SoFoNO: Arbitrary-scale image super-resolution via Sobolev Fourier neural operator

Accurately reconstructing fine textures and sharp edges remains a significant challenge in Single Image Super-Resolution (SISR) tasks, often resulting in overly smooth and less realistic images. To alleviate this issue we propose a novel SISR framework named Sobolev Fourier Neural Operator (SoFoNO). Central to our approach is a specialized architecture featuring a Sobolev Branch, which effectively captures detailed structures in the frequency domain via a learnable Sobolev exponent. Importantly, the learned Sobolev exponent is directly employed as derivative order parameters within the Sobolev loss function, enabling more precise and visually coherent reconstructions. Unlike conventional pixel-level loss functions, the Sobolev loss explicitly incorporates frequency-domain penalties, significantly enhancing the reconstruction quality of detailed image structures. Extensive experiments conducted on multiple datasets under both in-scale and out-scale scenarios demonstrate that our SoFoNO provides robust and effective performance in arbitrary-scale super-resolution, consistently outperforming representative existing methods across various tested scale factors without relying on attention mechanisms. • We propose SoFoNO, a novel SISR framework with a Sobolev Branch operating in the frequency domain. • We introduce a learnable Sobolev exponent that directly guides the Sobolev loss, enabling precise and visually coherent reconstructions. • We validate SoFoNO through extensive in-scale and out-scale experiments, showing robust arbitrary-scale performance.