Borgia G C, Brown R J, Fantazzini P
Department of ICMA, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy.
J Magn Reson. 2000 Dec;147(2):273-85. doi: 10.1006/jmre.2000.2197.
The basic method of UPEN (uniform penalty inversion of multiexponential decay data) is given in an earlier publication (Borgia et al., J. Magn. Reson. 132, 65-77 (1998)), which also discusses the effects of noise, constraints, and smoothing on the resolution or apparent resolution of features of a computed distribution of relaxation times. UPEN applies negative feedback to a regularization penalty, allowing stronger smoothing for a broad feature than for a sharp line. This avoids unnecessarily broadening the sharp line and/or breaking the wide peak or tail into several peaks that the relaxation data do not demand to be separate. The experimental and artificial data presented earlier were T(1) data, and all had fixed data spacings, uniform in log-time. However, for T(2) data, usually spaced uniformly in linear time, or for data spaced in any manner, we have found that the data spacing does not enter explicitly into the computation. The present work shows the extension of UPEN to T(2) data, including the averaging of data in windows and the use of the corresponding weighting factors in the computation. Measures are implemented to control portions of computed distributions extending beyond the data range. The input smoothing parameters in UPEN are normally fixed, rather than data dependent. A major problem arises, especially at high signal-to-noise ratios, when UPEN is applied to data sets with systematic errors due to instrumental nonidealities or adjustment problems. For instance, a relaxation curve for a wide line can be narrowed by an artificial downward bending of the relaxation curve. Diagnostic parameters are generated to help identify data problems, and the diagnostics are applied in several examples, with particular attention to the meaningful resolution of two closely spaced peaks in a distribution of relaxation times. Where feasible, processing with UPEN in nearly real time should help identify data problems while further instrument adjustments can still be made. The need for the nonnegative constraint is greatly reduced in UPEN, and preliminary processing without this constraint helps identify data sets for which application of the nonnegative constraint is too expensive in terms of error of fit for the data set to represent sums of decaying positive exponentials plus random noise.
UPEN（多指数衰减数据的统一惩罚反演）的基本方法在早期的一篇出版物中给出（博尔贾等人，《磁共振杂志》132卷，65 - 77页（1998年）），该出版物还讨论了噪声、约束和平滑对计算得到的弛豫时间分布特征的分辨率或表观分辨率的影响。UPEN对正则化惩罚应用负反馈，对于宽特征允许比尖锐线更强的平滑处理。这避免了不必要地加宽尖锐线和/或将宽峰或尾部分解为几个弛豫数据并不要求分开的峰。早期呈现的实验数据和人工数据是T(1)数据，并且所有数据间距固定，在对数时间上是均匀的。然而，对于通常在线性时间上均匀间隔的T(2)数据，或者对于以任何方式间隔的数据，我们发现数据间距并不明确进入计算过程。目前的工作展示了UPEN对T(2)数据的扩展，包括窗口内数据的平均以及在计算中使用相应的加权因子。实施了一些措施来控制计算分布中超出数据范围的部分。UPEN中的输入平滑参数通常是固定的，而不是依赖于数据。当UPEN应用于由于仪器不理想或调整问题而存在系统误差的数据集时，会出现一个主要问题，特别是在高信噪比情况下。例如，宽线的弛豫曲线可能会因弛豫曲线的人为向下弯曲而变窄。生成诊断参数以帮助识别数据问题，并且在几个例子中应用了这些诊断方法，特别关注弛豫时间分布中两个紧密间隔峰的有意义分辨率。在可行的情况下，几乎实时地使用UPEN进行处理应该有助于识别数据问题，同时仍可进行进一步的仪器调整。在UPEN中对非负约束的需求大大减少，并且在没有此约束的情况下进行初步处理有助于识别那些对于数据集而言应用非负约束在拟合误差方面代价过高的数据集合，因为这些数据集要表示衰减正指数之和加上随机噪声。
