Reconstruction algorithm for single photon emission computed tomography and its numerical implementation

Evaluation of the spline reconstruction technique for preclinical PET imaging

BACKGROUND AND OBJECTIVE

The Spline Reconstruction Technique (SRT) is a fast algorithm based on a novel numerical implementation of an analytic representation of the inverse Radon transform. The purpose of this study is to provide a comparison between SRT, Filtered Back-Projection (FBP), Ordered Subset Expectation Maximization 2D (2D-OSEM), and the Tera-Tomo 3D algorithm, using phantom data at various acquisition durations as well as small-animal data obtained from the Mediso nanoScan® PET/CT scanner.

METHODS

For this purpose, the "NEMA NU 4-2008 standards" protocol was employed at five different realizations and acquisition durations. In addition to the image quality metrics described by the NEMA protocol, Cold Region Contrast was also considered as a figure-of-merit. Furthermore, Cold Region Contrast was measured in the myocardial infarction region of six male Wistar rats. The volumetric defect quantification was assessed with dedicated computer software. Lastly, plots of Recovery Coefficient and Spill-Over Ratio as a function of the Percentage Standard Deviation were generated, after smoothing the phantom reconstructions with four different Gaussian filters. Statistical significance was determined by employing the Kruskal-Wallis test or One-way Analysis of Variance depending on the normality of the variable's distribution.

RESULTS

The present study revealed that, at the expense of slightly increased noise in the reconstructed images, SRT resulted in higher Recovery Coefficient values for small hot regions of interest, when compared with FBP and 2D-OSEM at all acquisition durations. Furthermore, SRT reconstructed images exhibit higher Recovery Coefficient values, for all hot regions of interest, when compared to the other 2D algorithms at short acquisition durations. In both phantom and animal studies, SRT achieved a significant improvement over 2D-OSEM for the Spill-Over Ratio and the Cold Region Contrast. These advantages were maintained even after comparing the algorithms at equal noise levels. The Tera-Tomo 3D algorithm (4 subsets, iterations≥ 13) performed significantly better compared to the other algorithms for all figures-of-merit. No statistically significant differences regarding the myocardial defect size were observed between the algorithms investigated.

CONCLUSIONS

Overall, SRT appears that could be useful for the quantification of small hot regions of interest, cold regions of interest, as well as in low-count imaging applications.

Reconstruction of Preclinical PET Images via Chebyshev Polynomial Approximation of the Sinogram

Over the last decades, there has been an increasing interest in dedicated preclinical imaging modalities for research in biomedicine. Especially in the case of positron emission tomography (PET), reconstructed images provide useful information of the morphology and function of an internal organ. PET data, stored as sinograms, involve the Radon transform of the image under investigation. The analytical approach to PET image reconstruction incorporates the derivative of the Hilbert transform of the sinogram. In this direction, in the present work we present a novel numerical algorithm for the inversion of the Radon transform based on Chebyshev polynomials of the first kind. By employing these polynomials, the computation of the derivative of the Hilbert transform of the sinogram is significantly simplified. Extending the mathematical setting of previous research based on Chebyshev polynomials, we are able to efficiently apply our new Chebyshev inversion scheme for the case of analytic preclinical PET image reconstruction. We evaluated our reconstruction algorithm on projection data from a small-animal image quality (IQ) simulated phantom study, in accordance with the NEMA NU 4-2008 standards protocol. In particular, we quantified our reconstructions via the image quality metrics of percentage standard deviation, recovery coefficient, and spill-over ratio. The projection data employed were acquired for three different Poisson noise levels: 100% (NL1), 50% (NL2), and 20% (NL3) of the total counts, respectively. In the uniform region of the IQ phantom, Chebyshev reconstructions were consistently improved over filtered backprojection (FBP), in terms of percentage standard deviation (up to 29% lower, depending on the noise level). For all rods, we measured the contrast-to-noise-ratio, indicating an improvement of up to 68% depending on the noise level. In order to compare our reconstruction method with FBP, at equal noise levels, plots of recovery coefficient and spill-over ratio as functions of the percentage standard deviation were generated, after smoothing the NL3 reconstructions with three different Gaussian filters. When post-smoothing was applied, Chebyshev demonstrated recovery coefficient values up to 14% and 42% higher, for rods 1–3 mm and 4–5 mm, respectively, compared to FBP, depending on the smoothing sigma values. Our results indicate that our Chebyshev-based analytic reconstruction method may provide PET reconstructions that are comparable to FBP, thus yielding a good alternative to standard analytic preclinical PET reconstruction methods.

The attenuated spline reconstruction technique for single photon emission computed tomography

We present the attenuated spline reconstruction technique (aSRT) which provides an innovative algorithm for single photon emission computed tomography (SPECT) image reconstruction. aSRT is based on an analytic formula of the inverse attenuated Radon transform. It involves the computation of the Hilbert transforms of the linear attenuation function and of two sinusoidal functions of the so-called attenuated sinogram These computations are achieved by employing the attenuation information provided by computed tomography (CT) scans and by utilizing custom-made cubic spline interpolation. The purpose of this work is: (i) to present the mathematics of aSRT, (ii) to reconstruct simulated and real SPECT/CT data using aSRT and (iii) to evaluate aSRT by comparing it to filtered backprojection (FBP) and to ordered subsets expectation minimization (OSEM) reconstruction algorithms. Simulation studies were performed by using an image quality phantom and an appropriate attenuation map. Reconstructed images were generated for 45, 90 and 180 views over 360 degrees with 20 realizations and involved Poisson noise of three different levels (NL), namely 100% (NL1), 50% (NL2) and 10% (NL3) of the total counts, respectively. Moreover, real attenuated SPECT sinograms were reconstructed from a real study of a Jaszczak phantom, as well as from a real clinical myocardial SPECT/CT study. Comparisons between aSRT, FBP and OSEM reconstructions were performed using contrast, bias and image roughness. The results suggest that aSRT can efficiently produce accurate attenuation-corrected reconstructions for simulated and real phantoms, as well as for clinical data. In particular, in the case of the clinical myocardial study, aSRT produced reconstructions with higher cold contrast than both FBP and OSEM. aSRT, by incorporating the attenuation correction within itself, may provide an improved alternative to FBP. This is particularly promising for 'cold' regions as those occurring in myocardial ischaemia.

A box spline calculus for the discretization of computed tomography reconstruction problems

B-splines are attractive basis functions for the continuous-domain representation of biomedical images and volumes. In this paper, we prove that the extended family of box splines are closed under the Radon transform and derive explicit formulae for their transforms. Our results are general; they cover all known brands of compactly-supported box splines (tensor-product B-splines, separable or not) in any number of dimensions. The proposed box spline approach extends to non-Cartesian lattices used for discretizing the image space. In particular, we prove that the 2-D Radon transform of an N-direction box spline is generally a (nonuniform) polynomial spline of degree N-1. The proposed framework allows for a proper discretization of a variety of tomographic reconstruction problems in a box spline basis. It is of relevance for imaging modalities such as X-ray computed tomography and cryo-electron microscopy. We provide experimental results that demonstrate the practical advantages of the box spline formulation for improving the quality and efficiency of tomographic reconstruction algorithms.

Exact analytic reconstruction in x-ray fluorescence CT and approximated versions

X-ray fluorescence computed tomography (XFCT) is a relatively new synchrotron-based imaging modality aiming at reconstructing the distribution of nonradiative elements within a sample irradiated with high-intensity monochromatic x-rays. In a recent paper La Rivière (2004 Phys. Med. Biol. 49 2391-405) presented an approximated inversion method based on reducing the problem to the inversion of the exponential radon transform. In this paper we compare La Rivière's results with recently derived 'exact' analytic formulae for the generalized attenuated radon transform. We present numerical experiments with real and simulated data.