The V-line transform (VLT) maps a function to its integrals along V-shaped trajectories with a vertex inside the support of the function. This transform and its various generalizations appear in mathematical models of several imaging techniques utilizing scattered particles. The talk presents recent results on inversion of generalized VLTs defined on vector fields and symmetric 2-tensor fields in the plane.

Spectral CT is an x-ray transmission imaging technique that uses the energy dependence of x-ray photon attenuation to determine elemental composition of an object of interest. Mathematically, forward spectral CT measurements are modeled by a nonlinear integral transform for which no analytical inversion is available. In this talk, I will present some of our recent results on the global uniqueness and the stability of spectral CT reconstructions. These analyses are useful for designing optimal scan parameters, which will be demonstrated using numerical simulations. This is joint work with G. Bal and E. Sidky.

For in-situ gamma ray spectrometry, Compton cameras are an efficient imaging tool that operate without collimation and therefore attain large sensitivities. Conventionally, Compton cameras are built with separated scattering and absorbing layers. This setup allows detector materials to be tailored to maximize sensitivity and have good energetic or spatial resolution, but often sacrifices the camera's ability to produce spatially resolved images in the whole $4\pi$ field of view resulting in a de facto collimation. We propose the mathematical model of a Compton camera whose detectors are all considered to both scatter and absorb the incoming gamma rays. Since the measurements of the camera do not give any information about the direction of a coincidence of scattering and absorption, we talk of a bidirectional Compton camera. The additional uncertainty is reflected in the operator describing the forward model, which is the weighted sum of two conical Radon transforms. We demonstrate the ability of the system to efficiently image gamma radiation by numerical results on simulated and measured data.

The aim of multi-energy CT is to reconstruct the distribution of a known set of substances inside a sample by performing CT measurements at different energies. The measurements can be achieved either by using different tube voltages at the source or by means of energy sensitive detectors (e.g. photon counting detectors). In any case the energy dependent absorption of the materials under consideration is used to distinguish the substances in the sample which leads to a nonlinear reconstruction problem. The majority of reconstruction algorithms can be divided into those performing the material decomposition in the sinogram domain and those decomposing the image after inversion of the Radon transform for each energy bin. Both types of algorithms can be implemented very efficiently but also suffer from specific artefacts. More recently one-step algorithms performing decomposition and inversion in one pass have become an active research area. While they eliminate most of the problems of two-step approaches, they are usually computationally costy because they are iterative in nature and the relative similarity of absorption coefficients often leads to poor convergence. We present a method that combines preconditioning in the sinogram domain and an efficient numerical method for the nonlinear problem with a simple and thus fast iteration for the linear part of the problem. Our hybrid method does not suffer from systematic problems like beam hardening or difficulties with not perfectly aligned images for different energy bins. It is iterative but convergence is fast and the computational cost of each iteration is modest.