逐点多尺度纹理算子：分析基础与函数表征
The pointwise multiscale texture operator: analytical foundations and functional characterization
AI总结：
该研究从高斯尺度空间视角提出多尺度纹理算子，建立其函数分析性质，将经典贝索夫与索伯列夫正则性等价表征为多尺度纹理持续性，搭建了相关理论间的严谨桥梁。
AI中文摘要：
纹理通常通过统计描述子、滤波器组响应或多尺度表示来描述，但其函数分析结构仍在很大程度上未被探索。在这项工作中，我们从高斯尺度空间演化的视角研究纹理，将其解释为局部结构在不同尺度下的持续性。这一视角自然引出了一种多尺度纹理算子，用于量化高斯扩散下局部图像结构的演化。我们建立了该算子的基本分析性质，证明其在$L^p(\boldsymbol{R}^n)$上的有界性、希尔伯特空间设定下的无穷阶平滑效应，以及显式谱表征，表明其作为局部高斯带通滤波器发挥作用。受高斯尺度的对数组织方式启发，我们引入了一种$r$进制高斯差分（$r$-adic Difference-of-Gaussians）离散化方法，并证明了其小伍德-佩利型稳定性估计及显式重构公式。基于该多尺度表示，我们定义了多尺度纹理范数，用于量化局部结构在不同尺度下的持续性。这些范数诱导出一个自然的巴拿赫空间族，其中包含一个特殊的希尔伯特情形，我们证明它们对经典的贝索夫（Besov）和索伯列夫（Sobolev）尺度提供了等价表征。因此，经典的函数正则性可通过高斯尺度演化下的纹理持续性进行内在描述。这些结果建立了高斯尺度演化下纹理持续性的首个函数分析表征，揭示了经典贝索夫和索伯列夫正则性可通过多尺度纹理持续性获得等价描述，为高斯尺度空间理论、调和分析与基于纹理的变分模型之间搭建了严谨的桥梁。
英文摘要：
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.
