ShearLab: a rational design of a digital parabolic scaling algorithm. Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is threefold: We first develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implies that shearlet theory provides a unified treatment of both the continuum and digital realms. Second, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying MATLAB toolbox called ShearLab ( is provided. And, third, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is exemplarily demonstrated for the digital shearlet transform.

This software is also referenced in ORMS.

References in zbMATH (referenced in 53 articles , 2 standard articles )

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  1. Vera, Daniel: Two simple shearlet-based inverses for the multidimensional Radon and John transforms (2022)
  2. Andrushia, A. Diana; Anand, N.; Prince Arulraj, G.: Evaluation of thermal cracks on fire exposed concrete structures using Ripplet transform (2021)
  3. Córdova, Santiago; Vera, Daniel: A simple shearlet-based 2D Radon inversion with an application to computed tomography (2021)
  4. Héctor Andrade-Loarca, Gitta Kutyniok: tfShearlab: The TensorFlow Digital Shearlet Transform for Deep Learning (2020) arXiv
  5. Han, Bin; Mo, Qun; Zhao, Zhenpeng; Zhuang, Xiaosheng: Directional compactly supported tensor product complex tight framelets with applications to image denoising and inpainting (2019)
  6. Pfeifer, Lienhard: Shearlet features for pedestrian detection (2019)
  7. Sun, Guomin; Leng, Jinsong; Cattani, Carlo: A new type of multilevel system for image sparse recovery (2019)
  8. Bubba, Tatiana A.; Porta, Federica; Zanghirati, Gaetano; Bonettini, Silvia: A nonsmooth regularization approach based on shearlets for Poisson noise removal in ROI tomography (2018)
  9. Czaja, Wojciech; Manning, Benjamin; Murphy, James M.; Stubbs, Kevin: Discrete directional Gabor frames (2018)
  10. Mousavi, Zohre; Lakestani, Mehrdad; Razzaghi, Mohsen: Combined shearlet shrinkage and total variation minimization for image denoising (2018)
  11. Sun, Guomin; Leng, Jinsong; Cattani, Carlo: A framework for circular multilevel systems in the frequency domain (2018)
  12. Guo, Kanghui; Labate, Demetrio: Microlocal analysis of edge flatness through directional multiscale representations (2017)
  13. Kutyniok, Gitta; Mehrmann, Volker; Petersen, Philipp C.: Regularization and numerical solution of the inverse scattering problem using shearlet frames (2017)
  14. Kutyniok, Gitta; Petersen, Philipp: Classification of edges using compactly supported shearlets (2017)
  15. O’Connor, Daniel; Vandenberghe, Lieven: Total variation image deblurring with space-varying kernel (2017)
  16. Averbuch, Amir; Shabat, Gil; Shkolnisky, Yoel: Direct inversion of the three-dimensional pseudo-polar Fourier transform (2016)
  17. Guo, Kanghui; Labate, Demetrio: Characterization and analysis of edges in piecewise smooth functions (2016)
  18. Han, Bin; Zhao, Zhenpeng; Zhuang, Xiaosheng: Directional tensor product complex tight framelets with low redundancy (2016)
  19. Kutyniok, Gitta; Lim, Wang-Q; Reisenhofer, Rafael: ShearLab 3D: faithful digital shearlet transforms based on compactly supported shearlets (2016)
  20. Lobos, Rodrigo; Silva, Jorge F.; Ortiz, Julián M.; Díaz, Gonzalo; Egaña, Alvaro: Analysis and classification of natural rock textures based on new transform-based features (2016)

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