ON COSINE-MODULATED WAVELET ORTHONORMAL BASES

Recently, multiplicity M, K-regular, orthonormal wavelet bases (that have implications in transform coding applications) have been constructed by several authors. This paper describes and parameterizes the cosine-modulated class of multiplicity M wavelet tight frames (WTF's). In these WTF's, 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 WTF's are described and their relative merits discussed. Wavelets in K-regular WTF's may or may not be smooth, Since coding applications use WTFs with short length scaling and wavelet vectors (since long filters produce ringing artifacts, which is undesirable in, say, image coding), many smooth designs of K regular WTF's 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 define smoothness of filter banks and WTF's using the concept of total variation and give several useful designs based on this smoothness criterion. Optimal design of cosine-modulated WTF's for signal representation is also described. All WTF's constructed in this paper are orthonormal bases, and we believe that this is always the case.