Eduardo X. Miqueles

Iterative Reconstruction in X-ray Fluorescence Tomography Based on Radon Inversion

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.

C-library raft: Reconstruction algorithms for tomography. Applications to X-ray fluorescence tomography ☆ ☆☆

Abstract There are many reconstruction algorithms for tomography, raft for short, and some of them are considered “classic” by researchers. The so-called raft library, provide a set of useful and basic tools, usually needed in many inverse problems that are related to medical imaging. The subroutines in raft are free software and written in C language; portable to any system with a working C compiler. This paper presents source codes written according to raft routines, applied to a new imaging modality called X-ray fluorescence tomography. Program summary Program title: raft Catalogue identifier: AEJY_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEJY_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: GNU General Public Licence, version 2 No. of lines in distributed program, including test data, etc.: 218 844 No. of bytes in distributed program, including test data, etc.: 3 562 902 Distribution format: tar.gz Programming language: Standard C. Computer: Any with a standard C compiler Operating system: Linux and Windows Classification: 2.4, 2.9, 3, 4.3, 4.7 External routines: raft: autoconf 2.60 or later – http://www.gnu.org/software/autoconf/ GSL scientific library – http://www.gnu.org/software/gsl/ Confuse parser library – http://www.nongnu.org/confuse/ raft-fun: gengetopt – http://www.gnu.org/software/gengetopt/gengetopt.html Nature of problem: Reconstruction algorithms for tomography, specially in X-ray fluorescence tomography. Solution method: As a library, raft covers the standard reconstruction algorithms like filtered backprojection, Novikovʼs inversion, Hoganʼs formula, among others. The input data set is represented by a complete sinogram covering a determined angular range. Users are allowed to set solid angle range for fluorescence emission at each algorithm. Running time: 1 second to 15 minutes, depending on the data size.

Fluorescence tomography: Reconstruction by iterative methods

X-Ray fluorescence computed tomography (XFCT) aims at reconstructing fluorescence density from emission data given the measured X-Ray attenuation. In this paper, inspired by emission tomography (ECT) reconstruction literature, we propose and compare different reconstruction methods for XFCT, based on iteratively inverting the generalized attenuated Radon transform. We compare the different approaches using simulated and real data as well.

Generalized Backprojection Operator: Fast Calculation

The inverse Radon transform and his straightforward implementation, known as filtered backprojection (also known as FBP), has become a powerful algorithm for solving a tomographic inverse problem. It has a wide range of applications, including geophysics, medicine and synchrotrons, and from kilo to centi to micro scale respectively. Such a classical inversion has a major computational disadvantage: increasing slowness proportionally to the data size. An ordinary implementation of this algorithm relies on a simple integral that has to be done pixelwise. Many accelerating techniques were proposed in the literature so as to make this part of the inversion as fast as possible. One the most promising strategies is converting the backprojection as a convolution operator (at log-polar coordinates). The generalized backprojector has many applications, for instance in the analytical inversion of single-photon emission tomography or x-ray fluorescence tomography. Our aim in this paper is to show how these ideas can be used for other inversion methods, the iterative ones; which deal much better with noise.

A Computer Library for Ray Tracing in Analytical Media

Ray tracing technique is an important tool not only for forward but also for inverse problems in Geophysics, which most of the seismic processing steps depends on. However, implementing ray tracing codes can be very time consuming. This article presents a computer library to trace rays in 2.5D media composed by stack of layers. The velocity profile inside each layer is such that the eikonal equation can be analitically solved. Therefore, the ray tracing within such profile is made fast and accurately. The great advantage of an analytical ray tracing library is the numerical precision of the quantities computed and the fast execution of the implemented codes. Although ray tracing programs already exist for a long time, for example the seis package by Cervený, with a numerical approach to compute the ray. Regardless of the fact that numerical methods can solve more general problems, the analytical ones could be part of a more sofisticated simulation process, where the ray tracing time is completely relevant. We demonstrate the feasibility of our codes using numerical examples.

Grazing-incidence XRF analysis of layered samples: Detailed study of amplitude calculation

Abstract In this article, we propose a new mathematical approach for the computation of electromagnetic wave amplitudes in grazing incidence X-ray fluorescence ( gixrf )—an analytical method for surface and near-surface layer analysis. The new contribution comes from an applied point of view, in order to have stable and fast algorithms to simulate the fluorescence intensity from a stacking of thin layer films. The calculation of transmitted/reflected amplitudes is an important part of the direct and/or inverse problem. An analysis of the amplitude versus layer thickness is also given.

Numerical aspects of grazing incidence XRF

In this article, we give a different mathematical approach for background aspects of grazing incidence x-ray fluorescence, GIXRF for short. Our contribution comes from an applied point of view, in order to have a computer program to simulate the fluorescence intensity from a stacking of thin layer films. A typical ill-posed inverse problem is formulated. Our aim is to reconstruct the fluorescence intensity for a variety of grazing angle measurements. We rederive some classical equations pointing out the numerical aspects of the inversion procedure and giving new directions for direct in inverse algorithms.

Fast Continuous Iterative Methods in X-Rays Fluorescence Computed Tomography

Fluorescence computed tomography is a synchrotron imaging technique that reconstructs the fluorescence emission within a sample object. For a monochromatic source hitting the object, the amount of fluorescence detected is given as a linear equation. Iterative methods based on the inversion of the Radon transform were introduced. These methods were compared with the Expectation Maximization algorithm, implemented in a continuous setting. This implementation provided better quality results, but with higher computational cost. Recently, a faster OS-EM algorithm was applied in XFCT, in a discrete setting. In this manuscript we further improve on previous results by considering a relaxed version of OS-EM, in a continuous setting (faster implementation per iteration).

Fluorescence Computed Tomography with Polychromatic Source Data

Fluorescence computed tomography is a synchrotron imaging technique aiming at reconstructing the fluorescence emission within a sample object. For a polychromatic source hitting the object, the amount of fluorescence detected is defined by a linear equation. For the monochromatic case, the operator is a Generalized Attenuated Radon Transform (GART). The main goal is to reconstruct the density function, given the sinogram data and the weight function. An eficient iterative algorithm for the inversion of the GART was presented recently by the authors. This inversion can only be performed if the weight function is previously known, which means that ? = ?(?, ) and ? are also known. For monochromatic XFCT (acronym for x-rays fluorescence computed tomography), the determination of ? is a dificult task, and we have considered the approximation ? ? ?, which is valid for low energies ranging from 3Kev to 10Kev. So, for solving our problem, the first step is to find ? given the polychromatic sinogram. There are different approaches for this in the literature. Recently, an elegant and efficient method for solving this problem was introduced, using a fixed point algorithm. Opposite to this, where ?(?, ) needs to be computed for all E, we claim that the integral of ?(?, ) for all has a physical meaning and provides a good aproximation for the solution. Also we present fast algorithm for computations.

Xfct inversion by generalized ridge functions

X-Ray fluorescence computed tomography (xfct) aims at reconstructing fluorescence density from emission data given the measured x-ray attenuation. In this paper, inspired by the classical results from Logan & Shepp [3], we briefly discuss the existence of generalized ridge functions providing the minimal norm solution of the inverse problem. An algorithm to construct such functions is presented, based on results from Kazantsev [4]. Numerical results are also shown, with real and simulated data.