Smooth Affine Shear Tight Frames: Digitization and Applications

摘要

In this paper, we mainly discuss one of the recent developed directional multiscale representation systems: smooth affine shear tight frames. A directional wavelet tight frame is generated by isotropic dilations and translations of directional wavelet generators, while an affine shear tight frame is generated by anisotropic dilations, shears, and translations of shearlet generators. These two tight frames are actually connected in the sense that the affine shear tight frame can be obtained from a directional wavelet tight frame through subsampling. Consequently, an affine shear tight frame indeed has an underlying filter bank from the MRA structure of its associated directional wavelet tight frame. We call such filter banks affine shear filter banks, which can be designed completely in the frequency domain. We discuss the digitization of affine shear filter banks and their implementations: the forward and backward digital affine shear transforms. Redundancy rate and computational complexity of digital affine shear transforms are also investigated in this paper. Numerical experiments and comparisons in image/video processing show the advantages of digital affine shear transforms over many other stat-of-art directional multiscale representation systems.