Tight and sibling frames originated from discrete splines

Abstract

We present a new family of frames, which are generated by perfect reconstruction filter banks. The filter banks are based on the discrete interpolatory splines and are related to Butterworth filters. Each filter bank comprises one interpolatory symmetric low-pass filter, one band-pass and one high-pass filters. In the sibling frames case, all the filters are linear phase and generate symmetric scaling functions with analysis and synthesis pairs of framelets. In the tight frame case, all the analysis waveforms coincide with their synthesis counterparts. In the sibling frame, we can vary the framelets making them different for synthesis and analysis cases. This enables us to swap vanishing moments between the synthesis and the analysis framelets or to add smoothness to the synthesis framelets. We construct dual pairs of frames, where all the waveforms are symmetric and the framelets may have any number of vanishing moments. Although most of the designed filters are IIR, they allow fast implementation via recursive procedures. The waveforms are well localized in time domain despite their infinite support.