Structure Preserving Diffusion Models

In many generation tasks, the data involved often exhibit inherent "structures" that result in their distributions remaining invariant - or the mappings between them being equivariant - under a set of transformations. For example, it is commonly assumed that the distribution of photographic images remains invariant under horizontal flipping. In tasks such as image denoising or inpainting, where the orientation of an image is not provided, it is natural to require the denoised or inpainted image to retain the same orientation as the input. Namely, the denoising or inpainting processes should exhibit equivariance under rotations and flipping. Importantly, in certain critical applications, these properties are not merely desired but should be theoretically guaranteed to ensure consistency of the given model outputs or to prevent the introduction of additional biases or errors. A prime example can be found in the field of medical imaging analysis, where X-ray images are used to achieve diagnostic results. However, obtaining high-resolution X-ray scans often requires subjecting patients to higher doses of radiation, which presents a risk to patient safety (Goldman 2007; Huda 2002). Consequently, in order to improve patient safety, noise reduction techniques are critical to the use of lower resolution X-ray images by preserving image quality (Siemund et al. 2012).