Acceleration of HOTV-PAPA was achieved using ROS. This was accompanied by an improved RMSE metric and perceptual image quality that were both superior to that obtained with either clinical or optimized OSEM. This may allow up to a four-fold reduction of the radiation dose to the patients in a PET study, as compared with current clinical practice. The proposed unsupervised parameter selection method provided useful estimates of the penalty weights for the selected phantoms' and patients' PET studies. In sum, the outcomes of this research indicate that ROS-HOTV-PAPA is an appropriate candidate for clinical applications and warrants further research.

This paper presents a preconditioned Krasnoselskii-Mann (KM) algorithm with an improved EM preconditioner (IEM-PKMA) for higher-order total variation (HOTV) regularized positron emission tomography (PET) image reconstruction. The PET reconstruction problem can be formulated as a three-term convex optimization model consisting of the Kullback-Leibler (KL) fidelity term, a nonsmooth penalty term, and a nonnegative constraint term which is also nonsmooth. We develop an efficient KM algorithm for solving this optimization problem based on a fixed-point characterization of its solution, with a preconditioner and a momentum technique for accelerating convergence. By combining the EM precondtioner, a thresholding, and a good inexpensive estimate of the solution, we propose an improved EM preconditioner that can not only accelerate convergence but also avoid the reconstructed image being "stuck at zero." Numerical results in this paper show that the proposed IEM-PKMA outperforms existing state-of-the-art algorithms including, the optimization transfer descent algorithm and the preconditioned L-BFGS-B algorithm for the differentiable smoothed anisotropic total variation regularized model, the preconditioned alternating projection algorithm, and

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We estimate convergence rates for fixed-point iterations of a class of nonlinear operators which are partially motivated from solving convex optimization problems. We introduce the notion of the generalized averaged nonexpansive (GAN) operator with a positive exponent, and provide a convergence rate analysis of the fixed-point iteration of the GAN operator. The proposed generalized averaged nonexpansiveness is weaker than the averaged nonexpansiveness while stronger than nonexpansiveness. We show that the fixed-point iteration of a GAN operator with a positive exponent converges to its fixed-point and estimate the local convergence rate (the convergence rate in terms of the distance between consecutive iterates) according to the range of the exponent. We prove that the fixed-point iteration of a GAN operator with a positive exponent strictly smaller than 1 can achieve an exponential global convergence rate (the convergence rate in terms of the distance between an iterate and the solution). Furthermore, we establish the global convergence rate of the fixed-point iteration of a GAN operator, depending on both the exponent of generalized averaged nonexpansiveness and the exponent of the Hölder regularity, if the GAN operator is also Hölder regular. We then apply the established theory to three types of convex optimization problems that appear often in data science to design fixed-point iterative algorithms for solving these optimization problems and to analyze their convergence properties.

We investigated the imaging performance of a fast convergent ordered-subsets algorithm with subiterationdependent preconditioners (SDPs) for positron emission tomography (PET) image reconstruction. In particular, we considered the use of SDP with the block sequential regularized expectation maximization (BSREM) approach with the relative difference prior (RDP) regularizer due to its prior clinical adaptation by vendors. Because the RDP regularization promotes smoothness in the reconstructed image, the directions of the gradients in smooth areas more accurately point toward the objective function's minimizer than those in variable areas. Motivated by this observation, two SDPs have been designed to increase iteration stepsizes in the smooth areas and reduce iteration step-sizes in the variable areas relative to a conventional expectation maximization preconditioner. The momentum technique used for convergence acceleration can be viewed as a special case of SDP. We have proved the global convergence of SDP-BSREM algorithms by assuming certain characteristics of the preconditioner. By means of numerical experiments using both simulated and clinical PET data, we have shown that our SDP-BSREM substantially improved the convergence rate, as compared to conventional BSREM and a vendor's implementation as Q.Clear. Specifically, in numerical experiments with a synthetic brain phantom, both proposed algorithms outperformed the conventional BSREM by a factor of two, while in experiments with clinical whole-body J.

Nesterov's accelerated forward-backward algorithm (AFBA) is an efficient algorithm for solving a class of two-term convex optimization models consisting of a differentiable function with a Lipschitz continuous gradient plus a nondifferentiable function with a closed form of its proximity operator. It has been shown that the iterative sequence generated by AFBA with a modified Nesterov's momentum scheme converges to a minimizer of the objective function with an o 1 k 2 convergence rate in terms of the function value (FV-convergence rate) and an o 1 k convergence rate in terms of the distance between consecutive iterates (DCI-convergence rate). In this paper, we propose a more general momentum scheme with an introduced power parameter ω ∈ (0, 1] and show that AFBA with the proposed momentum scheme converges to a minimizer of the objective function with an oFV-convergence rate and an o 1 k ω DCI-convergence rate. The generality of the proposed momentum scheme provides us a variety of parameter selections for different scenarios, which makes the resulting algorithm more flexible to achieve better performance. We then employ AFBA with the proposed momentum scheme to solve the smoothed hinge loss 1 -support vector machine model. Numerical results demonstrate that the proposed generalized momentum scheme outperforms two existing momentum schemes.

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