Sinogram Blurring Matrix Estimation From Point Sources Measurements With Rank-One Approximation for Fully 3-D PET

An accurate system matrix is essential in positron emission tomography (PET) for reconstructing high quality images. To reduce storage size and image reconstruction time, we factor the system matrix into a product of a geometry projection matrix and a sinogram blurring matrix. The geometric projection matrix is computed analytically and the sinogram blurring matrix is estimated from point source measurements. Previously, we have estimated a 2-D blurring matrix for a preclinical PET scanner. The 2-D blurring matrix only considers blurring effects within a transaxial sinogram and does not compensate for inter-sinogram blurring effects. For PET scanners with a long axial field of view, inter-sinogram blurring can be a major problem influencing the image quality in the axial direction. Hence, the estimation of a 4-D blurring matrix is desirable to further improve the image quality. The 4-D blurring matrix estimation is an ill-conditioned problem due to the large number of unknowns. Here, we propose a rank-one approximation for each blurring kernel image formed by a row vector of the sinogram blurring matrix to improve the stability of the 4-D blurring matrix estimation. The proposed method is applied to the simulated data as well as the real data obtained from an Inveon microPET scanner. The results show that the newly estimated 4-D blurring matrix can improve the image quality over those obtained with 2-D blurring matrix and requires less point source scans to achieve similar image quality compared with an unconstrained 4-D blurring matrix estimation.