The pointwise multiscale texture operator: analytical foundations and functional characterization

Abstract

Texture is commonly described through statistical descriptors, filter-bank responses, or multiscale representations, yet its functional-analytic structure remains largely unexplored. In this work we investigate texture from the viewpoint of Gaussian scale-space evolution, interpreting it as the persistence of local structures across scales. This perspective naturally leads to a multiscale texture operator that quantifies the evolution of local image structures under Gaussian diffusion. We establish the fundamental analytical properties of this operator, proving its boundedness on $L^p(\mathbb{R}^n)$, its infinite-order smoothing effect in the Hilbert setting, and an explicit spectral characterization showing that it acts as a localized Gaussian band-pass filter. Motivated by the logarithmic organization of Gaussian scales, we introduce an $r$-adic Difference-of-Gaussians discretization and prove Littlewood--Paley-type stability estimates together with an explicit reconstruction formula. Building on this multiscale representation, we define multiscale texture norms that quantify the persistence of local structures across scales. These norms induce a natural family of Banach spaces, with a distinguished Hilbertian case, and we prove that they provide an equivalent characterization of the classical Besov and Sobolev scales. Consequently, classical functional regularity can be described intrinsically through the persistence of texture under Gaussian scale evolution. These results establish the first functional-analytic characterization of texture persistence under Gaussian scale evolution, revealing that classical Besov and Sobolev regularity admit an equivalent description in terms of multiscale texture persistence. This provides a rigorous bridge between Gaussian scale-space theory, harmonic analysis, and variational models based on texture.