Sinogram recovery with the method of convex projections for limited-data reconstruction in computed tomography

Image reconstruction from incomplete projection data is strongly required in widespread applications of computed tomography. This problem can be formulated as a sinogram–recovery problem. The sinogram–recovery problem is to find a complete sinogram that is compatible with the Helgason–Ludwig consistency condition, the measured incomplete sinogram, and other a priori knowledge about the sinogram in question. The direct use of the Helgason–Ludwig consistency condition considerably reduces computational requirements and the accumulation of digital-processing errors over the conventional iterative reconstruction–reprojection method. Most research for solving the sinogram–recovery problem is based on directly inverting systems of linear equations associated with the Helgason–Ludwig consistency condition. However, these noniterative techniques cannot be applied to various different types of limited-data situations in a unified way. Moreover, nonlinear a priori constraints such as the nonnegativity and the amplitude limit are not easily incorporated. We solve the sinogram–recovery problem by using an iterative signal-recovery technique known as the method of projection onto convex sets. Once an estimation of the complete sinogram is obtained, the conventional convolution–backprojection method can be utilized to reconstruct an image. The performance of the proposed method is investigated both with numerical phantoms and with actual x-ray data.