|Date:||Tue, May 24, 2011|
|Place:||IRC East Wing Seminar Room|
Abstract: Directional representative systems provide sparse approximation of anisotropic features are highly desired in both theory and application. The shearlet system is a novel system developed by Guo, Kutyniok, Labate, Lim, Weiss, which achieves these properties.
In this talk, we first develop a digital shearlet theory which is rationally designed in the sense that it is the digitalization of the existing shearlet theory for continuum data. This shows that shearlet theory indeed provides a unified treatment of both the continuum and digital realms. Our implementation of the digital shearlet transform is based on the utilization of pseudo-polar grids and the pseudo-polar Fourier transform, which provide a natural implementation for digital shearlets on the discrete image domain. However, the pseudo-polar Fourier transform is generally not an isometry, hence its adjoint transform cannot be used directly for reconstruction. Isometry can be can be achieved by careful weighting of the pseudo-polar grid, yet it is generally difficult to obtain such a weight function. We show how efficient weight functions can be designed and obtained on the pseudo-polar grids so that almost isometry can be achieved. In addition, we shall discuss the software package ShearLab that implements the digital shearlet transform. The ShearLab in addition provides various quantitative measures allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations.
Numerically results and examples will be provided to illustrate our digital shearlet transform.