The approximate inverse and conjugate gradient: non-symmetrical algorithms for fast attenuation correction in SPECT

A priori selection of the regularization parameters in emission tomography by Fourier synthesis

This paper describes how the stability of the inverse problem underlying emission tomography can be measured and controlled in clinical settings. We show how the Lanczos approximation provides a way to regularize a certain class of iterative reconstruction algorithms through a given level of noise or resolution in the slices and for a given acquisition protocol. Moreover, we show how the same Lanczos approximation can be used to decide when the iterative reconstruction algorithm actually converges for a given machine precision. These ideas are illustrated by means of reconstructions of simulated and actual emission datasets.

Rectification for cone-beam projection and backprojection

The purpose of this paper is to derive a technique for accelerating the computation of cone-beam forward and backward projections that are the basic steps of tomographic reconstruction. The cone-beam geometry of C-arm systems is commonly described with projection matrices. Such matrices provide a continuous framework for analyzing the flow of operations needed to compute backprojection for analytical reconstruction, as well as the combination of forward and backward projections for iterative reconstruction. The proposed rectification technique resampies the original data to planes that are aligned with two of the reconstructed volume main axes, so that the original cone-beam geometry can be replaced by a simpler geometry, where succession of plane magnifications are involved only. Rectification generalizes previous independent results to the cone-beam backprojection of preprocessed data as well as to cone-beam iterative reconstruction. The memory access pattern of simple magnifications provides superior predictability and is, therefore, easier to optimize, independently of the choice of the interpolation technique. Rectification is also shown to provide control over interpolation errors through oversampling, allowing tradeoffs between computation speed and precision to be set. Experimental results are provided for linear and nearest neighbor interpolations, based on simulations, as well as phantom and patient data acquired on a digital C-arm system.

Experimental study of stochastic noise propagation in SPECT images reconstructed using the conjugate gradient algorithm

Thanks to an experimental study based on simulated and physical phantoms, the propagation of the stochastic noise in slices reconstructed using the conjugate gradient algorithm has been analysed versus iterations. After a first increase corresponding to the reconstruction of the signal, the noise stabilises before increasing linearly with iterations. The level of the plateau as well as the slope of the subsequent linear increase depends on the noise in the projection data.

Development and validation of an X-ray tomograph for two-phase flow

This paper describes the development and validation of a high spatial resolution X-ray tomograph designed for the investigation of air-water two-phase flow. The device hardware mainly comprises a 60 keV X-ray source, a detector, and an accurate mechanical bench. Our study concentrated on accurate quantification with emphasis on the reconstruction procedure. As is well known, absorption gradients induce reconstruction artifacts when using standard algorithms based on uniform regularization. In the particular case of two-phase flow in a pipe, this leads to poor measurement accuracy in the vicinity of the walls. To overcome such effects, improved algorithms were developed during this study that involve spatially adaptive regularization methods. Preliminary calibration performed on static phantoms clearly exhibited the benefits of the advanced reconstruction algorithms. A validation procedure was carried out on an air-water bubble column, equipped with an optical probe, which could be translated in order to explore the 80 mm x 80 mm square cross section. Comparisons of local void fraction measurements were performed pixel by pixel. They demonstrate the accuracy improvement induced by the advanced reconstruction algorithms.

Iterative reconstruction for helical CT: a simulation study

Iterative reconstruction algorithms for helical CT are presented. The algorithms are derived from two-dimensional reconstruction algorithms, by adapting the projector/backprojector to the helical orbit of the source, and by constraining the axial frequencies with a Gaussian sieve. Simulations have been carried out and the performance of the iterative algorithms is compared to that of filtered backprojection of synthetic (interpolated) two-dimensional sinograms. The iterative algorithms produce superior bias-noise curves. Axial resolution is superior, but disturbing edge-artefacts are introduced.

Minimal residual cone-beam reconstruction with attenuation correction in SPECT

This paper presents an iterative method based on the minimal residual algorithm for tomographic attenuation compensated reconstruction from attenuated cone-beam projections given the attenuation distribution. Unlike conjugate-gradient based reconstruction techniques, the proposed minimal residual based algorithm solves directly a quasisymmetric linear system, which is a preconditioned system. Thus it avoids the use of normal equations, which improves the convergence rate. Two main contributions are introduced. First, a regularization method is derived for quasisymmetric problems, based on a Tikhonov-Phillips regularization applied to the factorization of the symmetric part of the system matrix. This regularization is made spatially adaptive to avoid smoothing the region of interest. Second, our existing reconstruction algorithm for attenuation correction in parallel-beam geometry is extended to cone-beam geometry. A circular orbit is considered. Two preconditioning operators are proposed: the first one is Grangeat's inversion formula and the second one is Feldkamp's inversion formula. Experimental results obtained on simulated data are presented and the shadow zone effect on attenuated data is illustrated.