Digital Shearlet Transforms
Over the past years, various representation systems which sparsely approximate functions governed by anisotropic features such as edges in images have been proposed. We exemplarily mention the systems of contourlets, curvelets, and shearlets. Alongside the theoretical development of these systems, algorithmic realizations of the associated transforms were provided. However, one of the most common shortcomings of these frameworks is the lack of providing a unified treatment of the continuum and digital world, i.e., allowing a digital theory to be a natural digitization of the continuum theory. In fact, shearlet systems are the only systems so far which satisfy this property, yet still deliver optimally sparse approximations of cartoon-like images. In this chapter, we provide an introduction to digital shearlet theory with a particular focus on a unified treatment of the continuum and digital realm. In our survey we will present the implementations of two shearlet transforms, one based on band-limited shearlets and the other based on compactly supported shearlets. We will moreover discuss various quantitative measures, which allow an objective comparison with other directional transforms and an objective tuning of parameters. The codes for both presented transforms as well as the framework for quantifying performance are provided in the Matlab toolbox ShearLab.
Key wordsDigital shearlet system Fast digital shearlet transform Performance measures Pseudo-polar Fourier transform Pseudo-polar grid ShearLab Software package Tight frames
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The first author would like to thank David Donoho and Morteza Shahram for many inspiring discussions on topics in this area. She also acknowledges support by the Einstein Foundation Berlin and by Forschungsgemeinschaft (DFG) Grant SPP-1324 KU 1446/13 and DFG Grant KU 1446/14. The second author was supported by DFG Grant SPP-1324 KU 1446/13, and the third author was supported by DFG Grant KU 1446/14.
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