Liow J S, Strother S C
Department of Radiology, University of Minnesota, Minneapolis.
Phys Med Biol. 1993 Jan;38(1):55-70. doi: 10.1088/0031-9155/38/1/005.
Study of the maximum likelihood by EM algorithm (ML) with a reconstruction kernel equal to the intrinsic detector resolution and sieve regularization has demonstrated that any image improvements over filtered backprojection (FBP) are a function of image resolution. Comparing different reconstruction algorithms potentially requires measuring and matching the image resolution. Since there are no standard methods for describing the resolution of images from a nonlinear algorithm such as ML, we have defined measures of effective local Gaussian resolution (ELGR) and effective global Gaussian resolution (EGGR) and examined their behaviour in FBP images and in ML images using two different measurement techniques. For FBP these two resolution measures are equal and exhibit the standard convolution behaviour of linear systems. For ML, the FWHM of the ELGR monotonically increased with decreasing Gaussian object size due to slower convergence rates for smaller objects. For the simple simulated phantom used, this resolution dependence is independent of object position. With increasing object size, number of iterations and sieve size the object size dependence of the ELGR decreased. The FWHM of the EGGR converged after approximately 200 iterations, masking the fact that the ELGR for small objects was far from convergence. When FBP is compared to a nonlinear algorithm such as ML, it is recommended that at least the EGGR be matched; for ML this requires more than the number of iterations (e.g., < 100) that are typically run to minimize the mean square error or to satisfy a feasibility or similar stopping criterion. For many tasks, matching the EGGR of ML to FBP images may be insufficient and >> 200 iterations may be needed, particularly for small objects in the ML image because their ELGR has not yet converged.
使用等于固有探测器分辨率的重建核和筛法正则化的期望最大化（EM）算法进行最大似然估计（ML）的研究表明，相对于滤波反投影（FBP）的任何图像改进都是图像分辨率的函数。比较不同的重建算法可能需要测量并匹配图像分辨率。由于没有标准方法来描述来自诸如ML之类的非线性算法的图像分辨率，我们定义了有效局部高斯分辨率（ELGR）和有效全局高斯分辨率（EGGR）的度量，并使用两种不同的测量技术研究了它们在FBP图像和ML图像中的行为。对于FBP，这两种分辨率度量是相等的，并且表现出线性系统的标准卷积行为。对于ML，由于较小物体的收敛速度较慢，ELGR的半高宽（FWHM）随着高斯物体尺寸的减小而单调增加。对于所使用的简单模拟体模，这种分辨率依赖性与物体位置无关。随着物体尺寸、迭代次数和筛孔尺寸增加，ELGR的物体尺寸依赖性降低。EGGR的FWHM在大约200次迭代后收敛，掩盖了小物体的ELGR远未收敛的事实。当将FBP与诸如ML之类的非线性算法进行比较时，建议至少匹配EGGR；对于ML，这需要的迭代次数比通常为最小化均方误差或满足可行性或类似停止准则而运行的迭代次数（例如，<100）更多。对于许多任务，将ML的EGGR与FBP图像匹配可能不够，可能需要>> 200次迭代，特别是对于ML图像中的小物体，因为它们的ELGR尚未收敛。
