An Investigation of Stochastic Variance Reduction Algorithms for Relative Difference Penalized 3D PET Image Reconstruction

Abstract

Penalised PET image reconstruction algorithms are often accelerated during early iterations with the use of subsets. However, these methods may exhibit limit cycle behaviour at later iterations due to variations between subsets. Desirable converged images can be achieved for a subclass of these algorithms via the implementation of a relaxed step size sequence, but the heuristic selection of parameters will impact the quality of the image sequence and algorithm convergence rates. In this work, we demonstrate the adaption and application of a class of stochastic variance reduction gradient algorithms for PET image reconstruction using the relative difference penalty and numerically compare convergence performance to BSREM. The two investigated algorithms are: SAGA and SVRG. These algorithms require the retention in memory of recently computed subset gradients, which are utilised in subsequent updates. We present several numerical studies based on Monte Carlo simulated data and a patient data set for fully 3D PET acquisitions . The impact of the number of subsets, different preconditioners and step size methods on the convergence of regions of interest values within the reconstructed images is explored. We observe that when using constant preconditioning, SAGA and SVRG demonstrate reduced variations in voxel values between subsequent updates and are less reliant on step size hyper-parameter selection than BSREM reconstructions. Furthermore, SAGA and SVRG can converge significantly faster to the penalised maximum likelihood solution than BSREM, particularly in low count data.

Recent advances in combined Positron Emission Tomography and Magnetic Resonance Imaging

“…In terms of noise models, the weighted least squares can be viewed as the log-likelihood of a gaussian model of the noise. Inspired by the use of gradient-descent Stochastic Variant Reduction (SVR) algorithms in machine learning [222], many authors have recently studied the application of these methods in PET imaging to avoid limit cycles when ordered subsets are implemented [223][224][225][226][227].…”

Modular-Proximal Gradient Algorithms in Variable Exponent Lebesgue Spaces

“…Roughly speaking, this is done by defining a suitable decomposition of the original problem and implementing an iterative scheme where only a batch of data, typically one, is used to compute the current update. Note that the use of SGD schemes has recently attracted the attention of the mathematical imaging community [10,13] due to its applicability in large-scale applications such as medical imaging [9,17,23]. However, its extension to variable exponent Lebesgue setting is not trivial due to some structural difficulties (e.g., non-separability of the norm), making the adaptation a challenging task.…”