Detection of Geometric Structures by an Optimally Subsampled Shearlet System in Noisy Digital Images

Compactly Supported Shearlets are Optimally Sparse

Cartoon-like images, i.e., C^2 functions which are smooth apart from a C^2 discontinuity curve, have by now become a standard model for measuring sparse (non-linear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit (almost) optimally sparse approximation within this model. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications. In this paper, we now present the first com…

Dualizable Shearlet Frames and Sparse Approximation

Shearlet systems have been introduced as directional representation systems, which provide optimally sparse approximations of a certain model class of functions governed by anisotropic features while allowing faithful numerical realizations by a unified treatment of the continuum and digital realm. They are redundant systems, and their frame properties have been extensively studied. In contrast to certain band-limited shearlets, compactly supported shearlets provide high spatial localization, but do not constitute Parseval frames. Thus reconstruction of a signal from shearlet coefficients requ…

Compactly Supported Shearlets

Shearlet theory has become a central tool in analyzing and representing 2D data with anisotropic features. Shearlet systems are systems of functions generated by one single generator with parabolic scaling, shearing, and translation operators applied to it, in much the same way wavelet systems are dyadic scalings and translations of a single function, but including a precise control of directionality. Of the many directional representation systems proposed in the last decade, shearlets are among the most versatile and successful systems. The reason for this being an extensive list of desirable…

Shearlets on Bounded Domains

Shearlet systems have so far been only considered as a means to analyze $L^2$-functions defined on $\R^2$, which exhibit curvilinear singularities. However, in applications such as image processing or numerical solvers of partial differential equations the function to be analyzed or efficiently encoded is typically defined on a non-rectangular shaped bounded domain. Motivated by these applications, in this paper, we first introduce a novel model for cartoon-like images defined on a bounded domain. We then prove that compactly supported shearlet frames satisfying some weak decay and smoothness …

紧支撑剪切波框架对三维函数的最优稀疏逼近

Optimally sparse approximations of 3D functions by compactly supported shearlet frames

针对三维各向异性结构，引入广义三维卡通图像模型，通过金字塔自适应混合剪切波系统构造L^2(R^3)框架，在光滑参数1<α≤β≤2范围内实现近乎最优的稀疏逼近率。

Shearlets与最优稀疏逼近

Shearlets and Optimally Sparse Approximations

本文综述了利用带限和紧支撑剪切波框架对卡通图像进行最优稀疏逼近的理论与方法，证明了剪切波系统在二维和三维中唯一能实现此类逼近的最优性。