# Measuring Identification and Quantification Errors in Spectral CT Material Decomposition
## Materials and Methods
### Image Processing and Reconstruction  
Flat field measurements (1440 for each camera position and energy bin) were taken after the sample scan to correct for fixed-pattern variations in individual pixel responses. The raw data in DICOM format was transferred from the scanner's server to the scanner's inbuilt patient archive and communication system (PACS), where an automated image processing was performed. The image processing has three stages: pre-reconstruction processing, reconstruction into attenuation volumes, and decomposition into material volumes (see Section 2.5). In this case, the image processing produced slices with 0.1 mm\({}^{2}\) voxels on a 480 \(\times\) 480 matrix. The pre-reconstruction processing consists of three steps; the first step is pixel masking to remove any bad pixels; the second step is flat field correction [32; 46]; the final step is ring filtration. The ring filter is based on the work by [PERSON] and [PERSON] [47], but applied to the projection data and adapted into 3D to exploit more information provided by the detector (x, y, and theta). The reconstruction uses a low-resolution version of the polychromatic form of the Beer-Lambert Law. What this means is that the overlapping, low-threshold counters that measure the data are processed simultaneously to produce attenuation volumes representing non-overlapping energy bins across the measured spectrum. For example, four counters representing the energy range 30-120 keV, 45-120 keV, 60-120 keV and 78-120 keV would simultaneously produce four attenuation volumes representing the energy ranges 30-45 keV, 45-60 keV, 60-78 keV and 78-120 keV. Processing the counters simultaneously exploits all photons available in the scan. Consider the attenuation volume for 78-120 keV. All four counters contribute to this volume. The counter for 60-120 keV provides specific information about the energy range but has fewer photons (higher statistical noise). The counter for 30-120 keV is less specific but contains all the photons in the scan (lower statistical noise). Given that the counter's measurements are acquired simultaneously, we know each counter maps the same structures. This reconstruction process uses all photons available in the wider counters to reconstruct clean boundaries and structures, while using the narrower counters to guarantee that the correct energy response is obtained. The reconstruction algorithm itself is a statistical iterative technique. It uses multiplicative correction terms similar to OSEM [48], or MART [49]. The statistical element is introduced by weighting the corrections to the volume by a normalized Poisson distribution to slow down convergence when approaching the solution, thereby reducing noise. The reconstruction algorithm also adopts a  
Figure 2: (**a**) Spectroscopic phantom with three 6 mm diameter hydroxyapatite calibration rods (54.3, 211.7 and 808.5 mg/mL) and 6 mm diameter vials of gadolinium (1, 2, 4, 8 mg/mL), oil (canola oil) and distilled water; (**b**) CT image of the phantom. multi-stage approach where it initially reconstructs voxels that are eight times larger than requested. Later on, this is repeatedly subdivided for a total of four stages until the requested voxel size is reached [50]. This approach allows for the reconstruction to proceed quickly. It is also a weak form of a sparsity constraint as a large voxel is the same as a set of small voxels with the same value. Lastly, the larger the voxel, the more pixels from the projection images will contribute to it. This reduces the effect of dead regions. This is particularly useful during the initial reconstruction stages where the effects of dead regions are the most significant. To evaluate the spectroscopic response, CT slices corresponding to a thickness of \(\sim\)0.5 mm were averaged and then analyzed.