The pointwise multiscale texture operator: analytical foundations and functional characterization

What the math gives to ML

The paper treats texture as persistence of local Gaussian averages across scale, and claims that an r-adic Difference-of-Gaussians decomposition gives a stable characterization equivalent to Besov and Sobolev regularity. The transferable asset is not Gaussian blurring itself, which is standard, but the use of scale-indexed band-pass coefficients and their persistence as a functional regularity measure. A practical neural-network adaptation is to regularize intermediate feature maps by the energy and cross-scale persistence of their Gaussian DoG coefficients, with weights chosen to approximate a Besov norm. This could suppress unstable fine-scale artifacts while preserving genuinely persistent structure, especially in CNNs, image restoration, diffusion models, and vision transformers with spatial feature maps.