Ochs M F, Stoyanova R S, Arias-Mendoza F, Brown T R
NMR and Medical Spectroscopy, Fox Chase Cancer Center, Philadelphia, PA, USA.
J Magn Reson. 1999 Mar;137(1):161-76. doi: 10.1006/jmre.1998.1639.
A frequent problem in analysis is the need to find two matrices, closely related to the underlying measurement process, which when multiplied together reproduce the matrix of data points. Such problems arise throughout science, for example, in imaging where both the calibration of the sensor and the true scene may be unknown and in localized spectroscopy where multiple components may be present in varying amounts in any spectrum. Since both matrices are unknown, such a decomposition is a bilinear problem. We report here a solution to this problem for the case in which the decomposition results in matrices with elements drawn from positive additive distributions. We demonstrate the power of the methodology on chemical shift images (CSI). The new method, Bayesian spectral decomposition (BSD), reduces the CSI data to a small number of basis spectra together with their localized amplitudes. We apply this new algorithm to a 19F nonlocalized study of the catabolism of 5-fluorouracil in human liver, 31P CSI studies of a human head and calf muscle, and simulations which show its strengths and limitations. In all cases, the dataset, viewed as a matrix with rows containing the individual NMR spectra, results from the multiplication of a matrix of generally nonorthogonal basis spectra (the spectral matrix) by a matrix of the amplitudes of each basis spectrum in the the individual voxels (the amplitude matrix). The results show that BSD can simultaneously determine both the basis spectra and their distribution. In principle, BSD should solve this bilinear problem for any dataset which results from multiplication of matrices representing positive additive distributions if the data overdetermine the solutions.
分析中一个常见的问题是需要找到两个与基础测量过程密切相关的矩阵，它们相乘后能重现数据点矩阵。这类问题在整个科学领域都会出现，例如在成像中，传感器的校准和真实场景可能都未知；在局部光谱学中，任何光谱中可能存在多种成分且含量各异。由于这两个矩阵都是未知的，所以这种分解是一个双线性问题。我们在此报告针对这种情况的一个解决方案，即分解得到的矩阵元素来自正加法分布。我们在化学位移成像（CSI）上展示了该方法的威力。新方法，即贝叶斯光谱分解（BSD），将CSI数据简化为少量的基础光谱及其局部幅度。我们将这种新算法应用于对人肝脏中5 - 氟尿嘧啶分解代谢的19F非局部研究、对人头和小腿肌肉的31P CSI研究以及展示其优缺点的模拟。在所有情况下，数据集被视为一个矩阵，其行包含各个NMR光谱，它是由一个通常非正交的基础光谱矩阵（光谱矩阵）与各个体素中每个基础光谱幅度的矩阵（幅度矩阵）相乘得到的。结果表明，BSD能够同时确定基础光谱及其分布。原则上，如果数据能确定解，BSD应该能解决任何由表示正加法分布的矩阵相乘产生的双线性问题。
