The stability of images taken in computed tomography by back-projection method

Abstract

Background: The back-projection method is quite widespread in the modern computed tomography and there are not principled critical comments addressed to it in the scientific literature. But there are reasons for such comments. The main one of them, in our opinion, is that the method does not use the theory of ill-posed problems in any way, despite the fact that the problem of the reconstruction is ill-posed. Objectives: The aim of the work is a development of a method of the reconstruction of tomograms, which is a modification of the back-projection method taking into account the theory of ill-posed problems. Materials and methods: In the back-projection method, the value of the filter parameter is chosen practically arbitrarily. In the method proposed in the article, this choice receives a justification: the filter parameter is identified with the regularization parameter that allows us to use the theory of ill-posed problems for its determination, and thereby to ensure the stability of the reconstructed image. Results: The dependence of the quality of the reconstruction on the selected filter width was obtained. The value of the filter at the given error level of the initial data and the given geometry corresponding to the minimum error of the reconstructed image was found. The value of the filter width depends on the scanning settings and the noise level on the projections, so the result is not the specific value of the filter but it’s the way to select the optimal value. Conclusions: In this paper we show that it is possible without completely abandon the back-projection method with all its positive sides to modify this method supplementing it with the approaches used in the theory of ill-posed problems. Such an approach must ensure the stability of the reconstructed image. This can be, for example, the identification of the filter parameter with the regularization parameter that formed the basis for the method of image reconstruction proposed in this article. It’s possible there are other ways of using the theory of ill-posed problems in the of back-projection method.