Department of Control Engineering and Information Technology, Budapest University of Technology and Economics, Budapest, Hungary.
IEEE Trans Med Imaging. 2013 Mar;32(3):589-600. doi: 10.1109/TMI.2012.2231693. Epub 2012 Dec 4.
Iterative positron emission tomography (PET) reconstruction computes projections between the voxel space and the lines of response (LOR) space, which are mathematically equivalent to the evaluation of multi-dimensional integrals. The dimension of the integration domain can be very high if scattering needs to be compensated. Monte Carlo (MC) quadrature is a straightforward method to approximate high-dimensional integrals. As the numbers of voxels and LORs can be in the order of hundred millions and the projection also depends on the measured object, the quadratures cannot be precomputed, but Monte Carlo simulation should take place on-the-fly during the iterative reconstruction process. This paper presents modifications of the maximum likelihood, expectation maximization (ML-EM) iteration scheme to reduce the reconstruction error due to the on-the-fly MC approximations of forward and back projections. If the MC sample locations are the same in every iteration step of the ML-EM scheme, then the approximation error will lead to a modified reconstruction result. However, when random estimates are statistically independent in different iteration steps, then the iteration may either diverge or fluctuate around the solution. Our goal is to increase the accuracy and the stability of the iterative solution while keeping the number of random samples and therefore the reconstruction time low. We first analyze the error behavior of ML-EM iteration with on-the-fly MC projections, then propose two solutions: averaging iteration and Metropolis iteration. Averaging iteration averages forward projection estimates during the iteration sequence. Metropolis iteration rejects those forward projection estimates that would compromise the reconstruction and also guarantees the unbiasedness of the tracer density estimate. We demonstrate that these techniques allow a significant reduction of the required number of samples and thus the reconstruction time. The proposed methods are built into the Teratomo system.
迭代正电子发射断层扫描（PET）重建计算体素空间和响应线（LOR）空间之间的投影，这在数学上等效于多维积分的评估。如果需要补偿散射，则积分域的维数可能非常高。蒙特卡罗（MC）求积是一种直接逼近高维积分的方法。由于体素和 LOR 的数量可以达到数千万，并且投影还取决于测量对象，因此不能预先计算求积，而是应该在迭代重建过程中实时进行蒙特卡罗模拟。本文提出了对最大似然、期望最大化（ML-EM）迭代方案的修改，以减少由于正向和反向投影的实时 MC 逼近而导致的重建误差。如果 MC 样本位置在 ML-EM 方案的每个迭代步骤中都相同，则近似误差将导致修改后的重建结果。然而，当 MC 样本位置在 ML-EM 方案的每个迭代步骤中都相同，则近似误差将导致修改后的重建结果。然而，当 MC 样本位置在 ML-EM 方案的每个迭代步骤中都相同，则近似误差将导致修改后的重建结果。然而，当 MC 样本位置在不同的迭代步骤中是统计独立的，则迭代可能会发散或在解周围波动。我们的目标是在保持随机样本数量低的同时，提高迭代解的准确性和稳定性。我们首先分析了实时 MC 投影的 ML-EM 迭代的误差行为，然后提出了两种解决方案：平均迭代和 Metropolis 迭代。平均迭代在迭代序列中平均正向投影估计。Metropolis 迭代拒绝那些会影响重建的正向投影估计，同时保证示踪剂密度估计的无偏性。我们证明了这些技术可以显著减少所需的样本数量，从而缩短重建时间。所提出的方法已被集成到 Teratomo 系统中。
