A Theory of General Difference in Continuous and Discrete Domain

TL;DR

The paper tackles noise-susceptible numerical differentiation on digital data by bridging the gap between infinitesimal calculus and finite-interval processing with Tao General Difference (TGD). It introduces Tao Derivative (TD) as a finite-interval generalization of differentiation, and extends this to full general differences via first- and second-order operators $T$ and $R$ that realize $f'_{\mathrm{TGD}} = C_1 (T * f)$ and $f''_{\mathrm{TGD}} = C_2 (R * f)$, respectively. Directional and high-dimensional extensions are achieved through Rotational Construction (averaging 1D TGD over directions with rotation weights) and Orthogonal Construction (separable, axis-aligned operators with smoothing in orthogonal directions), including Laplace-of-TGD (LoT) isotropic variants. The work provides extensive kernel-function analyses (Gaussian, Linear, Exponential, Landau, Weibull), counterexamples to highlight constraint importance, and empirical demonstrations in 1D/2D/3D showing improved noise suppression and precise edge localization, suggesting practical impact in denoising, feature detection, and high-dimensional signal processing.

Abstract

Though a core element of the digital age, numerical difference algorithms struggle with noise susceptibility. This stems from a key disconnect between the infinitesimal quantities in continuous differentiation and the finite intervals in its discrete counterpart. This disconnect violates the fundamental definition of differentiation (Leibniz and Cauchy). To bridge this gap, we build a novel general difference (Tao General Difference, TGD). Departing from derivative-by-integration, TGD generalizes differentiation to finite intervals in continuous domains through three key constraints. This allows us to calculate the general difference of a sequence in discrete domain via the continuous step function constructed from the sequence. Two construction methods, the rotational construction and the orthogonal construction, are proposed to construct the operators of TGD. The construction TGD operators take same convolution mode in calculation for continuous functions, discrete sequences, and arrays across any dimension. Our analysis with example operations showcases TGD's capability in both continuous and discrete domains, paving the way for accurate and noise-resistant differentiation in the digital era.

Figures (21)

• Figure 1: The smooth operator, first- and second-order TGD operators constructed based on the Gaussian kernel function.
• Figure 2: The 2D First- and Second-order directional TGD operator (a) $\widetilde{T}_{\mathbf{v}}$ and its projection map, cross-section projection (b). Gaussian kernel function is used, and the derivation directions include 0' and 45', $\widetilde{w}(\theta)$ is a cosine function.
• Figure 3: The decomposition diagram of the convolution calculation process of an image and 2D TGD operators obtained by the orthogonal construction method.
• Figure 4: The decomposition diagram of the convolution calculation process of a series of images and 3D TGD operators obtained by the orthogonal construction method.
• Figure 5: 2D first- and second-order TGD operators based on the orthogonal construction method. Gaussian kernel function is used for both TGD and filtering.

Theorems & Definitions (10)

• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1
• Theorem 3
• proof
• Definition 1
• Theorem 4
• proof