$\sigma$: Sigmoid Modulation for Ultra High Resolution Diffusion

Diffusion Transformers (DiTs) can synthesize high-fidelity images, but training at ultra-high resolutions is expensive, making inference-time extrapolation essential. Existing methods are typically \textit{scale-agnostic}, applying the same positional-modulation schedule regardless of target resolution. We show that this misses a scale-sensitive property of denoising: resizing shifts a fixed semantic pattern toward lower \emph{normalized} spatial frequencies while enlarging the spatial support over which global structure must be coordinated. The latter delays structural lock-in at high resolutions, so schedules may either relax guidance before global layout has stabilized, causing \textit{structural collapse}, or retain excessive intervention into late denoising, causing \textit{textural degradation}. We introduce \textbf{SigMa ($\sigma$)}, a training-free framework that uses Sigmoid Modulation for \textit{scale-adaptive} extrapolation through two scaling laws: \textit{Decoupled Geometric Center Alignment} and \textit{Iso-Variance Rate Adaptation}. Experiments show that SigMa reduces this mismatch, enabling training-free extrapolation up to 16 megapixels and achieving the best or competitive performance among the tested training-free extrapolation baselines. Code is available at \href{https://github.com/bxuanz/SigMa.git}{github.com/bxuanz/SigMa}.

AI image generators naturally build images by transitioning from a broad global layout to fine local textures over time. However, existing stretching methods apply a rigid intervention strength that remains identical regardless of the target resolution. This ignores a critical fact: as images get larger, their internal structural patterns change how they evolve and mature across frequencies. Because of this size-blind approach, previous methods often fail to synchronize with the scale-dependent structural formation window, leading to structural collapse or severe layout distortions. To bridge this gap, we introduce SigMa, a plug-and-play framework that links the timing and pace of the image-building process directly to the requested scale. By recognizing that larger images require a longer phase to secure their overarching geometry, SigMa mathematically tailors the generation timeline for width and height independently. It adapts the structural guidance dynamically, ensuring a stable global scaffold before allowing the model to refine local details. Our approach allows existing AI models to generate flawless, high-fidelity images up to 16 megapixels and seamless panoramas without any retraining or extra computing costs. By resolving the structural mismatch while preserving sharp textures, this research makes eco-friendly, professional-grade visual creation highly accessible.