M\"obius strip as the simplest example.

De problematis inversis

The Light Ray transform in Stationary and Static Lorentzian geometries

Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar function on a globally hyperbolic stationary Lorentzian manifold and prove injectivity holds if either a convex foliation condition is satisfied on a Cauchy surface on the manifold or the manifold is real analytic and null geodesics do not have cut points. Next, we consider the light ray transform on tensor fields of arbitrary rank in the more restrictive class of static Lorentzia…

Tensor tomography in periodic slabs

Abstract The X-ray transform on the periodic slab [ 0 , 1 ] × T n , n ≥ 0 , has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless n = 0 . We characterize the kernel of the geodesic X-ray transform for L 2 -regular m -tensors for any m ≥ 0 . The characterization extends to more general manifolds, twisted slabs, including the Mobius strip as the simplest example.

Geodesic ray transform with matrix weights for piecewise constant functions

We show injectivity of the geodesic X-ray transform on piecewise constant functions when the transform is weighted by a continuous matrix weight. The manifold is assumed to be compact and nontrapping of any dimension, and in dimension three and higher we assume a foliation condition. We make no assumption regarding conjugate points or differentiability of the weight. This extends recent results for unweighted transforms.

On Radon transforms on compact Lie groups

We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to $S^1$ nor to $S^3$. This is true for both smooth functions and distributions. The key ingredients of the proof are finding totally geodesic tori and realizing the Radon transform as a family of symmetric operators indexed by nontrivial homomorphisms from $S^1$.

Optimal recovery of a radiating source with multiple frequencies along one line

We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam hardening. If measurements are carried out with frequencies ranging in an open set, we show that the source density is uniquely determined by these measurements up to averaging over levelsets of the integrated attenuation. This leads to a generalized Laplace transform. We also discuss some numerical approaches and demonstrate the results with several examples.

On the broken ray transform

Abel transforms with low regularity with applications to X-ray tomography on spherically symmetric manifolds

We study ray transforms on spherically symmetric manifolds with a piecewise $C^{1,1}$ metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of $L^2$ functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds with a $C^{1,1}$ metric. To make these problems tractable in low regularity, we introduce and study a class of generalized Abel transforms and study their properties. This low regularity setting is relevant for geophysical applications.

Boundary reconstruction for the broken ray transform

We reduce boundary determination of an unknown function and its normal derivatives from the (possibly weighted and attenuated) broken ray data to the injectivity of certain geodesic ray transforms on the boundary. For determination of the values of the function itself we obtain the usual geodesic ray transform, but for derivatives this transform has to be weighted by powers of the second fundamental form. The problem studied here is related to Calder\'on's problem with partial data.

X-ray Tomography of One-forms with Partial Data

If the integrals of a one-form over all lines meeting a small open set vanish and the form is closed in this set, then the one-form is exact in the whole Euclidean space. We obtain a unique continuation result for the normal operator of the X-ray transform of one-forms, and this leads to one of our two proofs of the partial data result. Our proofs apply to compactly supported covector-valued distributions.

Profiiliksi profiilittomuus

On Radon Transforms on Tori

We show injectivity of the X-ray transform and the $d$-plane Radon transform for distributions on the $n$-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvi\`ere. We also show solenoidal injectivity of the X-ray transform on the $n$-torus for tensor fields of any order, allowing the tensors to have distribution valued coefficients. These imply new injectivity results for the periodic broken ray transform on cubes of any dimension.

Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds

We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

Coherence in neutrino oscillations

The theory of neutrino oscillations has turned out to be the most reasonable explanation to the observed violations in lepton number conservation of solar and atmospheric neutrino fluxes. A derivation of the most important results of this theory is first given using a plane wave treatment and subsequently using a three-dimensional shape-independent wave packet approach. Both methods give the same oscillation patterns, but only the latter one serves as a decent starting point for analyzing coherence in neutrino oscillations. A numerical analysis of the oscillation patterns on various distance scales is also given to graphically illustrate the phenomenon of neutrino oscillation and loss of co…

Broken ray transform on a Riemann surface with a convex obstacle

We consider the broken ray transform on Riemann surfaces in the presence of an obstacle, following earlier work of Mukhometov. If the surface has nonpositive curvature and the obstacle is strictly convex, we show that a function is determined by its integrals over broken geodesic rays that reflect on the boundary of the obstacle. Our proof is based on a Pestov identity with boundary terms, and it involves Jacobi fields on broken rays. We also discuss applications of the broken ray transform.

Superconductive and insulating inclusions for linear and non-linear conductivity equations

We detect an inclusion with infinite conductivity from boundary measurements represented by the Dirichlet-to-Neumann map for the conductivity equation. We use both the enclosure method and the probe method. We use the enclosure method to prove partial results when the underlying equation is the quasilinear $p$-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation $\operatorname{div}(\sigma\lvert\nabla u\rvert^{p-2}\nabla u)=0$ where the measurable conductivity $\sigma\colon\Omega\to[0,\infty]$ is zero or infinity in large sets and $1<p<\infty$.

A Eulogy in Honour of Anders Johan Lexell, an 18th Century Finnish Mathematician

We present a poem written in the honour of Anders Johan Lexell (1740–1784), a mathematician of Finnish origin, who became a collaborator and successor of Leonhard Euler. The poem was composed in Latin by Fredrik Pryss (1741–1767) in the honour of the 18-year-old promising young man in 1759. We discuss the poem itself and its connections to ancient poetic tradition as well as the foresight of Pryss in seeing the career that lay ahead of Lexell. We find that the poem is of excellent quality as a piece of art following ancient style in form, language and content. Discussing Lexell’s life in light of the poem reveals that Pryss did see that Lexell would rise to fame, but not how. peerReviewed

RECOVERY OF THE SOUND SPEED FOR THE ACOUSTIC WAVE EQUATION FROM PHASELESS MEASUREMENTS

We recover the higher order terms for the acoustic wave equation from measurements of the modulus of the solution. The recovery of these coefficients is reduced to a question of stability for inverting a Hamiltonian flow transform, not the geodesic X-ray transform encountered in other inverse boundary problems like the determination of conformal factors. We obtain new stability results for the Hamiltonian flow transform, which allow to recover the higher order terms.

Suomalaisnuoret menestyivät 46. kansainvälisissä fysiikkaolympialaisissa Intiassa

Coherent Quantum Tomography

We discuss a quantum mechanical indirect measurement method to recover a position dependent Hamilton matrix from time evolution of coherent quantum mechanical states through an object. A mathematical formulation of this inverse problem leads to weighted X-ray transforms where the weight is a matrix. We show that such X-ray transforms are injective with very rough weights. Consequently, we can solve our quantum mechanical inverse problem in several settings, but many physically relevant problems we pose also remain open. We discuss the physical background of the proposed imaging method in detail. We give a rigorous mathematical treatment of a neutrino tomography method that has been previous…

Integral geometry on manifolds with boundary and applications

We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on manifolds with boundary.

The Geodesic Ray Transform on Spherically Symmetric Reversible Finsler Manifolds

We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography.

Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: dynamic ray tracing in Cartesian coordinates

With a Hamilton–Jacobi equation in Cartesian coordinates as a starting point, it is common to use a system of ordinary differential equations describing the continuation of first-order derivatives of phase-space perturbations along a reference ray. Such derivatives can be exploited for calculating geometrical spreading on the reference ray and for establishing a framework for second-order extrapolation of traveltime to points outside the reference ray. The continuation of first-order derivatives of phase-space perturbations has historically been referred to as dynamic ray tracing. The reason for this is its importance in the process of calculating amplitudes along the reference ray. We exte…

X-ray transforms in pseudo-Riemannian geometry

We study the problem of recovering a function on a pseudo-Riemannian manifold from its integrals over all null geodesics in three geometries: pseudo-Riemannian products of Riemannian manifolds, Minkowski spaces and tori. We give proofs of uniqueness anc characterize non-uniqueness in different settings. Reconstruction is sometimes possible if the signature $(n_1,n_2)$ satisfies $n_1\geq1$ and $n_2\geq2$ or vice versa and always when $n_1,n_2\geq2$. The proofs are based on a Pestov identity adapted to null geodesics (product manifolds) and Fourier analysis (other geometries). The problem in a Minkowski space of any signature is a special case of recovering a function in a Euclidean space fro…