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

Abstract

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.

Fig. 1.

Numerical phantom used in simulations. a) Brain phantom. b) Uniform phantom: uniform background (1 ROI with radius 25 pixels is shown) with 6 uniform spheres of different radii (2 cold spheres and 4 hot spheres).

Fig. 2.

Angle (left) and average angle (right) between the gradients of the successive subiterations vs. iterations projected in the smooth areas and variable areas in the reconstructed images, respectively, for the brain phantom with high count data. Top row: SDP-P1 (24). Bottom row: SDP-P2(24).

Fig. 3.

Comparison of performance of preconditioners investigated in this study. Objective function vs. elapsed CPU time in reconstructions performed with SDP-BSREM algorithm with four preconditioners: M1, M2, P1, and P2, and with 12 subsets (left) and 24 subsets (right), respectively, for the brain phantom with high count data. Preconditioners P1 and P2 were generalized from M1 and M2, respectively.

Fig. 4.

Global NRMSD vs. Iterations in reconstructions performed with different algorithms: BSREM(12), SDP-P1(12), SDP-P2(12), BSREM(24), SDP-P1(24), SDP-P2(24) for the brain phantom with low (top row) and high (bottom row) count data, respectively.

Fig. 5.

Comparison of performance of SDP-BSREM vs. BSREM algorithm. Objective function vs. elapsed CPU time in reconstructions performed with BSREM, SDP-P1, and SDP-P2, with 12 (left) and 24 (right) subsets for the brain phantom with low (top row) and high (bottom low) count data, respectively. The dash lines represent the objective function values of BSREM at 20 seconds CPU time.

Fig. 6.

Comparison of local convergence performance of SDP-BSREM vs. BSREM algorithms for the uniform phantom with high count data. ROI based normalized root mean square difference (NRMSD) vs. iterations is shown. The eight ROIs are the 4 hot spheres, 2 cold spheres, 1 background spheres, and the region consisting of all the former 7 ROIs, named “all ROIs”.

Fig. 7.

Clinical PET patient and ACR phantom. a) Clinical PET patient: Coronal maximum intensity projection (MIP) of a clinical whole-body 18F-FDG PET patient image acquired on a GE D710 PET/CT and reconstructed using the Q.Clear clinical method [3]. b) Clinical ACR quality assurance phantom showing the regions of interest for cold/hot cylinders, 0:1 and 2.5:1 activity concentration ratios, respectively, and background [34].

Fig. 8.

Comparison of performance of SDP-BSREM vs. Q.Clear (β = 350) algorithms. A whole-body 18F-FDG clinical PET patient scan was used. Eight patient bed positions separated by dashed lines are shown in the coronal maximum intensity projection (MIP) image. The objective function vs. iterations is shown for PET scanner patient bed positions 4 and 6 (red arrows) for nonTOF (left) and TOF data (right). The dashed lines represent the objective function values at the final Q.Clear iterations.

Fig. 9.

Local convergence is assessed using the 8 regions of interest from an ACR quality assurance test with TOF data [34]. These regions of interest can be seen in Fig. 7b. Each subplot represents one of the eight regions, which are from left to right and top to bottom: background, cold Teflon/air/water, and hot 8/12/16/25 mm cylinders, respectively.