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

### W. Zhu *et al.* (1995)

#### Summary

This paper presents a new algorithm for solving the perturbation equation of the form **W**Δ**x** =
Δ**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 **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 [**W**|Δ**I**] 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.