Zhu W, Wang Y, Zhang J
Department of Electrical Engineering, Polytechnic University, Brooklyn, New York 11201, USA.
J Opt Soc Am A Opt Image Sci Vis. 1998 Oct;15(10):2639-50. doi: 10.1364/josaa.15.002639.
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.
在之前的一篇论文[朱等人，《美国光学学会杂志A》14，799（1997）]中，提出了一种基于瑞利商公式获取线性系统总体最小二乘（TLS）解的迭代算法。在此，我们推导出据我们所知该解的首批统计特性。结果表明，当数据和算子元素中的噪声项均为独立同分布的高斯分布时，瑞利商形式的TLS（RQF - TLS）估计器等同于最大似然估计器。推导了RQF - TLS解的扰动分析，并由此以封闭形式获得了RQF - TLS解的均方误差，该误差在低噪声水平下有效。然后，我们提出了一种基于小波的多分辨率方案来获取TLS解。该方法与多重网格算法一起用于求解光学层析成像中遇到的线性扰动方程。数值模拟结果表明，该方法所需的计算量比先前报道的单网格TLS算法少得多。该方法还允许从粗级别重建中快速识别感兴趣区域，并将后续精细分辨率下的重建限制在这些区域。最后，该方法对噪声的敏感度低于单网格TLS和多重网格最小二乘算法。
