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 desc
ribe 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 new non-parametric approach to density estimation and classification derived from theory in Radon transforms and image reconstruction. We start by constructing a forward problem in which the unknown density is mapped to a set of one
dimensional empirical distribution functions computed from the raw input data. Interpreting this mapping in terms of Radon-type projections provides an analytical connection between the data and the density with many very useful properties including stable invertibility, fast computation, and significant theoretical grounding. Using results from the literature in geometric inverse problems we give uniqueness results and stability estimates for our methods. We subsequently extend the ideas to address problems in manifold learning and density estimation on manifolds. We introduce two new algorithms which can be readily applied to implement density estimation using Radon transforms in low dimensions or on low dimensional manifolds embedded in $mathbb{R}^d$. We test our algorithms performance on a range of synthetic 2-D density estimation problems, designed with a mixture of sharp edges and smooth features. We show that our algorithm can offer a consistently competitive performance when compared to the state-of-the-art density estimation methods from the literature.
