Power laws for image quality measures in PET penalized-likelihood image reconstruction

Most image reconstruction methods have parameters for users to determine: for example, an iteration number and post-reconstruction filter parameters in OSEM, or a regularization parameter in penalized-likelihood (PL). To optimize such reconstruction parameters, one needs to quantitatively understand the relationship among those parameters and image quality. However, image quality is a function of not only the reconstruction parameters but also patient, scanner and imaging protocol. A major advantage of PL over OSEM is the availability of a computationally efficient algebraic procedure to predict resolution and noise properties as a function of the regularization parameter for a given sinogram data set while taking into account the dependence on patient, scanner and protocol. But the procedure, which is based on discrete-space matrix and vector operations, despite its usefulness, lacks intuitive insights such as can be obtained from studying continuous-space Radon transforms. Here, by continuous-space analysis of PL with quadratic (or Gaussian) penalties, we derive approximate yet insightful closed-form expressions for functional relationships, which turn out to be power laws, among the regularization parameter and such image quality measures as resolution, variance and spatial correlation. The expressions we derive provide intuitive insights into how a regularization parameter affects the image quality measures. As a by-product, we develop an understanding of why the ensemble voxel variance in PL is a function of the regularization parameter only and is, somewhat surprisingly, independent of other factors including patient size, scan time and dose.