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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 (www.ShearLab.org) 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.<\/jats:p>","DOI":"10.1137\/110854497","type":"journal-article","created":{"date-parts":[[2012,11,20]],"date-time":"2012-11-20T15:00:38Z","timestamp":1353423638000},"page":"1291-1332","source":"Crossref","is-referenced-by-count":81,"title":["ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm"],"prefix":"10.1137","volume":"5","author":[{"given":"Gitta","family":"Kutyniok","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Morteza","family":"Shahram","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaosheng","family":"Zhuang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,11,20]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/060650283"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/1033097"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/05064182X"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1999.0444"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.10116"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2005.02.003"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2005.02.004"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"O. Christensen,<i>An Introduction to Frames and Riesz Bases<\/i> Birkh\u00e4user Boston, Boston, 2003.","DOI":"10.1007\/978-0-8176-8224-8"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2005.859376"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1109\/MCSE.2009.15"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2007.09.003"},{"key":"atypb13","unstructured":"P. Grohs and G. Kutyniok,<i>Parabolic Molecules<\/i> preprint, arXiv:1206.1958v1 [math.FA], 2012."},{"key":"atypb14","first-page":"189","author":"Guo K.","year":"2006","journal-title":"TN"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/060649781"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/090780912"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1111\/j.1365-246X.2007.03698.x"},{"key":"atypb18","unstructured":"E. Hewitt and K. A. Ross,<i>Abstract Harmonic Analysis. Vol.I:Structure of Topological Groups. Integration Theory, Group Representations<\/i> Academic Press, New York; Springer-Verlag, Berlin, G\u00f6ttingen, Heidelberg, 1963."},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"E. Hewitt and K. A. Ross,<i>Abstract Harmonic Analysis. Vol.II:Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups<\/i> Springer-Verlag, New York, Berlin, 1970.","DOI":"10.1007\/978-3-662-26755-4_3"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-011-9142-y"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-08-04700-4"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/j.jat.2011.06.005"},{"key":"atypb23","doi-asserted-by":"crossref","unstructured":"G. Kutyniok, J. Lemvig, and W.Q Lim,<i>Compactly supported shearlets<\/i> in Approximation Theory XIII (San Antonio, TX, 2010), Springer Proc. 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