Accelerated iterative image reconstruction methods based on block-circulant system matrix derived from a cylindrical image representation

Iterative image reconstruction methods based on an accurate and fully three-dimensional (3D) system probability matrix are well-known to provide images of higher quality. However, the size of the system matrix and the computation burden often make such methods impractical. To address this problem, we proposed to use a cylindrical image representation that preserves both in-plane and axial symmetries between the tubes of response for a given camera, leading to a system matrix having a block-circulant structure. For 3D image reconstruction, such a system matrix can be structured into a block-circulant matrix where blocks are themselves block-circulant. By storing only non-redundant parts of the block-circulant matrix, memory requirements can be reduced by a factor equivalent to the total number of system symmetries. The block-circulant system matrix can be stored in the Fourier domain representation to accelerate the forward and back projection steps of the iterative image reconstruction methods. When represented in the Fourier domain, the system matrix sparsity is reduced compared to the spatial domain representation, but some null values are still preserved.