LPNSR: Optimal Noise-Guided Diffusion Image Super-Resolution Via Learnable Noise Prediction

Diffusion-based image super-resolution (SR) aims to reconstruct high-resolution (HR) images from low-resolution (LR) observations. However, the inherent randomness injected during the reverse diffusion process causes the performance of diffusion-based SR models to vary significantly across different sampling runs, particularly when the sampling trajectory is compressed into a limited number of steps. A critical yet underexplored question is: what is the optimal noise to inject at each intermediate diffusion step? In this paper, we establish a theoretical framework that derives the closed-form analytical solution for optimal intermediate noise in diffusion models from a maximum likelihood estimation perspective, revealing a consistent conditional dependence structure that generalizes across diffusion paradigms. We instantiate this framework under the residual-shifting diffusion paradigm and accordingly design an LR-guided multi-input-aware noise predictor to replace random Gaussian noise. We further mitigate initialization bias with a high-quality pre-upsampling network. The compact 4-step trajectory uniquely enables end-to-end optimization of the entire reverse chain, which is computationally prohibitive for conventional long-trajectory diffusion models. Extensive experiments demonstrate that LPNSR achieves state-of-the-art perceptual performance on both synthetic and real-world datasets, without relying on any large-scale text-to-image priors. The source code of our method can be found at https://github.com/Faze-Hsw/LPNSR.