OPTICAL REVIEW Vol. 7, No. 2 (2000) 132-137 An Acceleration Algorithm for Ilnage Reconstruction Based on Continuous-Discrete Mapping Model Taiga YAMAYA, Takashi OBI, Masahiro YAMAGUCHI and Nagaaki OHYAMA* Imaging Scielece and Elegi7eeerileg Laboratory. Tokyo 1lestitute of Technologv, 4259. Nagatsuta. Midori-ku. Yokohama, 226- 8503 Japan (Received September 30, 1999; Accepted February 1, 2000) In image reconstruction for X-ray computed tomography, images reconstructed by filtered backprojection (FBP) include systematic errors because the FBP method does not take into account some of the properties of the actual imaging system such as the divergence of X-ray beam. One solution to this problem is to use algebraic reconstruction methods, such as generalized analylic reconstruction from discrete samples and natural pixel decomposition. However, in the process of reconstruction using these methods, it is necessary to solve the linear algebraic equations which have a large coefiicient matrix. In this paper, we propose a method to accelerate the iteration solving these equations by preconditioning the coe~icient matrix using a polynomial function. The results of the computer simulations show the effectiveness of the proposed method. Key words: computed tomography, GARDS, NPD, continuous-discrete mapping model, conjugate gradient algorithm, preconditioning, acceleration 1. Introductioll Computed tomography (CT), such as X-ray CT, single photon emission CT (SPECT) and positron emission tomography (PET), are being widely used in medical di- agnosis, and these imaging systems usually use a recon- struction algorithm based on the filtered backprojection (FBP) method. The FBP method, however, does not take into account some of the properties of the actual imaging system such as the divergence of X-ray beam because it assumes that projection data are obtained by line in- tegral. Images reconstructed by the FBP method thus sometimes include error due to the inaccurate modeling of the imaging system, and such error causes, for exam- ple, an artifact called the edge effect. One solution to this problem is to use algebraic reconstruction methods, which can improve image quality by accurate modeling of the measurement system. Among them, methods based on a continuous-discrete (C-D) mapping model, in which the object space is defined as continuous while the observation space is discrete, are known to be effective in combating this problem.1) Methods proposed based on the C-D model are generalized analyiic reconstruction from discrete samples (GARDS)2) and natural pixel decomposition (NPD),3) and their advantages have been reported.4-7) However, in the process of reconstruction using these techniques, it is necessary to solve linear algebraic equa- tions which have a large coefiicient matrix of the same dimension as that of the observed data. For example, in the present X-ray CT system, the dimensions of ob- served data will be over 250,000 for a one slice image. Consequently, solving these equations requires a huge amount of computation. * E-mail: yama@isl.titech.ac.jp Iterative methods, such as the conjugate gradient (CG) algorithm,8) are convenient to use for solving large sets of linear equations. It is also known that the convergence of the CG algorithm depends on the numerical condition of the coefiicient matrix. In this paper, we propose a means of accelerating the convergence of the CG algorithm solv- ing these equations by preconditioning the coefficient matrix using a polynomial function. GARDS and NPD are mathematically equivalent except for their deriva- tion, and we deal here with GARDS as the reconstruction method based on the C-D model. In Sect. 2, after a brief review of GARDS, the theory of the acceleration al- gorithm is introduced. In Sect. 3, computer simulations of X-ray CT are carried out to examine the performance of the proposed method. Finally, concluding remarks are presented in Sect. 4. 2. Method In this section, GARDS is briefly reviewed, and the theory of the acceleration algorithm is described. 2. I A Review of GARDS In the C-D mapping model of linear imaging systems, the object space and the observation space are treated as continuous and discrete, respectively, and the continu- ous object f(r) is mapped to a set of observed data g, which is represented by a column vector. For an element gi of the vector g, the C-D mapping model gives the equa- tion: gi=J f(r)h,(r)dr (i=1-M), where r indicates the position in the object space. M is the number of observed data, and hi(r) is the sensitivity function defining both the area and the gain factors con- tributing to the i-th detector element. The integral is res- tricted to the object support v, which means the region l 32