Discrete Shearlet Transform

The shearlet transform, is a very recent sibling in the family of geometric image representations and provides a traditional multiresolution analysis. By a specific design of the discrete shearlet transform in [1] that we use, a lower redundancy factor (around 2.66 or 8/3) is possible than with most other multiresolution representations, while offering an excellent directional analysis and even shift invariance.

Below, a visualization of different shearlet basis functions (spatial domain + frequency domain) is given. Click on the frequency tiling on the left, to select a particular shearlet or scaling function.

Your browser is not Java enabled. Instead you can download and unzip the software and execute the applets locally through the provided batch scripts: click here. Note that this still requires a local Java installation.

Brief explanation of the parameters:

• Scales: the number of frequency scales for the transform.
• Max. orient: the number of orientations for the finest scale of the Shearlet transform (the number of orientations halves at every coarser scale).
• Angular cutoff: determines the angular bandwidth of the shearlet frequency responses (or consequently the size of the spatial support of the basis functions)
• Show frequency spectrum: if checked, the Discrete Fourier spectrum of the basis function is shown. Otherwise, the basis function is shown in spatial domain.

References

• [1] B. Goossens, J. Aelterman, H. Q. Luong, A. Pizurica and W. Philips, "Efficient Design of a Low Redundant Discrete Shearlet Transform, " in Proc. 2009 International Workshop on Local and Non-Local Approximation in Image Processing (LNLA2009), August 19-21, 2009, Tuusula, Finland (invited paper), p. 112-124