A Total Least Squares Approach for the Solution of the Perturbation Equation

W. Zhu et al. (1995)


This paper presents a new algorithm for solving the perturbation equation of the form WΔx = ΔI encountered in optical tomographic image reconstruction. The methods we developed previously are all based on the least squares formulation, which finds a solution that best fits the measurement Δx while assuming the weight matrix W is accurate. In imaging problems, usually errors also occur in the weight matrix W. In this paper, we propose an interative total least squares (ITLS) method which minimizes the errors in both weights and detector readings. Theoretically the total least squares (TLS) solution is given by the singular vector of the matrix [WI] associated with the minimal singular value. The proposed ITLS method obtains this solution using a conjugate gradient method which is particularly suitable for very large matrices. Experimental results have shown that the TLS method can yield a significantly more accurate result than the LS method when the perturbation equation is overdetermined.