Image reconstruction by means of chaotic iterative algorithm

The image reconstruction problem from projection data for some particular schemes of reconstruction is considered. There are used the Algebraic Reconstruction Technique (ART) and the chaotic algebraic iterative algorithm CHART to reconstruct high-construct objects from incomplete data. The influence of various parameters of these algorithms, such as the relaxation coefficient λ, the number of iterations, the number of projection data, and the noise in the projection data on reconstruction quality for different schemes of reconstruction is investigated. Numerical simulation results for a number of modeling high-contrast objects are presented and discussed. Introduction Computed tomography allows to research the internal structure of an object without its destruction. This technique makes possible to reconstruct this structure using the projection data collected outside the object. In the recent years computed tomography found a wide application not only in medicine but also for solving many technical problems. In all these applications we deal with inverse problems which are those of image reconstruction from projections. In spite of the wide different physical methods for collecting projection data there is a common mathematical nature of reconstruction problems: there is an unknown distribution of some physical parameter; a finite number of line integrals of this parameter can be estimated from physical measurements, and an estimate of the distribution of the original parameter is desired. Theoretically it is possible to reconstruct the internal structure of an object uniquely without errors by means of the Radon inversion formula. If the number of projections is large enough and projections can be obtained for any orientation (in medicine, for example) then it is more preferable to use the analytical methods based on the inverse Radon transform or the Fourier transform [1-3]. However in practice projections are obtained by physical measurements and therefore they are given with some errors. Sometimes they cannot be available for each orientation and/or their number is very small, they may be sparse or missing at certain orientations (the limited angle problem). In all these cases one says that there is a problem of image reconstruction with incomplete projection data. In particular, such kind of N. Gubareni, M. Pleszczynski 60 problems arise in mineral industries and engineering geophysics connected with acid drainage, stability of mine workers, mineral exploration and others [4, 5]. In these applications the algorithms based on analytical methods cannot be used [6]. The alternative reconstruction method which can be used under these conditions is the Algebraic Reconstruction Technique (ART), i.e. the iterative algebraic algorithms [7]. In this case the problem of image reconstruction is reduced to solving a system (possibly inconsistent) of linear algebraic equations, one for each projection ray, which has a few main characteristics: it is rectangular (not square as a rule), and it has a very large dimension. The problem of solving a large nonsingular system of algebraic equations is the most wide-spread and arises in many other applications. For solving such systems the different kinds of algebraic iterative algorithms are often used, the best known from which is the algorithm ART [7-10]. They are rather simple, flexible and permit to use a priori knowledge of the object before its reconstruction that is very important in many practical applications with incomplete projection data. Recently, these algorithms are also used in magnetic resonance imaging (MRI) [11, 12]. The outline of this paper is as follows. Section 1 considers the problem of image reconstruction from projections. For solving this problem the main algebraic iterative algorithm ART is used. In this section we also consider a chaotic iterative algebraic algorithm CHART, which is based on asynchronous methods and can been realized on a sequential computer. Section 2 is devoted to study the quality and convergence of the algebraic algorithms ART and CHART for image reconstruction of some modeling objects from incomplete projection data by means of computer simulation. Some numerical results of computer simulation which shows the comparing evaluations of errors and speed of convergence of these algorithms are presented in section 3. The influence of various parameters of these algorithms, such as the relaxation coefficient, the number of iterations, the number of projections, and the noise in the projections, on the reconstruction quality for different schemes of reconstruction are investigated. It is shown that the best results are obtained by means of the chaotic iterative algorithm CHART, which has the same robustness as the algorithm ART. 1. Algebraic algorithms of reconstruction Let f(x,y) be a function which represents the spatial distribution of a physical parameter. If θ θ sin cos : y x l L + = is a line (ray) in the plane then the line integral