Symmetric wavelets dyadic sibling and dual frames

Abstract

We introduce a family of symmetric dyadic wavelets arising from dual and sibling frames. Each of the frames consists of three generators obtained using spectral factorization. We describe two cases of dual frames: symmetric frames with redundant highpass filters and symmetric frames with redundant bandpass filters. We present design methods and examples for both types of dual frames. We additionally consider the design of symmetric frames where the analysis and synthesis filterbanks have the same lowpass filter, leading to sibling frames. In the proposed sibling frames the non-redundant filters are identical in the synthesis and analysis filterbanks. Only the redundant filters are different, thus obtaining a dual frame approximating tight frames. The filters are simple to construct, and offer smooth scaling functions and wavelets, as well as dense time-scale grid. Examples of sibling frames are discussed. The filters are all FIR of even and odd lengths, linear phase, and possess at least one vanishing moment each. A denoising application compares the proposed wavelets with published frames.