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 34 articles , 2 standard articles )

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  1. Averbuch, Amir; Shabat, Gil; Shkolnisky, Yoel: Direct inversion of the three-dimensional pseudo-polar Fourier transform (2016)
  2. Guo, Kanghui; Labate, Demetrio: Characterization and analysis of edges in piecewise smooth functions (2016)
  3. Kutyniok, Gitta; Lim, Wang-Q; Reisenhofer, Rafael: ShearLab 3D: faithful digital shearlet transforms based on compactly supported shearlets (2016)
  4. 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)
  5. Zhuang, Xiaosheng: Digital affine shear transforms: fast realization and applications in image/video processing (2016)
  6. Aneja, Ruchira: Emergence of shearlets and its applications (2015)
  7. Cotronei, Mariantonia; Ghisi, Daniele; Rossini, Milvia; Sauer, Tomas: An anisotropic directional subdivision and multiresolution scheme (2015)
  8. Grohs, Philipp: Optimally sparse data representations (2015)
  9. Han, Bin; Zhuang, Xiaosheng: Smooth affine shear tight frames with MRA structure (2015)
  10. Kutyniok, Gitta: Geometric separation by single-pass alternating thresholding (2014)
  11. Tan, Chaoqiang; Zhuang, Xiaosheng: The common Hardy space and BMO space for singular integral operators associated with isotropic and anisotropic homogeneity (2014)
  12. Dahlke, Stephan; Oswald, Peter; Raasch, Thorsten: A note on quarkonial systems and multilevel partition of unity methods (2013)
  13. Donoho, David; Kutyniok, Gitta: Microlocal analysis of the geometric separation problem (2013)
  14. Hinrichs, Aicke; Novak, Erich; Woźniakowski, Henryk: Discontinuous information in the worst case and randomized settings (2013)
  15. Rohwedder, Thorsten: The continuous coupled cluster formulation for the electronic Schrödinger equation (2013)
  16. Bender, Christian; Steiner, Jessica: Least-squares Monte Carlo for backward SDEs (2012)
  17. Czaja, Wojciech; King, Emily J.: Isotropic shearlet analogs for $L^2(\BbbR)^k$ and localization operators (2012)
  18. Görner, Torsten; Hielscher, Ralf; Kunis, Stefan: Efficient and accurate computation of spherical mean values at scattered center points (2012)
  19. Harbrecht, Helmut; Peters, Michael; Schneider, Reinhold: On the low-rank approximation by the pivoted Cholesky decomposition (2012)
  20. Heinen, Dennis; Plonka, Gerlind: Wavelet shrinkage on paths for denoising of scattered data (2012)

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