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Here we present a novel microlocal analysis of generalized Radon transforms which describe the integrals of $L^2$ functions of compact support over surfaces of revolution of $C^{infty}$ curves $q$. We show that the Radon transforms are elliptic Fourier Integral Operators (FIO) and provide an analysis of the left projections $Pi_L$. Our main theorem shows that $Pi_L$ satisfies the semi-global Bolker assumption if and only if $g=q/q$ is an immersion. An analysis of the visible singularities is presented, after which we derive novel Sobolev smoothness estimates for the Radon FIO. Our theory has specific applications of interest in Compton Scattering Tomography (CST) and Bragg Scattering Tomography (BST). We show that the CST and BST integration curves satisfy the Bolker assumption and provide simulated reconstructions from CST and BST data. Additionally we give example sinusoidal integration curves which do not satisfy Bolker and provide simulations of the image artefacts. The observed artefacts in reconstruction are shown to align exactly with our predictions.
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Here we present a novel microlocal analysis of a new toric section transform which describes a two dimensional image reconstruction problem in Compton scattering tomography and airport baggage screening. By an analysis of two separate limited data problems for the circle transform and using microlocal analysis, we show that the canonical relation of the toric section transform is 2--1. This implies that there are image artefacts in the filtered backprojection reconstruction. We provide explicit expressions for the expected artefacts and demonstrate these by simulations. In addition, we prove injectivity of the forward operator for $L^infty$ functions supported inside the open unit ball. We present reconstructions from simulated data using a discrete approach and several regularizers with varying levels of added pseudo-random noise.
We study the different horospherical Radon transforms that arise by regarding a homogeneous tree T as a simplicial complex whose simplices are vertices V, edges E or flags F (flags are oriented edges). The ends (infinite geodesic rays starting at a reference vertex) provide a boundary $Omega$ for the tree. Then the horospheres form a trivial principal fiber bundle with base $Omega$ and fiber $mathZ$. There are three such fiber bundles, consisting of horospheres of vertices, edges or flags, but they are isomorphic: however, no isomorphism between these fiber bundles maps special sections to special sections (a special section consists of the set of horospheres through a given vertex, edge or flag). The groups of automorphisms of the fiber bundles contain a subgroup $A$ of parallel shifts, analogous to the Cartan subgroup of a semisimple group. The normalized eigenfunctions of the Laplace operator on T are boundary integrals of complex powers of the Poisson kernel, that is characters of $A$, and are matrix coefficients of representations induced from $A$ in the sense of Mackey, the so-called spherical representations. The vertex-horospherical Radon transform consists of summation over V in each vertex-horosphere, and similarly for edges or flags. We prove inversion formulas for all these Radon transforms, and give applications to harmonic analysis and the Plancherel measure on T. We show via integral geometry that the spherical representations for vertices and edges are equivalent. Also, we define the Radon back-projections and find the inversion operator of each Radon transform by composing it with its back-projection. This gives rise to a convolution operator on T, whose symbol is obtained via the spherical Fourier transform, and its reciprocal is the symbol of the Radon inversion formula.
An analysis of the stability of the spindle transform, introduced in (Three dimensional Compton scattering tomography arXiv:1704.03378 [math.FA]), is presented. We do this via a microlocal approach and show that the normal operator for the spindle transform is a type of paired Lagrangian operator with blowdown--blowdown singularities analogous to that of a limited data synthetic aperture radar (SAR) problem studied by Felea et. al. (Microlocal analysis of SAR imaging of a dynamic reflectivity function SIAM 2013). We find that the normal operator for the spindle transform belongs to a class of distibutions $I^{p,l}(Deltacupwidetilde{Delta},Lambda)$ studied by Felea and Marhuenda (Microlocal analysis of SAR imaging of a dynamic reflectivity function SIAM 2013 and Microlocal analysis of some isospectral deformations Trans. Amer. Math.), where $widetilde{Delta}$ is reflection through the origin, and $Lambda$ is associated to a rotation artefact. Later, we derive a filter to reduce the strength of the image artefact and show that it is of convolution type. We also provide simulated reconstructions to show the artefacts produced by $Lambda$ and show how the filter we derived can be applied to reduce the strength of the artefact.
Intersection bodies represent a remarkable class of geometric objects associated with sections of star bodies and invoking Radon transforms, generalized cosine transforms, and the relevant Fourier analysis. The main focus of this article is interre
lation between generalized cosine transforms of different kinds in the context of their application to investigation of a certain family of intersection bodies, which we call $lam$-intersection bodies. The latter include $k$-intersection bodies (in the sense of A. Koldobsky) and unit balls of finite-dimensional subspaces of $L_p$-spaces. In particular, we show that restrictions onto lower dimensional subspaces of the spherical Radon transforms and the generalized cosine transforms preserve their integral-geometric structure. We apply this result to the study of sections of $lam$-intersection bodies. New characterizations of this class of bodies are obtained and examples are given. We also review some known facts and give them new proofs.
Here we introduce a new forward model and imaging modality for Bragg Scattering Tomography (BST). The model we propose is based on an X-ray portal scanner with linear detector collimation, currently being developed for use in airport baggage screening. The geometry under consideration leads us to a novel two-dimensional inverse problem, where we aim to reconstruct the Bragg scattering differential cross section function from its integrals over a set of symmetric $C^2$ curves in the plane. The integral transform which describes the forward problem in BST is a new type of Radon transform, which we introduce and denote as the Bragg transform. We provide new injectivity results for the Bragg transform here, and describe how the conditions of our theorems can be applied to assist in the machine design of the portal scanner. Further we provide an extension of our results to $n$-dimensions, where a generalization of the Bragg transform is introduced. Here we aim to reconstruct a real valued function on $mathbb{R}^{n+1}$ from its integrals over $n$-dimensional surfaces of revolution of $C^2$ curves embedded in $mathbb{R}^{n+1}$. Injectivity proofs are provided also for the generalized Bragg transform.
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