Numerical difference computation is one of the cores and indispensable in the modern digital era. Tao general difference (TGD) is a novel theory and approach to difference computation for discrete sequences and arrays in multidimensional space. Built on the solid theoretical foundation of the general difference in a finite interval, the TGD operators demonstrate exceptional signal processing capabilities in real-world applications. A novel smoothness property of a sequence is defined on the first- and second TGD. This property is used to denoise one-dimensional signals, where the noise is the non-smooth points in the sequence. Meanwhile, the center of the gradient in a finite interval can be accurately location via TGD calculation. This solves a traditional challenge in computer vision, which is the precise localization of image edges with noise robustness. Furthermore, the power of TGD operators extends to spatio-temporal edge detection in three-dimensional arrays, enabling the identification of kinetic edges in video data. These diverse applications highlight the properties of TGD in discrete domain and the significant promise of TGD for the computation across signal processing, image analysis, and video analytic.

翻译：数值差分计算是现代数字时代不可或缺的核心内容之一。通用差分（Tao General Difference, TGD）是一种针对多维空间中离散序列与数组进行差分计算的新型理论与方法。基于有限区间内通用差分坚实的理论基础，TGD算子在实际应用中展现出卓越的信号处理能力。通过一阶与二阶TGD定义了序列的新型平滑性属性，该属性可用于一维信号去噪——其中噪声即为序列中的非平滑点。与此同时，有限区间内梯度中心可通过TGD计算实现精确定位，从而解决了计算机视觉中传统难题：在噪声鲁棒性条件下实现图像边缘的精准定位。此外，TGD算子的能力还可拓展至三维数组的时空边缘检测，实现视频数据中运动边缘的识别。这些多样化应用凸显了TGD在离散域中的特性，以及其在信号处理、图像分析与视频分析领域展现出的重要计算潜力。