Fast image reconstruction techniques are becoming important with the increasing number of scientific cases in high resolution micro and nano tomography. The processing of the large scale 3D data demands new mathematical tools for the tomographic reconstruction. Due to the high computational complexity of most current algorithms, big data sizes demands powerful hardware and more sophisticated numerical techniques. Several reconstruction algorithms are dependent on a mathematical tool called backprojection (a transposition process). A conventional implementation of the backprojection operator has cubic computational complexity. In the present manuscript we propose a new fast backprojection operator for the processing of tomographic data, providing a low-cost algorithm for this task. We compare our formula against other fast transposition techniques, using real and simulated large data sets.

This paper proposes an accelerating process through the use of a fast and exact line search for the over-relaxed monotone fast iterative shrinkage-threshold algorithms (OMFISTA). This algorithm is applied to high resolution tomographic image reconstruction using data from the Brazilian Synchrotron Light Source (LNLS). The LNLS can capture lots of data from the scanned objects in each angle, but in order to reduce acquisition time, only few angles are captured. Due to the reduced angles, the l\ penalty is used as a prior to provide a proper reconstruction. Algorithms, such as the fast iterative shrinkage-threshold algorithms (FISTA) and the OMFISTA can be used with some success. However, due to the large computational time at each iteration, even with fast projection/backprojection operators, any reduction in the number of iterations is welcome. In this paper, the fast and exact unidimensional optimization for íi-í\ is used as line search to accelerate the OMFISTA. The results in this paper illustrate that this line search accelerated OMFISTA is faster than FISTA, MFISTA and the OMFISTA.

We describe a new approach for the inversion of the generalized attenuated radon transform in X-ray fluorescence computed tomography (XFCT). The approach consists of using the radon inverse as an approximation for the actual one, followed by an iterative refinement. Also, we analyze the problem of retrieving the attenuation map directly from the emission data, giving rise to a novel alternating method for the solution. We applied our approach to real and simulated XFCT data and compared its performance to previous inversion algorithms for the problem, showing its main advantages: better images than those obtained by other analytic methods and much faster than iterative methods in the discrete setting.