[Paper Review] Reconstruction Algorithms for Positron Emission Tomography and Single Photon Emission Computed Tomography and their Numerical Implementation

A. S. Fokas, Arieh Iserles+1 more|ArXiv.org|Dec 4, 2004

Medical Imaging Techniques and Applications33 references3 citations

TL;DR

This paper presents novel analytic reconstruction algorithms for PET and SPECT using mathematical techniques from nonlinear integrable systems, enabling exact inversion of the Radon and attenuated Radon transforms. Numerical implementation via cubic spline approximation achieves accurate reconstructions of standard phantoms like Shepp–Logan, with filtering reducing Gibbs artifacts and validating the method's robustness for clinical imaging applications.

ABSTRACT

The modern imaging techniques of Positron Emission Tomography and of Single Photon Emission Computed Tomography are not only two of the most important tools for studying the functional characteristics of the brain, but they now also play a vital role in several areas of clinical medicine, including neurology, oncology and cardiology. The basic mathematical problems associated with these techniques are the construction of the inverse of the Radon transform and of the inverse of the so called attenuated Radon transform respectively. We first show that, by employing mathematical techniques developed in the theory of nonlinear integrable equations, it is possible to obtain analytic formulas for these two inverse transforms. We then present algorithms for the numerical implementation of these analytic formulas, based on approximating the given data in terms of cubic splines. Several numerical tests are presented which suggest that our algorithms are capable of producing accurate reconstruction for realistic phantoms such as the well known Shepp--Logan phantom.

Motivation & Objective

• To develop exact analytic formulas for reconstructing radiopharmaceutical distributions in PET and SPECT using advanced mathematical techniques.
• To numerically implement these analytic formulas using cubic spline approximations for practical reconstruction.
• To validate the algorithms on realistic medical phantoms, including the Shepp–Logan phantom and thorax models, under realistic imaging conditions.
• To mitigate numerical artifacts such as the Gibbs–Wilbraham phenomenon through targeted filtering procedures.
• To demonstrate that coarse sampling of intermediate data (e.g., 10 points instead of 200) is sufficient for accurate reconstruction, reducing computational cost.

Proposed method

• The inverse Radon transform for PET is derived using mathematical methods from nonlinear integrable equations, yielding an exact analytic formula for reconstructing the radiopharmaceutical distribution g(x₁,x₂).
• The attenuated Radon transform for SPECT is similarly inverted using analytic techniques, accounting for attenuation f(x₁,x₂) measured via CT.
• Numerical implementation relies on approximating the measured data using cubic splines, enabling stable and accurate computation of the inverse transforms.
• The algorithm computes intermediate functions h(ρ,θ), fᶜᵐᵉ(ρ,θ), fˢᵐᵉ(ρ,θ), hᶜ(ρ,θ), hˢ(ρ,θ), and I(τ,ρ,θ) via spline interpolation and integral relations.
• Reconstruction is performed on a 2D grid using the inverse formula (3.10), with final image g(x₁,x₂) reconstructed from the computed function r(τ,ρ,θ).
• Post-processing includes filtering: averaging filter (a=0.005) applied five times with thresholding (values < 1/20 of max set to zero) for PET, and median filtering with thresholding for SPECT.

Experimental results

Research questions

• RQ1 Can analytic inversion formulas derived from nonlinear integrable systems be effectively applied to the inverse Radon and attenuated Radon transforms in PET and SPECT?
• RQ2 Can cubic spline-based numerical implementation achieve high-fidelity reconstruction of complex medical phantoms such as the Shepp–Logan phantom?
• RQ3 To what extent can coarse sampling of the Radon transform data (e.g., 10 points instead of 200) preserve reconstruction accuracy and reduce computational load?
• RQ4 How effective are filtering techniques (averaging and median filters with thresholding) in suppressing Gibbs–Wilbraham artifacts in reconstructed images?
• RQ5 Can the proposed algorithms produce accurate reconstructions for clinically relevant phantoms, including those modeling the human thorax and myocardium?

Key findings

• The proposed analytic reconstruction algorithms successfully reconstruct the Shepp–Logan phantom and clinical phantoms with high fidelity, as evidenced by visual comparison before and after filtering.
• The use of cubic spline approximation enables stable and accurate numerical implementation of the inverse transforms, even with limited data sampling.
• Filtering procedures—specifically averaging with thresholding (a=0.005) for PET and median filtering with thresholding for SPECT—significantly reduce Gibbs–Wilbraham artifacts in reconstructed images.
• Reconstruction accuracy is maintained even when intermediate data are sampled with only 10 equally spaced points for t, suggesting substantial reduction in computational cost without loss of fidelity.
• The method achieves accurate reconstruction on a 500×500 grid for PET and a 140×140 grid for SPECT, demonstrating scalability and robustness on standard imaging grids.
• The results confirm that the mathematical framework based on nonlinear integrable systems provides a viable and accurate path for clinical PET and SPECT reconstruction.