Borgia G C, Brown R J, Fantazzini P
Department of ICMA, University of Bologna, Italy.
J Magn Reson. 1998 May;132(1):65-77. doi: 10.1006/jmre.1998.1387.
NMR relaxation data and those from many other physical measurements are sums of exponentially decaying components, combined with some unavoidable measurement noise. When decay data are inverted in order to give quasi-continuous distributions of relaxation times, some smoothing of the distributions is normally implemented to avoid excess variation. When the same distribution has a sharp peak and a much broader peak or a "tail," as for many porous media saturated with liquids, an inversion program using a fixed smoothing coefficient may broaden the sharp peak and/or break the wide peak or tail into several separate peaks, even if the coefficient is adaptively chosen in accord with the noise level of the data. We deal with this problem by using variable smoothing, determined by iterative feedback in such a way that the smoothing penalty is roughly constant. This uniform-penalty (UP) smoothing can give sharp lines, not broadened more than is consistent with the noise, and in the same distribution it can show a tail decades long without breaking it up into several peaks. The noise level must be known approximately, but it can be determined more than adequately by a preliminary inversion. The same iterative procedure is used to implement constraints such as non-negative (NN) or monotonic-from-peak (MT). The significance of an additional resolved peak may be tested by finding the cost of using MT to force a unimodal solution. A bimodal constraint can be applied. Decay data representing sharp lines in contact with broad features can require substantial computing time and some controls to stabilize the iterative sequence. However, UP can be made to function smoothly for a very wide variety of decay curves, which can be processed without adjustment of parameters, including the dimensionless smoothing parameters. Extensive testing has been done with artificial data. Examples are shown for artificial data, biological tissues, ceramic technology, and sandstones. Expressions are given relating noise level to line width and for significance of increase or decrease in error of fit.
核磁共振弛豫数据以及许多其他物理测量得到的数据都是指数衰减分量的总和，并伴有一些不可避免的测量噪声。当对衰减数据进行反演以得到弛豫时间的准连续分布时，通常会对分布进行某种平滑处理，以避免过度波动。当同一分布有一个尖锐峰和一个宽得多的峰或“尾部”时，就像许多充满液体的多孔介质那样，使用固定平滑系数的反演程序可能会使尖锐峰变宽，和/或把宽峰或尾部拆分成几个单独的峰，即使该系数是根据数据的噪声水平自适应选择的。我们通过使用可变平滑来处理这个问题，可变平滑由迭代反馈确定，使得平滑惩罚大致恒定。这种均匀惩罚（UP）平滑可以给出尖锐的谱线，其展宽不超过与噪声相符的程度，并且在同一分布中，它可以显示出长达数十年的尾部而不会将其拆分成几个峰。噪声水平必须大致已知，但可以通过初步反演充分确定。相同的迭代过程用于实施诸如非负（NN）或从峰单调（MT）等约束。可以通过找到使用MT强制单峰解的代价来检验额外解析峰的显著性。可以应用双峰约束。表示与宽特征接触的尖锐谱线的衰减数据可能需要大量计算时间和一些控制来稳定迭代序列。然而，UP可以针对非常多种衰减曲线平稳运行，无需调整参数（包括无量纲平滑参数）即可对其进行处理。已经对人工数据进行了广泛测试。给出了人工数据、生物组织、陶瓷技术和砂岩的示例。给出了将噪声水平与谱线宽度相关的表达式以及拟合误差增加或减少的显著性表达式。
