An Error Evaluation of Iterative Image Reconstruction Methods Using Chi-Square (χ2) Statistic Minimization for Poisson-Distributed Projection Data

Abstract

[Purpose] Iterative image reconstruction (IR) methods using Neyman's chi-square statistic (χ2N) or Pearson's chi-square statistic (χ2P) have been investigated in nuclear medicine. However, these chi-square statistic-based image reconstructions have never been installed on clinical nuclear medicine instruments. Mighell developed another chi-square statistic (χ2M). Recently, Mighell's chi-square statistic has been incorporated into commercial SPECT instrument aiming at high accuracy in the iterative image reconstruction from low count projection data. However, the error evaluation for χ2M was not reported by the instrument manufacturer or the joint research group involved in the product development. Therefore, it is not certain to what extent χ2M is superior to χ2N or χ2P. In this study we investigated the accuracy of the chi-square statistic-based IR methods by computer simulation.[Methods] We used two kinds of numerical phantoms (256×256 pixels) for testing root mean square error (RMSE). Phantom A was a disk that was 18.4 cm in diameter and the count density was varied from 1 count/pixel to 10 counts/pixel at intervals of 1 count/pixel in each trial. Phantom B was a disk that was 18.4 cm in diameter and the count densities for the seven disk inserts (diameter 3 cm) which were investigated were 1, 2, 3, 4, 5, 6, and 7 counts/pixel. Poisson noise was added to the projection data with 256 linear samplings and 256 views over 180°. Projection data were assumed to be without attenuation and scatter effects, because we focused our evaluation on the noise propagation from projection data to the reconstructed image that was attributable to the mathematical equations of the different types of chi-square statistic. Minimization of the chi-square statistic-based IR methods was performed by conjugate gradient method.[Results] We found the noise was suppressed by including the variance of projection data in each chi-square statistic; however, it was not suppressed sufficiently by χ2P in comparison with χ2N and χ2M. For 1000 iterations, the RMSEs of Phantom A having the count density of 1 count/pixel were 21.46±2.75, 39.21±0.71, and 12.29±0.63, obtained by χ2N, χ2P, and χ2M in 20 trials, respectively. For 2 counts/pixel, RMSEs were 5.26±0.32, 19.89±1.29, and 4.23±0.08; and for 3 counts/pixel, they were 5.34±0.56, 10.27±0.38, and 4.03±0.07. With Phantom B, RMSEs of the 3 cm disk insert having the count density of 2 counts/pixel were 7.36±0.56, 21.21±1.52, and 6.79±0.54; for 3 counts/pixel it was 5.46±0.34, 14.43±1.08, and 4.84±0.32, for χ2N, χ2P, and χ2M, respectively.(View PDF for the rest of the abstract.).