Digital Affine Shear Transforms: Fast Realization and Applications in Image/Video Processing

In this paper, we discuss the digitization and applications of smooth affine shear tight frames, a recently developed new class of directional multiscale representation systems. An affine 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 an affine shear tight frame can be obtained from an affine wavelet tight frame through subsampling. Consequently, an affine shear tight frame has an underlying filter bank from the MRA structure of its associated affine wavelet tight frame. We discuss the digitization of digital affine shear filter banks associated with the affine shear tight frames. Moreover, we provide the detailed algorithmic steps for both the forward and backward digital affine shear transforms. Analysis of the redundancy rate and computational complexity shows that the redundancy rate of the digital affine shear transforms does not increase with respect to the number of directions and the computational complexity is proportional to the redundancy rate and the FFT time for a fixed size of input data. Numerical experiments and comparisons in image/video processing show the advantages of our digital affine shear transforms over many other state-of-the-art frame-based directional transforms.