A Fast Convergent Ordered-Subsets Algorithm With Subiteration-Dependent Preconditioners for PET Image Reconstruction.

We investigated the imaging performance of a fast convergent ordered-subsets algorithm with subiteration-dependent 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 step-sizes 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 the SDP-BSREM algorithms substantially improve the convergence rate, as compared to conventional BSREM and a vendor's implementation as Q.Clear. Specifically, SDP-BSREM algorithms converge 35%-50% faster in reaching the same objective function value than conventional BSREM and commercial Q.Clear algorithms. Moreover, we showed in phantoms with hot, cold and background regions that the SDP-BSREM algorithms approached the values of a highly converged reference image faster than conventional BSREM and commercial Q.Clear algorithms.

A Krasnoselskii-Mann Algorithm With an Improved EM Preconditioner for PET Image Reconstruction.

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 the alternating direction method of multipliers for the nondifferentiable HOTV regularized model. Encouraging initial experiments using clinical data are presented.

Relaxed ordered subset preconditioned alternating projection algorithm for PET reconstruction with automated penalty weight selection.

PURPOSE: Performance of the preconditioned alternating projection algorithm (PAPA) using relaxed ordered subsets (ROS) with a non-smooth penalty function was investigated in positron emission tomography (PET). A higher order total variation (HOTV) regularizer was applied and a method for unsupervised selection of penalty weights based on the measured data is introduced. METHODS: A ROS version of PAPA with HOTV penalty (ROS-HOTV-PAPA) for PET image reconstruction was developed and implemented. Two-dimensional PET data were simulated using two synthetic phantoms (geometric and brain) in geometry similar to GE D690/710 PET/CT with uniform attenuation, and realistic scatter (25%) and randoms (25%). Three count levels (high/medium/low) corresponding to mean information densities (ID¯s) of 125, 25, and 5 noise equivalent counts (NEC) per support voxel were reconstructed using ROS-HOTV-PAPA. The patients' brain and whole body PET data were acquired at similar ID¯s on GE D690 PET/CT with time-of-fight and were reconstructed using ROS-HOTV-PAPA and available clinical ordered-subset expectation-maximization (OSEM) algorithms. A power-law model of the penalty weights' dependence on ID¯ was semi-empirically derived. Its parameters were elucidated from the data and used for unsupervised selection of the penalty weights within a reduced search space. The resulting image quality was evaluated qualitatively, including reduction of staircase artifacts, image noise, spatial resolution and contrast, and quantitatively using root mean squared error (RMSE) as a global metric. The convergence rates were also investigated. RESULTS: ROS-HOTV-PAPA converged rapidly, in comparison to non-ROS-HOTV-PAPA, with no evidence of limit cycle behavior. The reconstructed image quality was superior to optimally post-filtered OSEM reconstruction in terms of noise, spatial resolution, and contrast. Staircase artifacts were not observed. Images of the measured phantom reconstructed using ROS-HOTV-PAPA showed reductions in RMSE of 5%-44% as compared with optimized OSEM. The greatest improvement occurred in the lowest count images. Further, ROS-HOTV-PAPA reconstructions produced images with RMSE similar to images reconstructed using optimally post-filtered OSEM but at one-quarter the NEC. CONCLUSION: 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.