2019 Volume 23 Issue 2 Pages 41-48
The basic mathematical problem in computed tomography (CT) is defined as finding a nonnegative solution to a rectangular linear system with a sparse coefficient vector and matrix. The coefficient vector and matrix respectively correspond to acquired projection data and a projection operator based on a discrete Radon transform. As an approach to find a nonnegative solution corresponding to a tomographic image, a continuous method using simultaneous ordinary differential equations has been proposed. The continuous method can produce high-quality images even from an insufficient number of projection data. However, it requires a huge computational cost to obtain a high-quality image because of numerical integration. In this paper, to reduce the image-reconstruction time, we proposed two iterative methods without numerical integration that enable us to reconstruct high-quality CT images without negative pixels. The methods were produced on the basis of a hierarchical alternating least-squares algorithm that is known as a fast solver for nonnegative matrix factorization and conjugate gradient methods that are effective for a linear system. Through numerical experiments, we also discussed the performances of the proposed methods in terms of both the quality of obtained images and the time required to obtain high-quality images.
