Applications of Tao General Difference in Discrete Domain

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.

Tao General Differential and Difference: Theory and Application

Modern numerical analysis is executed on discrete data, of which numerical difference computation is one of the cores and is indispensable. Nevertheless, difference algorithms have a critical weakness in their sensitivity to noise, which has long posed a challenge in various fields including signal processing. Difference is an extension or generalization of differential in the discrete domain. However, due to the finite interval in discrete calculation, there is a failure in meeting the most fundamental definition of differential, where dy and dx are both infinitesimal (Leibniz) or the limit of dx is 0 (Cauchy). In this regard, the generalization of differential to difference does not hold. To address this issue, we depart from the original derivative approach, construct a finite interval-based differential, and further generalize it to obtain the difference by convolution. Based on this theory, we present a variety of difference operators suitable for practical signal processing. Experimental results demonstrate that these difference operators possess exceptional signal processing capabilities, including high noise immunity.