Computer Tomography

Abstract

In this research, a stochastic model for attenuation in Computer Tomography is developed. This model gives rise to the idea of using path dependent variance of the measurements (instead of constant variance) to improve image reconstruction. The distribution of the measurements in this model is determined and a difference with the current literature is found, which leads to a refinement of the noise or measurement errors in the model. To use the information about the variance of the measurements in the image reconstruction, a numerical model is considered in which a discretization is made of the tomographic image that has to be reconstructed, i.e., the unknown attenuation coefficients. Incorporating weights to reflect the relation between the area that is traversed by an X-ray beam and the entire area of a pixel in the grid results in a linear system of equations. Because the measurements are not exact, noise is added to this linear system of equations, which leads to a perturbed problem. A transformation of the measurements is needed to obtain the desired linear system of equations. The Delta Method is used for this purpose. Another method used for the transformation of stochastic models, variance stabilization, is briefly considered. The log-likelihood of the unknown attenuation coefficients is determined under different assumptions for the mean and variance of the measurements. A connection is made between the log-likelihood and the (weighted) Least-Squares Estimation, leading to different ideas for the adjustment of the current reconstruction algorithm. Several new reconstruction algorithms are developed to improve the image reconstruction by using the path dependent variance of the noise. Most of these new algorithms result in a better reconstruction than the current algorithm, but a problem is found when the convergence of the iterative algorithms is considered. In addition to the relative error, the log-likelihood function and the weighted sum of squared errors are used to investigate the convergence of the iterative reconstruction algorithms. Relaxation is incorporated into the iterative reconstruction algorithm to improve the convergence. A slightly better convergence is obtained, but progress could be made if a convergent iterative algorithm is found.