A. Projection Accuracy
Projection accuracy for each of the three projectors was studied using four discretized versions of the Shepp–Logan phantom with different voxel sizes. The phantom was first discretized onto a 1024 × 1024 × 648 array with 0.4-mm voxels in-plane and a slice thickness of 0.25 mm. This was then collapsed in all directions by factors of 2, 4, and 8 to produce phantoms with successively larger voxel sizes and levels of partial-volume effect due to discretization error. Each phantom was projected with the ray-driven, distance-driven, and rotate-and-slant projectors onto 335 LOR × 81 slice × 336 angle projection arrays using the geometry of the TruePoint Biograph scanner. The RMSEs for a significantly oblique segment (ring difference δ = 20) are shown in , quoted as the percent of the mean nonzero projection value.
Projection Accuracya(%RMSE) for Three Projectors
For all matrix sizes, the distance-driven and rotate-and-slant projectors had similar accuracy, and the ray-driven projector had larger %RMSE. This was expected since the ray-driven projector uses a line-integral projection model, whereas the other two projectors are more volumetric. Comparing the different columns of provides an assessment of the contribution of projector error and discretization error, which dominates for the larger voxel sizes (e.g., 128 × 128 matrix) and decreases rapidly with decreasing voxel size. The distance-driven projector had slightly better accuracy for the 128 × 128 image, likely because it had slightly smaller interpolation error than the rotate-and-slant projector. On the other hand, the rotate-and-slant projector had slightly better accuracy for the 1024 × 1024 image, where the discretization error was smaller and the truly volumetric nature of the rotate-and-slant projector produced more accurate results than the area-of-overlap based distance-driven projector. Overall, the accuracy of the distance-driven and rotate-and-slant projectors was deemed to be very comparable.
B. Effect of Depth Compression
The use of a depth compression factor with the rotate-and-slant projector reduces the number of computations required for projection to multiple oblique sinograms. However, since the use of such factors results in some loss of depth information, there may be a consequent loss of projection accuracy for oblique LORs which cross different slices at different depths. Like SSRB, depth-compression creates a depth-dependent approximation for oblique LORs. However, while the error for SSRB increases with increasing depth to the edge of the field-of-view, the error for depth-compression only increases as one moves from the center to the edge of each depth-compressed slab (it does not successively increase across all slabs). For example, when using a depth-compression factor of 8 with a 128 × 128 image, we have 16 depth-compressed slabs. The depth-information at the center of each slab is exactly retained, and at the edge of the slab the error is similar to that of SSRB at 8/2 = 4 voxels away from the center of the image. Since depths map (in a sense) to radii in the reconstructed image, the errors map to concentric rings on the reconstructed image as shown in . The error is very small at the central radius of each ring and maximal (though still small for depth-compression factors of ~ 8 or less) between each ring. Note that a very similar analogous approximation exists for FORE, where small ring differences (less than δlim
as defined in [1
]) are binned using the SSRB approximation. A typical value of δlim
for FORE would be on the order of 4–5, which roughly corresponds to the same approximation encountered for a depth-compression factor of twice δlim
, or about 8.
Fig. 4 Example images showing the depth-dependent behavior of the approximations introduced by the use of depth-compression. These images show the absolute value percent error of a simulated disc phantom one slice thick, centered in the field-of-view, and reconstructed (more ...)
shows the CPU times for fully-3-D projection of a 128 × 128 image to LORs for the three scanner geometries (each having a different number of LORs, slices, angles, and segments as listed earlier) with different depth compression factors. Projection times are markedly reduced for depth compression factors up to about 8, beyond which they remain relatively stable. also shows two figures-of-merit related to projection accuracy—the reconstructed axial FWHM of the 13-mm sphere for the NEMA phantom experiment, and the %RMSE between analytical and projected values for the Shepp-Logan phantom. Both of these measures were largely unaffected by depth compression factors up to 8, beyond which accuracy was degraded due to the loss of depth information. As a reference for comparison, the FWHM of the 13-mm sphere when reconstructed by SSRB followed by 2-D-OSEM was 20.8 mm—somewhat higher that that for the rotate-and-slant projector with the very large depth compression factor of 64. These data suggest that use of a depth compression factor of 8 (for a 128 × 128 image) with the rotate-and-slant projector offers a significant (almost 3 ×) speedup with negligible loss of accuracy. In general, the optimal depth compression factor will depend upon the image dimensions, number of segments, and range of ring differences included in the projection data, and in general would be smaller if very large ring differences are included. All remaining results for the rotate-and-slant projector presented in this paper used a compression factor of 8.
Fig. 5 Effect of using depth compression factors upon projection time (left) and accuracy measures (right). Here, the FWHM characterizes the axial profile of the 13-mm sphere in the NEMA phantom reconstructed image, and %RMSE provides a measure of accuracy for (more ...)
C. Characteristics of Reconstructed Images
shows example images of the Deluxe Jaszczak Phantom for each of the five reconstruction schemes. The images are shown at 10 iterations OSEM with 21 subsets (16 angles per subset) for each scheme. The 4.8-mm rods are clearly resolved on this dataset, and the smallest 9.5-mm cold sphere is likewise clearly resolved. Small circles of radioactivity surrounding the support posts for the spheres can also be seen between the wedges of hot rods. The images for the rotate-and-slant projector and the other reconstruction methods show similar image quality, and the most significant differences noted were differences in reconstructed noise texture. Horizontal profiles across one row of the 7.9-mm-diameter and 11.1-mm-diameter rods show somewhat better peak-to-valley definition for the distance-driven and rotate-and-slant projectors as compared to the other cases.
Fig. 6 Example images of the Deluxe Jaszczak Phantom at 10 iterations for the five reconstruction methods studied. The smallest 4.8-mm-diameter rods are resolved for each reconstruction method, as is the smallest 9.5-mm-diameter cold sphere. Small circles of (more ...)
Results of the quantitative analysis of the Deluxe Jaszczak phantom experiment are shown in . Plots of the resolution (average peak/valley ratio of the 6.4-mm rods) and contrast (12.7-mm cold sphere) measures versus iteration reveal differences in the rate of iterative recovery of image features for the different reconstruction schemes. The plots on the bottom row of effectively normalize for this effect, permitting comparison of image noise at the same resolution or contrast. The data demonstrate a trend toward improved image quality measures when moving from the preprocessing reconstruction schemes to fully-3-D LOR-OSEM. This reflects the improved statistical models of the fully-3-D iterative methods, coupled with reduced degradation when arc correction is included in the projector (LOR-based) and more-volumetric projectors are used. It is not clear why FORE outperformed AW-OSEM3D in , though this may have to do with differences in noise correlations which are not well characterized by the noise measure that was used. Notably, the rotate-and-slant projector performed as well as, or slightly better than, the two other projectors studied for the LOR-OSEM 3-D reconstructions. In particular, the rotate-and-slant and the distance-driven projectors provided comparable results for the analysis presented. The rotate-and-slant projector computes the full volume-of-intersection of each LOR with each image voxel, whereas the distance-driven projector accounts for most, but not all, of the volume-of-intersection. However, the rotate-and-slant projector also has a slight amount of blurring associated with the first two steps of the rotation operation which the distance-driven projector does not have. Overall, the accuracy of the rotate-and-slant and distance-driven projectors were deemed very similar, with both projectors providing comparable reconstructions.
Fig. 7 Quantitative analysis of image quality measures for the five reconstruction schemes studied. The resolution and contrast measures (top row) demonstrate broad similarities for each reconstruction method, with some differences in the rate of iterative recovery (more ...)
D. Projection and Reconstruction Times
The CPU times required for fully-3-D projection and 3-D LOR-OSEM reconstruction on a single-CPU Linux workstation with no parallelization or multithreading are shown in and . The TruePoint Biograph scanner geometry was used for the projection time computations, and the advance geometry was used for the reconstruction time computations. Both sets of timings were measured as a function of the number of segments included in the reconstruction, where the typical operating points for 2-D and fully-3-D modes are marked. The cases with fewer segments correspond to using larger polar mashing (or axial compression), whereas the cases with the most segments correspond to no polar mashing. It can be seen from that the projection time for the rotate-and-slant projector includes a component associated with 2-D projection and increases slowly with the number of oblique segments. In contrast, the projection time for the other two projectors rises rapidly with increasing number of projection rays (LORs), and each oblique segment adds significant time to the computation.
Fig. 8 Projection CPU times for the TruePoint Biograph scanner, plotted as a function of the number of oblique segments, for the three projectors studied. A 128 × 128 × 81 slice image matrix was used, and the projectors mapped to 335 unevenly-spaced (more ...)
Fig. 9 Reconstruction CPU times for the Advance scanner, plotted as a function of the number of oblique segments, for 4 iterations OSEM with 14 subsets and four reconstruction schemes. Each sinogram of the raw projection data had 283 unevenly-spaced LORs and (more ...)
The total reconstruction times plotted in show similar trends. Notably, fully-3-D iterative reconstruction with the rotate-and-slant projector was accomplished nearly as fast as FORE followed by 2-D iterative reconstruction, whereas fully-3-D reconstruction with the conventional projectors took many times longer. Thus, the computational efficiency of the rotate-and-slant projector enables fully-3-D iterative reconstruction to be performed in timeframes approaching those for rebinning followed by 2-D reconstruction. Given the results of the quantitative analysis comparing image quality features, we conclude that the rotate-and-slant projector offers a viable solution for direct fully-3-D PET reconstruction which takes full advantage of statistical algorithms without associated limitations in processing time.