Abstract
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.
Abstract (translated)
数值差分计算是现代数字时代的一个核心和不可或缺的理论。Tao一般差分（TGD）是一种新理论和新方法，用于处理多维空间中离散序列和阵列的差分计算。TGD操作基于有限间隔中一般差分的理论基础，在现实应用中表现出卓越的信号处理能力。在第一和第二TGD中定义了一个序列的平滑性质。这个性质被用来去噪一维信号，其中噪声是序列中的非平滑点。同时，通过TGD计算可以准确地定位有限间隔中的梯度中心。这解决了传统计算机视觉中的一个挑战，即在噪声抗性的图像边缘精确定位。此外，TGD操作的力量还扩展到三维数组中的时空边缘检测，能够识别视频数据中的运动边缘。这些多样应用突出了TGD在离散领域的性质以及TGD在信号处理、图像分析和视频分析领域具有的显著前景。
URL
https://arxiv.org/abs/2401.15287
