Penumbra-effect induced spectral mixing in x-ray computed tomography: a multi-ray spectrum estimation model and subsampled weighting algorithm

<i>Objective.</i>With the development of spectral CT, various spectral imaging technologies have been proposed. Among these, filter-based spectral imaging methods have been greatly advanced in recent years, such as split filters used in clinical diagnose, spectral modulators studied for spectral imaging and scatter correction. However, due to the finite size of the focal spot of x-ray source, spectral filters cause spectral mixing in the penumbra region. Traditional spectrum estimation methods fail to account for it, resulting in reduced spectral accuracy. To address this challenge, we develop a multi-ray spectrum estimation model and propose an Adaptive Subsampled WeIghting of Filter Thickness (A-SWIFT) method.<i>Approach.</i>First, we estimate the unfiltered spectrum using traditional methods. Next, we model the final spectra as a weighted summation of spectra attenuated by multiple filters. The weights and equivalent lengths are obtained by x-ray transmission measurements taken with altered spectra using different kVp or flat filters. Finally, the spectra are approximated by using the multi-ray model. To mimic the penumbra effect, we used a spectral modulator (0.2 mm Mo, 0.6 mm Mo), a split filter (0.07 mm Au, 0.7 mm Sn), and the abdominal images of an XCAT phantom in simulations; in experiments, we used spectral modulators made by molybdenum or copper along with a pure water phantom, a Gammex multi-energy CT phantom and a Kyoto chest phantom for validation.<i>Main results.</i>Simulation results show that the mean energy bias in the penumbra region decreased from 7.43 keV using the previous Spectral Compensation for Modulator (SCFM) method to 0.72 keV using the A-SWIFT method for the split filter, and from 1.98 keV to 0.61 keV for the spectral modulator. In physics experiments, for the pure water phantom with a molybdenum modulator, the average error of the mean values (ERMSE) in selected regions of interests decreased from 77 to 7 Hounsfield units (HU) using the A-SWIFT method compared with SCFM method; for the Gammex phantom,ERMSEin iodine images was 0.2 mg ml^-1using A-SWIFT method, and 1.5 mg ml^-1using SCFM method; for the chest phantom with an added 5 mg ml^-1iodine cylinder, the estimated material density of the iodine inserts was 5.0 mg ml^-1using A-SWIFT method, and 5.9 mg ml^-1using SCFM method.<i>Significance.</i>Based on a multi-ray spectrum estimation model, the A-SWIFT method provides an accurate and robust spectrum estimation in the penumbra region, contributing to enhanced spectral imaging performance of CT systems utilizing spectral filters.