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1.INTRODUCTIONRecent algorithmic and computational advances have made MC-like scatter correction approaches much more practical. The Acuros platform1,2 rapidly solves the linear Boltzmann Transport Equation (LBTE) using finite element methods to determine the scatter distribution directly rather than stochastically as in conventional MC. Recently, we unveiled 3D VSHARP3,4 that uses Acuros to achieve accuracies comparable to MC methods in a tiny fraction of the time. Acuros’ accuracy and run-time are both highly dependent on the choice of sampling grid used for the finite element solution. If the grid is too coarse then results are inaccurate, and if the grid is too fine then time is wasted. Wang et al2 have addressed this issue for medical use by optimizing the Pareto front over the set of sampling parameters. In this work we perform a similar Pareto optimization for an industrial case of an aluminum motorcycle cylinder head. 2.METHODS2.1CST FrameworkThe correction utilizes CST, Varex’s CT reconstruction SDK, which allows for flexible connection of modular plugins to perform a reconstruction. CST includes over 30 bundled plugins, including those necessary to implement 2D VSHARP5, 2D VSHARP-ML6, and 3D VSHARP. CST also includes a Physics Library that contains user-selectable cross-sections as well as x-ray spectra and detector response files required by 3D VSHARP. 2.2Pipeline with 3D VSHARP and 2-pass FDK reconstructionAn example pipeline with 3D VSHARP is shown in Figure 1. Figure 1.Two-pass pipeline for FDK reconstruction with 3D VSHARP. The data flow green lines indicate projection data and purple lines indicate volume data.  Processing for 3D VSHARP is in 6 basic stages:
2.3Data acquisition and reconstructionA motorcycle cylinder head was scanned and reconstructed with the parameters shown in Table 1. Table 1.Acquisition and reconstruction parameters.
Reconstruction was performed on a PC with 2 Intel Xeon ES-2637v4 chips each containing 8 cores at 3.5 GHz, and an NVIDIA Titan RTX GPU with 4608 cores at 1.35 GHz. 2.4Pareto optimizationTo characterize the processing time-vs-error tradeoff, we used the NSGA2 algorithm7, a genetic algorithm (GA) for discovering Pareto fronts in multi-objective problems. The two objectives were time and error. To measure error, a “golden” reconstruction was performed with all parameters set for maximum accuracy, then for each operating point the root mean square (RMS) error versus the golden reconstruction was measured. The total search space included: 1. Primary Volume Matrix Size, 2 Scatter Volume Matrix Size, 3. Primary Detector Matrix Size, 4. Scatter Detector Matrix Size, 5. Number of Primary Projections, 6. Number of Scatter Projections. The constraints on the search were:
Table 2 shows the native search space used by the GA. Table 2.Native search space used by the GA.
We ran 25 generations of NSGA2 with a population of 50, then another 25 generations with a population of 75. 3.RESULTS3.1Motorcycle cylinder head reconstructionsFigure 2 shows example reconstructions of a sagittal slice including A) an “Uncorrected” reconstruction, B) the first pass reconstruction (from the unoptimized 2D VSHARP), and two 3D VSHARP reconstructions: C) the “Golden” reconstruction performed at maximum LBTE resolution for a run time of about 6 hours, and D) the “Operating Point F” reconstruction using a coarser LBTE grid, requiring only 21 seconds of BTP time. Significant improvements in crispness and homogeneity are seen in the 3D VSHARP images with Operating Point F retaining similar image quality to the Golden image. Figure 2.Central sagittal slice. The window for each is chosen so that the 10% and 90% gray values within the window correspond to the 5th and 95th percentiles of the voxel values. The red outline shows the zoomed-in region in Figure 4.  3.2Pareto resultsFigure 3 shows results from all generations of the NSGA2 run. Each operating point is shown as a blue dot, the Pareto front is shown as a red line, the convex hull of all the operating points is shown in green, and the set of points that were further studied is indicated by labeled circles. Table 2 shows search parameter and objective results for several operating points (A,D,F,I). As expected, RMS error decreases as 3D VSHARP runtime increases as do the sampling grid dimensions in general. Of note, is the exception that Scatter NVoxelsXY is relatively constant. This may partially reflect that its lower bound was 25. Also of note, Scatter NDexelsUV is more than 1.5x larger than Scatter NVoxelsXY which is greater than the 1.2x magnification one might have assumed. This may reflect the relatively high fidelity and angular resolution of each voxel’s computed scatter distribution which is enabled by the use of Legendre polynomials to describe the profile and propagate scatter across the grid1 which, in turn, might allow for coarser volume resolution than detector resolution. Finally of note is that FrameRateDownsampling for each operating point was the same for the primary and scatter computations even though higher voxel and dexel resolution was required for the primary estimate. 3.3Example images along the Pareto frontFigure 3 shows zoomed-in images, corresponding to the red outlined region in Figure 2c, for the Operating Points in Table 3. The red arrows point to inhomogeneities in the form of streaks or shading. The top image (Operating Point A) takes the least amount of BTP time, 3.9 seconds, but does show artifacts. Moving down the figure, artifacts decrease as execution time increases. While homogeneity steadily improves with increased BTP time, we note that for many applications, images D or F may be perfectly acceptable, requiring 13 and 21 seconds BTP time respectively. Table 3a.Results for selected operating points matching Figure 3.
Table 3b.Results for selected operating points matching Figure 3.
3.4Contrast analysisThe histograms of different reconstructed volumes are shown in Figure 4. In the Uncorrected Image, the air and aluminum peaks are poorly separated, with slightly better separation occurring in the 2D VSHARP image and good separation in 3D VSHARP images. To quantify the separation of air and aluminum, the contrast-to-noise ratio in each histogram was computed by segmenting the images into “Air” or “Object” voxels and using the equation The resulting CNRs for operating Points A, D, F, I and “Golden” were 8.1, 8.9, 9.0, 9.0, 9.0 respectively. The Uncorrected CNR of 4.7 was the lowest while the 2D VSHARP CNR was slightly improved at 5.3. Points F and I have CNRs comparable to the golden image, which again suggests that there is no significant benefit to spending more than 10-20 seconds on the LBTE solution. 4.DISCUSSIONMC or pseudo-MC scatter correction methods such as 3D VSHARP can produce highly accurate scatter estimates and, in fact, are used as a gold standard for training ML-scatter correction methods. A main advantage of MC methods is that they are versatile since all that is needed at runtime is the geometric specification of the CBCT system and a physics library which characterizes it. However, MC or even pseudo-MC methods are generally not as fast as Machine Learning or Kernel methods since a first pass reconstruction is required and the scatter transport calculation is computationally intensive. For this data set, it was found that 10 to 20 second LTBE run times are sufficient if using an optimized sampling grid. We expect this result to be somewhat problem dependent, and may change with object size, complexity, or material, as well as with scanner geometry. However, it is interesting to note that our optimal time is roughly in line with the results of2. Of note is that the 2D VSHARP calibration was not optimized for this setup. Although proper tuning may improve 2D VSHARP image quality, we chose to leave it unoptimized to show that the segmentation algorithm is fairly forgiving. For future work, there are still many interesting parameters left to optimize including looking into non-uniform angular sampling2, optimizing interpolation and segmentation methods, and optimizing intrinsic LBTE parameters such as energy grouping. 5.CONCLUSION3D VSHARP was shown to significantly reduce scatter artifacts and produce excellent results with 10-20 seconds of computation time. Although a first-pass reconstruction is still needed, for the second pass reconstruction the results show that the additional time added by 3D VSHARP is minimal especially given that CST permits the LBTE computation to be performed in parallel with other FDK operations such as filtering and backprojection. REFERENCESA. Maslowski et. al,
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