Clarkson Eric, Kupinski Meredith
J Opt Soc Am A Opt Image Sci Vis. 2021 Mar 1;38(3):387-394. doi: 10.1364/JOSAA.403016.
An upper bound is derived for a figure of merit that quantifies the error in reconstructed pixel or voxel values induced by the presence of null functions for any list-mode system. It is shown that this upper bound decreases as the region in attribute space occupied by the allowable attribute vectors expands. This upper bound allows quantification of the reduction in this error when this type of expansion is implemented. Of course, reconstruction error is also caused by system noise in the data, which has to be treated statistically, but we will not be addressing that problem here. This method is not restricted to pixelized or voxelized reconstructions and can in fact be applied to any region of interest. The upper bound for pixelized reconstructions is demonstrated on a list-mode 2D Radon transform example. The expansion in the attribute space is implemented by doubling the number of views. The results show how the pixel size and number of views both affect the upper bound on reconstruction error from null functions. This reconstruction error can be averaged over all pixels to give a single number or can be plotted as a function on the pixel grid. Both approaches are demonstrated for the example system. In conclusion, this method can be applied to any list-mode system for which the system operator is known and could be used in the design of the systems and reconstruction algorithms.
针对一个品质因数推导出了一个上限，该品质因数量化了在任何列表模式系统中由零函数的存在所导致的重建像素或体素值中的误差。结果表明，随着允许属性向量在属性空间中所占区域的扩大，这个上限会降低。当实现这种类型的扩展时，这个上限允许对这种误差的减小进行量化。当然，重建误差也由数据中的系统噪声引起，这必须进行统计处理，但我们在此不会讨论这个问题。该方法不限于像素化或体素化重建，实际上可以应用于任何感兴趣的区域。在一个列表模式二维拉东变换示例中展示了像素化重建的上限。通过将视图数量加倍来实现属性空间的扩展。结果显示了像素大小和视图数量如何同时影响来自零函数的重建误差上限。这个重建误差可以在所有像素上进行平均以得到一个单一数字，或者可以作为像素网格上的函数进行绘制。针对示例系统展示了这两种方法。总之，该方法可应用于任何已知系统算子的列表模式系统，并且可用于系统和重建算法的设计。
