Symmetric interpolatory dual wavelet frames

Abstract:

For any symmetry group ${\mathcal H}$ and any appropriate matrix dilation (compatible with ${\mathcal H}$), an explicit method is given for the construction of ${\mathcal H}$-symmetric interpolatory refinable masks that obey the sum rule of an arbitrary order $n$. Moreover, a description of all such masks is obtained. This type of mask is the starting point for the construction of symmetric wavelets and interpolatory subdivision schemes preserving symmetry properties of the initial data. For any given ${\mathcal H}$-symmetric interpolatory refinable mask, an explicit technique is suggested for the construction of dual wavelet frames such that the corresponding wavelet masks are mutually symmetric and have vanishing moments up to the order $n$. For an Abelian symmetry group ${\mathcal H}$, this technique is modified so that all the resulting wavelet masks have the ${\mathcal H}$-symmetry property.