# Total least–squares reconstruction with wavelets for optical tomography

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

#### Summary

In a previous paper [Zhu *et al.*, J. Opt. Soc. Am. A **14**, 799 (1997)] an iterative
algorithm for obtaining the total least–squares (TLS) solution of a linear system based on the
Rayleigh quotient formulation was presented. Here we derive what to our knowledge are the first
statistical properties of this solution. It is shown that the Rayleigh–quotient–form TLS
(RQF–TLS) estimator is equivalent to the maximum–likelihood estimator when noise terms in both
data and operator elements are independent and identically distributed Gaussian. A perturbation
analysis of the RQF–TLS solution is derived, and from it the mean square error of the RQF–TLS
solution is obtained in closed form, which is valid at small noise levels. We then present a
wavelet–based multiresolution scheme for obtaining the TLS solution. This method was employed
with a multigrid algorithm to solve the linear perturbation equation encountered in optical
tomography. Results from numerical simulations show that this method requries substantially less
than the previously reported one–grid TLS algorithm. The method also allows one to identify
regions of interest quickly from a coarse–level reconstruction and restrict the reconstruction in
the following fine resolutions to those regions. Finally, the method is less sensitive to noise
than the one–grid TLS and multi–grid least–squares algorithms.