Abstract
<title>Abstract</title> <p>Integer-order differential operators such as Sobel (α=1) and Laplacian (α=2) are widely used for image texture enhancement, yet they occupy two isolated points on a richer continuum. This paper establishes a unified framework in which both are proved to be special cases of the Grünwald-Letnikov (GL) fractional-order derivative. For non-integer 0 < α < 1 the GL operator sits in a spectral regime that integer-order filters cannot reach, providing greater mid-frequency texture gain than α=1 with lower noise amplification than α=2. The GL operator is proved to subsume Sobel and Laplacian as exact special cases, a stronger result than prior work tends to state. The scalar operator is extended to N uniformly-spaced directions with provably exact rotational isotropy, paired with a local-variance-driven adaptive order α(x,y) and an AlphaNet encoder-decoder for end-to-end order prediction. The α domain is widened to cover negative orders (fractional integration) and complex orders; the two-stage denoising–enhancement pipeline and the phase-selective complex-order frequency response in Section 2.3 are both new. A C++ implementation with AVX2 and OpenMP offers a 2.80× throughput gain at 2048×2048. Experiments across four benchmarks (Kylberg, ISIC 2018 dermoscopy, MVTec AD industrial defects, DTD natural textures) show consistent gains over classical baselines (Sobel, unsharp mask) in PSNR, SSIM, and region contrast, confirming generalisation across industrial, clinical, and natural-texture domains.</p>