SD-Retinex: A Stable Retinex Method for Low-Light Image Enhancement Designed Around the Sigmoid Derivative

Low-light image enhancement (LLIE) is an important pre-processing step in surveillance, medical, industrial-inspection, and disaster-response settings where cloud transmission is restricted, creating demand for lightweight, training-free, deterministic methods that run in real time on edge devices. We propose a Retinex-based method, SD-Retinex, built on one principle: the derivative of the sigmoid function. We first show that the illumination kernel of our prior method XCR is exactly this derivative sampled on the imaginary axis, and that its instability—in kernel radius and slope, with a saturating reflectance gain—is a direct consequence of the logistic poles lying there. Guided by this diagnosis, SD-Retinex relocates the derivative to where it is stable by construction: a real-axis separable illumination kernel (no poles, O(N) per pixel) and a perceptually motivated bounded sigmoid (Naka–Rushton) tone map that needs no hard clipping. Ablations isolate the contributions: the bounded tone map is the dominant quality factor, while the real-axis kernel matches a scale-matched Gaussian in quality and is adopted for its (a,N) stability. Among training-free CPU methods SD-Retinex is the strongest—surpassing classical Retinex (SSR, MSR, LIME), XCR, and the zero-reference deep methods Zero-DCE and SCI in PSNR and color error ΔE without any training—while the paired-trained URetinex-Net and Retinexformer serve as stronger GPU upper references that quantify the price of training.