TY - JOUR
T1 - ShearLab
T2 - A rational design of a digital parabolic scaling algorithm
AU - Kutyniok, Gitta
AU - Shahram, Morteza
AU - Zhuang, Xiaosheng
PY - 2012
Y1 - 2012
N2 - 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 (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. © 2012 Society for Industrial and Applied Mathematics.
AB - 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 (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. © 2012 Society for Industrial and Applied Mathematics.
KW - Curvelets
KW - Digital shearlet system
KW - Directional representation system
KW - Fast digital shearlet transform
KW - Parabolic scaling
KW - Performance measures
KW - Software package
KW - Tight frames
UR - http://www.scopus.com/inward/record.url?scp=84871768917&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84871768917&origin=recordpage
U2 - 10.1137/110854497
DO - 10.1137/110854497
M3 - 21_Publication in refereed journal
VL - 5
SP - 1291
EP - 1332
JO - SIAM Journal on Imaging Sciences
JF - SIAM Journal on Imaging Sciences
SN - 1936-4954
IS - 4
ER -