Volken Werner, Frei Daniel, Manser Peter, Mini Roberto, Born Ernst J, Fix Michael K
Division of Medical Radiation Physics, Inselspital and University of Bern, Switzerland.
Phys Med Biol. 2008 Nov 7;53(21):6245-63. doi: 10.1088/0031-9155/53/21/023. Epub 2008 Oct 15.
The problem of re-sampling spatially distributed data organized into regular or irregular grids to finer or coarser resolution is a common task in data processing. This procedure is known as 'gridding' or 're-binning'. Depending on the quantity the data represents, the gridding-algorithm has to meet different requirements. For example, histogrammed physical quantities such as mass or energy have to be re-binned in order to conserve the overall integral. Moreover, if the quantity is positive definite, negative sampling values should be avoided. The gridding process requires a re-distribution of the original data set to a user-requested grid according to a distribution function. The distribution function can be determined on the basis of the given data by interpolation methods. In general, accurate interpolation with respect to multiple boundary conditions of heavily fluctuating data requires polynomial interpolation functions of second or even higher order. However, this may result in unrealistic deviations (overshoots or undershoots) of the interpolation function from the data. Accordingly, the re-sampled data may overestimate or underestimate the given data by a significant amount. The gridding-algorithm presented in this work was developed in order to overcome these problems. Instead of a straightforward interpolation of the given data using high-order polynomials, a parametrized Hermitian interpolation curve was used to approximate the integrated data set. A single parameter is determined by which the user can control the behavior of the interpolation function, i.e. the amount of overshoot and undershoot. Furthermore, it is shown how the algorithm can be extended to multidimensional grids. The algorithm was compared to commonly used gridding-algorithms using linear and cubic interpolation functions. It is shown that such interpolation functions may overestimate or underestimate the source data by about 10-20%, while the new algorithm can be tuned to significantly reduce these interpolation errors. The accuracy of the new algorithm was tested on a series of x-ray CT-images (head and neck, lung, pelvis). The new algorithm significantly improves the accuracy of the sampled images in terms of the mean square error and a quality index introduced by Wang and Bovik (2002 IEEE Signal Process. Lett. 9 81-4).
将组织成规则或不规则网格的空间分布数据重新采样为更精细或更粗糙分辨率的问题是数据处理中的常见任务。此过程称为“网格化”或“重新分箱”。根据数据所代表的量，网格化算法必须满足不同要求。例如，诸如质量或能量等经过直方图统计的物理量必须重新分箱以保持总体积分。此外，如果该量是正定的，则应避免出现负采样值。网格化过程需要根据分布函数将原始数据集重新分布到用户要求的网格上。分布函数可以通过插值方法根据给定数据来确定。一般来说，对于波动剧烈的数据的多个边界条件进行精确插值需要二阶甚至更高阶的多项式插值函数。然而，这可能会导致插值函数与数据出现不切实际的偏差（过冲或下冲）。因此，重新采样的数据可能会显著高估或低估给定数据。本文提出的网格化算法就是为了克服这些问题而开发的。不是使用高阶多项式对给定数据进行直接插值，而是使用参数化的埃尔米特插值曲线来逼近积分数据集。确定一个单一参数，用户可以通过该参数控制插值函数的行为，即过冲和下冲的量。此外，还展示了该算法如何扩展到多维网格。将该算法与使用线性和三次插值函数的常用网格化算法进行了比较。结果表明，此类插值函数可能会使源数据高估或低估约10% - 20%，而新算法可以进行调整以显著减少这些插值误差。在一系列X射线CT图像（头部和颈部、肺部、骨盆）上测试了新算法的准确性。就均方误差以及Wang和Bovik（2002年，《IEEE信号处理快报》9卷，81 - 4页）引入的质量指标而言，新算法显著提高了采样图像的准确性。
