Wavelet Frames: Symmetry, Periodicity, and Applications

Abstract

We describe the advantages of frames over bases in applications for the multidimensional setting. Our focus is on the construction of symmetric and antisymmetric wavelet frames from a given set of existing wavelets using unitary transformations. First, we construct symmetric and antisymmetric compactly supported wavelet frames for the space of finite energy signals. Next, we study the frame properties of affine and quasi-affine systems, their periodic analogues and the connections between them. We derive the generalized oblique extension principle that contains the unitary extension principle and oblique extension principle as special cases and characterize finite energy wavelet frames in terms of their periodic analogues. Next we construct periodic scaling functions and tight frames and the corresponding filters with detailed proofs of the process. The flexibility in our constructions allow information at various predetermined frequency bands to be filtered away and information in the desired modulation range to be retained. Fast algorithms based on the constructions are derived. The algorithms uses frame representations and the stationary wavelet transform. We analyze various chirp signals with them and show that they provide sparser representation of the signals when compared to other wavelet transforms, short time Fourier transform, packet transforms and other time and frequency distributions.