Cosine - Modulated Orthonormal Wavelet Bases

To give exibility to the time-frequency resolution trade-o of orthonormal (ON) wavelet bases constructed by I.Daubechies [2, 1], recently multiplicityM ON wavelet bases(more generally tight frames (TFs)) have been constructedby several authors [3, 4, ?, 7, 12]. These generalizations ofthe multiplicity 2 ON wavelet bases of I. Daubechies, di erfrom the latter in that, while the multiresolution analysisis uniquely determined by the scaling function0(t), thewavelets are not. In this paper we give a parameterization ofeven-length multiplicityM cosine-modulated wavelet tightframes (CMTFs), where both the wavelets and the mul-tiresolution analysis are uniquely determined by0(t). Weaccomplish this using the recently developed theory of co-sine-modulated (CM) lter banks [8, ?, 9]. We also con-struct \regular" CMTFs by a solving a set of non-linearequations. We conjecture that for all regularities CMTFsexist. All the TFs constructed turn out to be orthonormalbases. We conjecture that this is always the case. We thendesign optimal CMTFs for the representation of a givensignal (or classes of signals). Since regular CMTFs tend tohave long support and have to be designed numerically, andsince regularity is not the same as smoothness of0(whichone desires), we design smooth CMTFs by designing opti-mal CMTFs for arbitrary smooth functions (based on theintuition that the optimal wavelet for a smooth functionwill probably be smooth).