March 6, 2012

WITS: Shearlets from MIA 2012

At the wonderful MIA 2012 (hey Gabriel, did i tell you how that was great?), two talks have been devoted to shearlets (Gitta Kutyniok, Gabriele Steidl), yet another geometric multiscale representation for images. And a brand new Matlab toolbox from Gabriele Steidl group named FFST (fast finite shearlet transform) and developed by Sören Haüser has been announced.

Shearlet-Zerlegung eines Auges.© Gitta Kutyniok

A good opportunity to update a little corner of WITS: Where is the Starlet, which was beginning to grow a few webs. Mmh, look like some familiar frequency domain partitioning... And now for something completely different: number 1, the shearlet... the shearlet! (Monty Python inside)

In short: Non-separable wavelets built out of parabolic scaling, shear, and translation operations
Etymology: From shear, a sheer distorsion
Origin: Labate, Demetrio and Lim, W-Q. and Kutyniok, Gitta and Weiss Guido, Sparse multidimensional representation using shearlets (local copy) A handful lot of papers is available here: shearlet papers. A first overview is given in Shearlets. The First Five Year (Oberwolfach Report, 2010, local copy).

Some properties: Unlike curvelets, shearlets form an affine system with a single generating mother shearlet function parameterized by a scaling, a shear, and a translation parameter. Provides the same approximation properties as curvelets, albeit with a different directional sensitivity. Exist in band-limited or compact support flavors. Possess natural, canonical smoothness spaces, called shearlet coorbit spaces, similar to Besov spaces for wavelets. Apparently extend to arbitrary any dimensions.

Usage: Image denoising, restoration, morphological component analysis
See also: The shearlet website, recently updated with ShearLab (... a rationally designed digital shearlet transform) For discrete implementation, there exists for instance a Digital Shearlet Transforms or Development of a Digital Shearlet Transform Based on Pseudo-Polar FFT. Shearlet Matlab toolboxes are available at ShearLab matlab toolboxes, local shearlet toolbox by G. Easley and FSST:
Comments: Potential a hard competitor, for years to come, to the quite oversold curvelets (IMHOlet: In My (little) Humble Opinion)

More on the topic:
2D wavelets: A panorama on multiscale geometric representations, intertwining spatial, directional and frequency selectivity