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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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We propose a new acquisition geometry for electron density reconstruction in three dimensional X-ray Compton imaging using a monochromatic source. This leads us to a new three dimensional inverse problem where we aim to reconstruct a real valued function $f$ (the electron density) from its integrals over spindle tori. We prove injectivity of a generalized spindle torus transform on the set of smooth functions compactly supported on a hollow ball. This is obtained through the explicit inversion of a class of Volterra integral operators, whose solutions give us an expression for the harmonic coefficients of $f$. The polychromatic source case is later considered, and we prove injectivity of a new spindle interior transform, apple transform and apple interior transform on the set of smooth functions compactly supported on a hollow ball. A possible physical model is suggested for both source types. We also provide simulated density reconstructions with varying levels of added pseudo random noise and model the systematic error due to the attenuation of the incoming and scattered rays in our simulation.
Here we present new $L^2$ injectivity results for 2-D and 3-D Compton scattering tomography (CST) problems in translational geometries. The results are proven through the explicit inversion of a new toric section and apple Radon transform, which describe novel 2-D and 3-D acquisition geometries in CST. The geometry considered has potential applications in airport baggage screening and threat detection. We also present a generalization of our injectivity results in 3-D to Radon transforms which describe the integrals of the charge density over the surfaces of revolution of a class of $C^1$ curves.
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.
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.
Several recent methods for tomographic reconstruction of stress and strain fields from Bragg-edge neutron strain images have been proposed in the literature. This paper presents an extension of a previously demonstrated approach based on Gaussian Process regression which enforces equilibrium in the method. This extension incorporates knowledge of boundary conditions, primarily boundary tractions, into the reconstruction process. This is shown to increase the rate of convergence and is more tolerant of systematic errors that may be present in experimental measurements. An exact expression for a central calculation in this method is also provided which avoids the need for the approximation scheme that was previously used. Convergence of this method for simulated data is compared to existing approaches and a reconstruction from experimental data is provided. Validation of the results to conventional constant wavelength strain measurements and comparison to prior methods shows a significant improvement.
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