Tight wavelet frames in low dimensions with canonical filters

This paper is to construct tight wavelet frame systems containing a set of canonical filters by applying the unitary extension principle of Ron and Shen (1997). A set of filters are canonical if the filters in this set are generated by flipping, adding a conjugation with a proper sign adjusting from one filter. The simplest way to construct wavelets of ss-variables is to use the 2s−12s−1 canonical filters generated by the refinement mask of a box spline. However, almost all wavelets (except Haar or the tensor product of Haar) defined by the canonical filters associated with box splines do not form a tight wavelet frame system. We consider how to build a filter bank by adding filters to a canonical filter set generated from the refinement mask of a box spline in low dimension, so that the wavelet system generated by this filter bank forms a tight frame system. We first prove that for a given low dimension box spline of ss-variables, one needs at least additional 2s2s filters to be added to the canonical filters from the refinement mask (that leads to the total number of highpass filters in the filter bank to be 2s+1−12s+1−1) to have a tight wavelet frame system. We then provide several methods with many interesting examples of constructing tight wavelet systems with the minimal number of framelets that contain canonical filters generated by the refinement masks of box splines. The supports of the resulting framelets are not bigger than that of the corresponding box spline whose refinement mask is used to generate the first 2s−12s−1 canonical filters in the filter bank. In many of our examples, the tight frame filter bank has the double-canonical property, meaning it is generated by adding another set of canonical filters generated from a highpass filter to the canonical filters generated by the refinement mask to make a tight frame system.