Matrixed-Spectrum Decomposition Accelerated Linear Boltzmann Transport Equation Solver for Fast Scatter Correction in Multi-Spectral CT

X-ray scatter has been a serious concern in computed tomography (CT), leading to image artifacts and distortion of CT values. The linear Boltzmann transport equation (LBTE) is recognized as a fast and accurate approach for scatter estimation. However, for multi-spectral CT, it is cumbersome to compute multiple scattering components for different spectra separately when applying LBTE-based scatter correction. In this work, we propose a Matrixed-Spectrum Decomposition accelerated LBTE solver (MSD-LBTE) that can be used to compute X-ray scatter distributions from CT acquisitions at two or more different spectra simultaneously, in a unified framework with no sacrifice in accuracy and nearly no increase in computation in theory. First, a matrixed-spectrum solver of LBTE is obtained by introducing an additional label dimension to expand the phase space. Then, we propose a ``spectrum basis'' for LBTE and a principle of selection of basis using the QR decomposition, along with the above solver to construct the MSD-LBTE. Based on MSD-LBTE, a unified scatter correction method can be established for multi-spectral CT. We validate the effectiveness and accuracy of our method by comparing it with the Monte Carlo method, including the computational time. We also evaluate the scatter correction performance using two different phantoms for fast-kV switching based dual-energy CT, and using an elliptical phantom in a numerical simulation for kV-modulation enabled CT scans, validating that our proposed method can significantly reduce the computational cost at multiple spectra and effectively reduce scatter artifact in reconstructed CT images.