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- In applied mathematical analysis, shearlets are a multiscale framework which allows efficient encoding of anisotropic features in multivariate problem classes. Originally, shearlets were introduced in 2006 for the analysis and sparse approximation of functions . They are a natural extension of wavelets, to accommodate the fact that multivariate functions are typically governed by anisotropic features such as edges in images, since wavelets, as isotropic objects, are not capable of capturing such phenomena. Shearlets are constructed by parabolic scaling, shearing, and translation applied to a few generating functions. At fine scales, they are essentially supported within skinny and directional ridges following the parabolic scaling law, which reads length² ≈ width. Similar to wavelets, shearlets arise from the affine group and allow a unified treatment of the continuum and digital situation leading to faithful implementations. Although they do not constitute an orthonormal basis for , they still form a frame allowing stable expansions of arbitrary functions . One of the most important properties of shearlets is their ability to provide optimally sparse approximations (in the sense of optimality in ) for cartoon-like functions . In imaging sciences, cartoon-like functions serve as a model for anisotropic features and are compactly supported in while being apart from a closed piecewise singularity curve with bounded curvature. The decay rate of the -error of the -term shearlet approximation obtained by taking the largest coefficients from the shearlet expansion is in fact optimal up to a log-factor: where the constant depends only on the maximum curvature of the singularity curve and the maximum magnitudes of , and . This approximation rate significantly improves the best -term approximation rate of wavelets providing only for such class of functions. Shearlets are to date the only directional representation system that provides sparse approximation of anisotropic features while providing a unified treatment of the continuum and digital realm that allows faithful implementation. Extensions of shearlet systems to are also available. A comprehensive presentation of the theory and applications of shearlets can be found in. (en)
- 在應用數學的分析方面，剪切小波(英語：Shearlet)是一個多尺度的架構，且在多變量問題中能高效率編碼有各向異性的特點。起初，為了分析及稀疏近似多維方程式，剪切小波在2006年被提出。剪切小波是小波分析的自然延伸，可以適應有各向異性特點的多元方程式，像是影像的輪廓、邊緣。然而，各向同性的小波是不能得到此現象。 把拋物線的縮放、剪切、平移施加在數個生成函數後可建構出剪切小波。雖然所建構出的剪切小波不能建構出在空間中的正交基底，它們仍然可以形成一個架構，且能允許任意函數的穩定擴張。 剪切小波具有以下幾個重要性質： 1.
* 良好的局部性。它在空間中具有快速的衰減性，在頻域內是緊支撐的。 2.
* 靈敏的方向性。通過一個剪切矩陣來控制方向，隨著尺度逐漸精細化，其方向性也會逐漸變得更靈敏。 3.
* 良好的稀疏逼近性。其逼近率與曲波变换和一樣，最高能達到。。其中常數只根據奇數曲線的最大曲率和, 及的最大振幅。這個逼近率顯著提高只使用這類函數的小波分析的最佳項估計率。 4.
* 多方辨性。它是由一個或一組函數的縮放平移生成的一個仿射系統，能設計出快速分解重設法。 5.
* 一致性。能一致處理連續和離散的情況，剪切小波不僅克服了輪廓波和典型小波的不足，也繼承了它們的優點。它既能接近最佳的表示一類高維函數，如類卡通圖像和其他高維分段平滑函數，又能有效地描述函數的幾何訊息。 剪切小波是迄今為止唯一有方向性且提供各向異性特點的稀疏估計的表現系統，可擴展至的剪切小波系統。理論的全面介紹和剪切小波的應用可以在中看到。 (zh)
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- Shearing (en)
- Classical shearlet frequency support (en)
- Classical shearlet frequency tiling (en)
- Parabolic scaling (en)
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- Frequency tiling of the classical shearlet system. (en)
- Trapezoidal frequency support of the classical shearlet. (en)
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- horizontal (en)
- vertical (en)
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- Geometric effects of parabolic scaling and shearing with several parameters a and s. (en)
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- Classshearsupp.svg (en)
- Classsheartiling.svg (en)
- shearletscaling.gif (en)
- shearletshearing.gif (en)
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- In applied mathematical analysis, shearlets are a multiscale framework which allows efficient encoding of anisotropic features in multivariate problem classes. Originally, shearlets were introduced in 2006 for the analysis and sparse approximation of functions . They are a natural extension of wavelets, to accommodate the fact that multivariate functions are typically governed by anisotropic features such as edges in images, since wavelets, as isotropic objects, are not capable of capturing such phenomena. (en)
- 在應用數學的分析方面，剪切小波(英語：Shearlet)是一個多尺度的架構，且在多變量問題中能高效率編碼有各向異性的特點。起初，為了分析及稀疏近似多維方程式，剪切小波在2006年被提出。剪切小波是小波分析的自然延伸，可以適應有各向異性特點的多元方程式，像是影像的輪廓、邊緣。然而，各向同性的小波是不能得到此現象。 把拋物線的縮放、剪切、平移施加在數個生成函數後可建構出剪切小波。雖然所建構出的剪切小波不能建構出在空間中的正交基底，它們仍然可以形成一個架構，且能允許任意函數的穩定擴張。 剪切小波具有以下幾個重要性質： 1.
* 良好的局部性。它在空間中具有快速的衰減性，在頻域內是緊支撐的。 2.
* 靈敏的方向性。通過一個剪切矩陣來控制方向，隨著尺度逐漸精細化，其方向性也會逐漸變得更靈敏。 3.
* 良好的稀疏逼近性。其逼近率與曲波变换和一樣，最高能達到。。其中常數只根據奇數曲線的最大曲率和, 及的最大振幅。這個逼近率顯著提高只使用這類函數的小波分析的最佳項估計率。 4.
* 多方辨性。它是由一個或一組函數的縮放平移生成的一個仿射系統，能設計出快速分解重設法。 5.
* 一致性。能一致處理連續和離散的情況，剪切小波不僅克服了輪廓波和典型小波的不足，也繼承了它們的優點。它既能接近最佳的表示一類高維函數，如類卡通圖像和其他高維分段平滑函數，又能有效地描述函數的幾何訊息。 (zh)
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