Total least squares approach for the solution of the perturbation equation

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Abstract
This paper presents a new algorithm for solving the perturbation equation of the form W(Delta) x equals (Delta) I 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 (Delta) x 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 iterative 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 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.

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