Smooth affine shear tight frames with MRA structure

Finding efficient directional representations is one of the most challenging and extensively sought problems in mathematics. Representation using shearlets recently receives a lot of attention due to their desirable properties in both theory and applications. Using the framework of frequency-based affine systems as developed in [16], in this paper we introduce and systematically study affine shear tight frames which include all known shearlet tight frames as special cases. Our results in this paper resolve several important questions on shearlets. We provide a complete characterization for an affine shear tight frame and then use it to construct smooth directional affine shear tight frames with all their generators in the Schwartz class. Though multiresolution analysis (MRA) together with filter banks is the foundation and key features of wavelet analysis for the fast numerical implementation of a wavelet transform, most papers on shearlets do not concern the underlying filter bank structure and its connection to MRA. In order to study affine shear tight frames with MRA structure, following the lines developed in [16], we introduce the notion of a sequence of affine shear tight frames and then we provide a complete characterization for such a sequence. Based on our characterizations, we present two different approaches, i.e., non-stationary and quasi-stationary, for the construction of sequences of directional affine shear tight frames with MRA structure such that all their generators are smooth (in the Schwartz class) and they have underlying filter banks. Consequently, their associated transforms can be efficiently implemented using filter banks and are very similar to the standard fast wavelet transform. Moreover, we provide concrete examples of directional affine shear tight frames with filter banks and apply them to the image denoising problem. Our numerical experiments show that our constructed directional affine shear tight frames perform better than known directional multiscale representation systems such as curvelets and shearlets for the image denoising problem.