# Iterative total least–squares image reconstruction algorithm for optical tomography by the
conjugate gradient method

### W. Zhu *et al.* (1997)

#### Summary

We present an iterative total least–squares algorithm for computing images of the interior of
highly scattering media by using the conjugate gradient method. For imaging of dense scattering
media in optical tomography, a perturbation approach has been described previously [Y. Wang *et
al.*, Proc SPIE **1641**, 58 (1992); R. L. Barbour *et al.*, in *Medical Optical
Tomography: Functional Imaging and Monitoring* (Society of Photo–Optical Instrumentation
Engineers, Bellingham, Wash., 1993), pp. 87–120], which solves a perturbation equation of the form
**W**Δ**x** = Δ**I**. In order to solve this equation, least–squares or
regularized least–squares solvers have been used in the past to determine best fits to the
measurement data Δ**I** while assuming that the operator matrix **W** is accurate. In
practice, errors also occur in the operator matrix. Here we propose an iterative total
least–squares (ITLS) method that 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 smallest singular value. The proposed ITLS
method obtains this solution by using a conjugate gradient method that is particularly suitable
for very large matrices. Simulation results have shown that the TLS method can yield a
significantly more accurate result than the least–squares method.