On Cosine - Modulated Wavelet Orthonormal

Recently multiplicity M, K-regular, orthonormal wavelet bases (that have implications in transform coding applications) have been constructed by several authors 1, 2, 3]. This paper describes and parameterizes the cosine-modulated class of multiplicity M wavelet tight frames (WTFs). In these WTFs (like in the general multiplicity 2 case in 4]), the scaling function uniquely determines the wavelets. This is in contrast to the general multiplicity M case, where one has to, for any given application, design the scaling function and the wavelets. Several design techniques for the design of K regular cosine-modulated WTFs are described and their relative merits discussed. Wavelets in K-regular WTFs may or may not be smooth. Since coding applications use WTFs with short length scaling and wavelet vectors (since long lters produce ringing artifacts which is undesirable in, say, image coding), many smooth designs of K regular WTFs of short lengths are presented. In some cases, analytical formulas for the scaling and wavelet vectors are also given. In many applications smoothness of the wavelets is more important than K regularity. We deene smoothness of lter banks and WTFs using the concept of total variation and give several useful designs based on this smoothness criterion. Optimal design of cosine-modulated WTFs for signal representation is also described. All WTFs constructed in this paper are orthonormal bases, and we believe that this is always the case.