Symmetry property and construction of wavelets with a general dilation matrix

In this note, we are interested in the symmetry property of a refinable function with a general dilation matrix. We investigate the symmetry group of a mask so that its associated refinable function with a general dilation matrix has a certain kind of symmetry. Given two dilation matrices which produce the same lattice, we demonstrate that if a mask has a certain kind of symmetry, then its associated refinable functions with respect to the two dilation matrices are the same; therefore, the two corresponding derived wavelet systems are essentially the same. Finally, we illustrate that for any dilation matrix, orthogonal masks, as well as interpolatory masks having nonnegative symbols, can be easily constructed with any preassigned order of sum rules by employing a linear transform. Without solving any equations, the method in this note on constructing masks with certain desirable properties is simple, painless and general. Examples of quincunx wavelets and wavelets with respect to the checkerboard lattice are presented to illustrate the general theory.