Applications of Tao General Difference in Discrete Domain

TL;DR

This work introduces Tao General Difference (TGD) as a robust discrete-domain framework for numerical difference, enabling 1D signal denoising with controllable smoothness via first- and second-order TGDs, and extending to 2D image edge detection with first- and second-order TGD operators. The 1D denoising is implemented as a lightweight learning-based method that optimizes a loss incorporating TGD continuity and data fidelity, achieving $C^2$ convergence and outperforming traditional denoisers in RMSE, PSNR, and SSIM. In 2D, first-order TGD-based edge detection provides accurate gradient localization and orientation, while second-order TGD-based LoT edge detection yields precise edge localization with favorable zero-crossing properties, particularly for text. The framework is further extended to 3D spatio-temporal edge detection to identify static and kinetic edges in video and CT data, illustrating the potential of TGD operators for multimedia and medical imaging tasks. Overall, the paper demonstrates that TGD-based operators offer robust noise immunity and precise localizing capabilities across 1D, 2D, and 3D discrete domains, with practical implications for signal processing, computer vision, and video analytics.

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.

Figures (29)

• Figure 1: LoG (Laplace of Gaussian) operator (Left) and LoT (Laplace of TGD) operator (Right) constructed with Gaussian kernel function.
• Figure 2: The first-order Gaussian derivative operator and TGD operator, whose kernel function is Gaussian.
• Figure 3: Schematic diagram of TGD-based signal denoise algorithm.
• Figure 4: The discrete TGD operators used in the denoising experiments with size of $51$, which are constructed based on the Gaussian kernel function.
• Figure 5: The training process for noisy signal $X_1$. The left column shows the current denoised signal. The middle column shows the first-order TGD results of the current denoised signal, and the rightmost column shows the second-order TGD results of the current denoised signal.