Framelets: MRA-based constructions of wavelet frames

Ingrid Daubechies, Bin Han, Amos Ron, Zuowei Shen


We are interested here in wavelet frames and their construction via multiresolution analysis (MRA); of particular interest to us are {\it tight} wavelet frames. The redundant representation offered by wavelet frames has already been put to good use for signal denoising, and is currently explored for image compression. Motivated by these and other applications, we explore in this article the theory of wavelet frames. We restrict our attention to wavelet frames constructed via multiresolution analysis, because this guarantees the existence of fast implementation algorithms. We shall explore the `power of redundancy' to establish general principles and specific algorithms for constructing framelets and tight framelets. In particular, we shall give several systematic constructions of spline and pseudo-spline tight frames and symmetric bi-frames with short supports and high approximation orders. Several explicit examples are discussed.

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