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. arXiv:1704.03378v1 [math.FA]

Here we present new L2 injectivity results for 2D and 3D 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 2D and 3D 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 3D to Radon transforms which describe the integrals of the charge density over the surfaces of revolution of a class of C1 curves.

An analysis of the stability of the spindle transform, introduced in [1], 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. [2]. We find that the normal operator for the spindle transform belongs to a class of distibutions I p,l (∆ ∪ ∆, Λ) studied by Felea and Marhuenda in [2,3], where ∆ is reflection through the origin, and Λ 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 Λ and show how the filter we derived can be applied to reduce the strength of the artefact.

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We present new joint reconstruction and regularization techniques inspired by ideas in microlocal analysis and lambda tomography, for the simultaneous reconstruction of the attenuation coefficient and electron density from x-ray transmission (i.e., x-ray CT) and backscattered data (assumed to be primarily Compton scattered). To demonstrate our theory and reconstruction methods, we consider the 'parallel line segment' acquisition geometry of Webber J and Miller E (2020 Inverse Problems 36 025007), which is motivated by system architectures currently under development for airport security screening. We first present a novel microlocal analysis of the parallel line geometry which explains the nature of image artefacts when the attenuation coefficient and electron density are reconstructed separately. We next introduce a new joint reconstruction scheme for low effective Z (atomic number) imaging (Z < 20) characterized by a regularization strategy whose structure is derived from lambda tomography principles and motivated directly by the microlocal analytic results. Finally we show the effectiveness of our method in combating noise and image artefacts on simulated phantoms.