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Register a new userMicrolocal analysis of a Compton tomography problem
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Added by
James Webber
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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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.
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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.
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.
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.
The recent development of energy-resolving cameras opens the way to new types of applications and imaging systems. In this work, we consider computerized tomography (CT) with fan beam geometry and equipped with such cameras. The measured radiation is then a function of the positions of the source and detectors and of the energy of the incoming photons. Due to the Compton effect, the variations in energy (or spectrum) of the measurement are modeled in terms of scattering events leading to the so-called Compton scattering tomography (CST). We propose an analysis of the spectral data in terms of modelling and mapping properties which results in a general reconstruction strategy. Thanks to the supplementary information given by the energy, this joint CT-CST scanner makes accurate reconstructions of characteristics of the sought-for object possible for very few source positions and a small number of detectors. The general reconstruction strategy is finally validated on synthetic data via a total variation iterative scheme. We further show how the method can be extended to high energetic polychromatic radiation sources. Also illustrative, this work motivates the potential of combining conventional CT and Compton scattering imaging (CSI) with various architectures in 2D and 3D.
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