Microlocal analysis and characterization of Sobolev wavefront sets using shearlets

使用连续剪切波的波前集的分辨率

Resolution of the Wavefront Set using Continuous Shearlets

已知函数$f$的连续小波变换在$f$光滑的点附近衰减非常快，而在不规则点附近衰减缓慢。这个性质允许人们精确地识别$f$的奇异支集。然而，连续的小波变换是无法提供额外的信息的几何形状的奇异点。在本文中，我们介绍了一种新的变换的函数和分布的$\R^2$，称为连续剪切波变换。这由$\mathcal{S}\mathcal{H}_f(a,s,t) = \ip{f}{ψ_{ast}}$定义，其中分析元素$ψ_{ast}$是单个生成函数$ψ$的扩张和平移副本，因此它们形成仿射系统。所得到的连续剪切波$ψ_{ast}$是连续尺度$a >0$、位置$t \in \R^2$上的光滑函数，并且在频域中沿着斜率线$s \in \R$取向。连续剪切波变换不仅能够识别分布$f$的奇异点的位置，而且能够识别它们的分布奇异点的方向。因此，我们可以使用该变换来精确地表征$f$的波前集。

Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings

The class of generalized shearlet dilation groups has recently been developed to allow the unified treatment of various shearlet groups and associated shearlet transforms that had previously been studied on a case-by-case basis. We consider several aspects of these groups: First, their systematic construction from associative algebras, secondly, their suitability for the characterization of wavefront sets, and finally, the question of constructing embeddings into the symplectic group in a way that intertwines the quasi-regular representation with the metaplectic one. For all questions, it is p…

连续Shearlet标架与波前集的分辨

Continuous Shearlet Frames and Resolution of the Wavefront Set

近年来，方向多尺度变换，如曲波或剪切波变换，得到了相当大的关注。其原因是这些变换-与小波等更传统的变换不同-能够有效地处理具有边缘特征的数据。[G. Kutyniok，D.拉贝特Resolution of the Wavefront Set using continuous Shearlets，Trans.AMS 361（2009），2719 - 2754] Kutyniok和Labate证实了剪切波的这一性质，其中表明，对于在紧凑锥形楔形体中具有频率支持的非常特殊的函数$ψ$，回火分布$f$相对于剪切波$ψ$的剪切波系数的衰减率可以解决这一问题。$f$的波前设置。我们证明了同样的结果可以验证下，更弱的假设$ψ$，即拥有足够多的各向异性消失矩。我们还展示了如何从任何这样的函数构建$L^2(\mathbb{R}^2)$的框架。为了证明我们的声明，我们开发了一种新的方法的基础上适应的Radon变换的剪切波结构。

Classification of Edges Using Compactly Supported Shearlets

We analyze the detection and classification of singularities of functions $f = χ_B$, where $B \subset \mathbb{R}^d$ and $d = 2,3$. It will be shown how the set $\partial B$ can be extracted by a continuous shearlet transform associated with compactly supported shearlets. Furthermore, if $\partial S$ is a $d-1$ dimensional piecewise smooth manifold with $d=2$ or $3$, we will classify smooth and non-smooth components of $\partial S$. This improves previous results given for shearlet systems with a certain band-limited generator, since the estimates we derive are uniform. Moreover, we will show t…

Colombeau函数广义拉回的微局部分析

Microlocal analysis of generalized pullbacks of Colombeau functions

在分布理论中，当$C^{\infty}$函数的法向丛不与分布的波前集相交时，$C^{\infty}$函数对一般分布的拉回是明确定义的。然而，广义函数的Colombeau理论允许由任意c-有界广义函数拉回。在以前的工作中已经表明，在Colombeau函数相乘的情况下（这是$C^{\infty}$拉回的特殊情况），广义波前集的产品满足相同的包含关系，在分布的情况下，如果因子有他们的波前集在有利的位置。我们证明了Colombeau函数的广义拉回（由一个c-有界广义映射）的微局部包含关系。这个结果的证明依赖于广义相函数的驻相定理，这在附录中给出。此外，我们研究了一个例子（由于赫德和Sattinger），拉回功能源于广义特征流的偏微分方程。

Resolution of the wavefront set using general continuous wavelet transforms

We consider the problem of characterizing the wavefront set of a tempered distribution $u\in\mathcal{S}'(\mathbb{R}^{d})$ in terms of its continuous wavelet transform, where the latter is defined with respect to a suitably chosen dilation group $H\subset{\rm GL}(\mathbb{R}^{d})$. In this paper we develop a comprehensive and unified approach that allows to establish characterizations of the wavefront set in terms of rapid coefficient decay, for a large variety of dilation groups. For this purpose, we introduce two technical conditions on the dual action of the group $H$, called microlocal admis…