### Abstract

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 requires substantially less computation 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 multigrid least-squares algorithms.

Original language | English (US) |
---|---|

Pages (from-to) | 2639-2650 |

Number of pages | 12 |

Journal | Journal of the Optical Society of America A: Optics and Image Science, and Vision |

Volume | 15 |

Issue number | 10 |

State | Published - 1998 |

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### ASJC Scopus subject areas

- Electronic, Optical and Magnetic Materials
- Computer Vision and Pattern Recognition

### Cite this

*Journal of the Optical Society of America A: Optics and Image Science, and Vision*,

*15*(10), 2639-2650.

**Total least-squares reconstruction with wavelets for optical tomography.** / Zhu, Wenwu; Wang, Yao; Zhang, Jun.

Research output: Contribution to journal › Article

*Journal of the Optical Society of America A: Optics and Image Science, and Vision*, vol. 15, no. 10, pp. 2639-2650.

}

TY - JOUR

T1 - Total least-squares reconstruction with wavelets for optical tomography

AU - Zhu, Wenwu

AU - Wang, Yao

AU - Zhang, Jun

PY - 1998

Y1 - 1998

N2 - 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 requires substantially less computation 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 multigrid least-squares algorithms.

AB - 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 requires substantially less computation 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 multigrid least-squares algorithms.

UR - http://www.scopus.com/inward/record.url?scp=0032186422&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032186422&partnerID=8YFLogxK

M3 - Article

C2 - 9768509

AN - SCOPUS:0032186422

VL - 15

SP - 2639

EP - 2650

JO - Journal of the Optical Society of America A: Optics and Image Science, and Vision

JF - Journal of the Optical Society of America A: Optics and Image Science, and Vision

SN - 0740-3232

IS - 10

ER -