Arrangement of projection angles in Mojette-transform based tomography

Abstract

<sec>Computed tomography (CT) is an effective tool for three-dimensional (3D) imaging by using optical detectors to capture the two-dimensional (2D) projections of tested parameters from multiple views and realizing 3D reconstruction through various algorithms. However, for practical applications, typically only a few detectors can be applied due to their high expense and the limited optical access of the test environment. The realization of high precision reconstruction with a few projections is of great significance for promoting the development and application of CT technology. The spatial arrangement of the detectors determines the amount of useful information collected by the system, which greatly affects the quality of CT reconstruction. Therefore, in this work we study the optimization method of projection arrangement based on the 3D Mojette transform theory.</sec><sec>Mojette transform is a special discrete form of Radon transform, which can realize projection sampling with minimum redundancy and accurate tomographic reconstruction from less projection angles. It provides a new way to realize the CT technology with fewer projections. However, the existing researches mainly focus on the reconstruction theories of 2D Mojette transform, which is used for realizing the 2D slice tomography. In order to realize the real 3D tomographic reconstruction, in this work we establish a mathematical model of 3D Mojette transform, and study its accurate reconstruction condition. The results show that the 3D Mojette transform is a combination of twice 2D Mojette transform in two directions. The accurate reconstruction condition of 3D Mojette transform is that the sum of the absolute values of projection vectors’ components in <i>x</i>, <i>y</i>, and <i>z</i> directions is greater than the number of discrete grids in each direction. The correctness of the mathematical model and the accurate reconstruction condition are verified by numerical simulations.</sec><sec>Considering the limitation of the pixels in the practical detectors, the method to determine the optimal arrangement of projection angles is proposed. The results indicate that the optimal arrangement is that all detectors are located in the same horizontal plane around the tested object, where the projection model is reduced to 2D Mojette transform. In this case, the minimum projection angles and pixels are required and the projection angles can be positioned in a smaller spatial range. If the condition cannot be satisfied in practice, projection vectors with smaller |<i>p_i</i>| and |<i>q_i</i>| should be chosen. This research provides the theoretical basis for establishing the actual CT system.</sec>