This paper presents an Accelerated Preconditioned Proximal Gradient Algorithm (APPGA) for effectively solving a class of Positron Emission Tomography (PET) image reconstruction models with differentiable regularizers. We establish the convergence of APPGA with the Generalized Nesterov (GN) momentum scheme, demonstrating its ability to converge to a minimizer of the objective function with rates of $o(1/k^{2\omega})$ and $o(1/k^{\omega})$ in terms of the function value and the distance between consecutive iterates, respectively, where $\omega\in(0,1]$ is the power parameter of the GN momentum. To achieve an efficient algorithm with high-order convergence rate for the higher-order isotropic total variation (ITV) regularized PET image reconstruction model, we replace the ITV term by its smoothed version and subsequently apply APPGA to solve the smoothed model. Numerical results presented in this work indicate that as $\omega\in(0,1]$ increase, APPGA converges at a progressively faster rate. Furthermore, APPGA exhibits superior performance compared to the preconditioned proximal gradient algorithm and the preconditioned Krasnoselskii-Mann algorithm. The extension of the GN momentum technique for solving a more complex optimization model with multiple nondifferentiable terms is also discussed.

翻译：本文提出了一种加速预条件近端梯度算法（APPGA），用于有效求解一类具有可微分正则化项的正电子发射断层扫描（PET）图像重建模型。我们建立了采用广义Nesterov（GN）动量方案的APPGA的收敛性，证明其能够以函数值的$o(1/k^{2\omega})$和相邻迭代点间距离的$o(1/k^{\omega})$速率收敛至目标函数的最小值点，其中$\omega\in(0,1]$为GN动量的幂参数。为针对高阶各向同性全变分（ITV）正则化PET图像重建模型获得具有高阶收敛率的高效算法，我们将ITV项替换为其平滑版本，随后应用APPGA求解平滑后模型。本文给出的数值结果表明，随着$\omega\in(0,1]$增大，APPGA以逐渐加快的速率收敛。此外，与预条件近端梯度算法及预条件Krasnoselskii-Mann算法相比，APPGA表现出更优越的性能。本文还讨论了将GN动量技术扩展应用于求解具有多个不可微项的更复杂优化模型。