Scale Space Diffusion

Creating images from noise is image generation; reconstructing fine details from coarse inputs is super-resolution. Despite their practical differences, both can be understood as reversing information loss across scales. We introduce $\textbf{SKILD}$, a $\textbf{S}$cale-invariant $\textbf{K}$-Space $\textbf{I}$mage $\textbf{L}$earning $\textbf{D}$iffusion model that unifies generation and continuous super-resolution within a single unconditional framework. Both natural images and critical physical systems exhibit scale invariance, and we leverage it to design a forward process that attenuates image content from fine to coarse scales while injecting spectrum-matched Gaussian noise, making scale an explicit coordinate of the diffusion dynamics. The same trained reverse process performs generation and continuous super-resolution by varying only the starting timestep: $\textit{no task-specific architecture, no conditioning branch, no classifier-free guidance, no retraining per scale factor}$. Empirically, SKILD reaches FID $2.65$ and Inception Score $9.63$ on unconditional CIFAR-10, performs $2\times$--$8\times$ super-resolution on ImageNet from a single unconditional checkpoint while outperforming conditional models across perceptual metrics, and reconstructs critical Ising models whose connected four-point correlations closely track the ground truth.

Diffusion models have demonstrated impressive image synthesis performance, yet many UNet-based models are trained at certain fixed resolutions. Their quality tends to degrade when generating images at out-of-training resolutions. We trace this issue to resolution-dependent parameter behaviors, where weights that function well at the default resolution can become adverse when spatial scales shift, weakening semantic alignment and causing structural instability in the UNet architecture. Based on this analysis, this paper introduces CR-Diff, a novel method that improves the cross-resolution visual consistency by pruning some parameters of the diffusion model. Specifically, CR-Diff has two stages. It first performs block-wise pruning to selectively eliminate adverse weights. Then, a pruned output amplification is conducted to further purify the pruned predictions. Empirically, extensive experiments suggest that CR-Diff can improve perceptual fidelity and semantic coherence across various diffusion backbones and unseen resolutions, while largely preserving the performance at default resolutions. Additionally, CR-Diff supports prompt-specific refinement, enabling quality enhancement on demand.

Pixel-space diffusion models directly model image distributions but remain difficult to optimize. Recent methods alleviate this challenge through target reparameterization, while still relying on a fixed clean-image target throughout denoising. Through empirical analysis, we identify a scale-time mismatch: image structures become predictable from coarse to fine as noise decreases, whereas existing models are forced to predict the full image even under high noise, resulting in low-SNR gradients that hinder optimization. To resolve this mismatch, we propose Observation Operator Diffusion, a unified framework that aligns both the supervision trajectory and feature refinement with the intrinsic recovery order of image structures. Specifically, we replace fixed full-image supervision along the standard flow path with a time-indexed observation trajectory that evolves from coarse structures to the full image during denoising. This trajectory is instantiated with a family of Gaussian-Lanczos operators at varying observation scales, yielding a path-consistent training objective. We further introduce GL-CoDA, a decoder that injects scale-specific Gaussian-Lanczos observations across decoding stages for coarse-to-fine feature refinement. Extensive experiments show that the proposed approach converges substantially faster while consistently improving generation quality, achieving an FID of 1.52 on ImageNet-256.