Irregular Filters for Semi-regular Dubuc-Deslauriers Wavelet Tight Frames
Description
We present filters for the irregular framelets of semi-regular Dubuc-Deslauriers 2n-point wavelet tight frames with mesh parameters h_\ell = 1 and h_r > 0 for the cases: n = 2, h_r = 1.5, 2, 2.5, 3 n = 3, h_r = 1.5, 2, 2.25, 2.5 n = 4, h_r = 1.5, 2, 2.15, 2.3 n = 5, h_r = 1.5, 2, 2.1, 2.2 These filters have been computed using the method described in "Semi-regular Dubuc-Deslaurier wavelet tight frames" submitted to Journal of Computational and Applied Mathematics Special Issue for SMART 2017. The filters are the columns of the matrix Q_irr where R_irr = Q_irr * transpose(Q_irr). To avoid numerical fluctuations Q_irr is computed via singular value decomposition, with threshold on the singular values set to 10^-8. The filters depend only on the ratio h_\ell over h_r and, when this ratio is inverted, it is sufficient to flip the filters. Therefore there is no loss of generality in considering h_\ell = 1 and h_r greater than or equal to 1 only. Moreover, for any fixed natural number n and h_\ell = 1, there is an interval of availability for h_r of the form ( 1/c, c ), where h_r = 1 reduces to the regular case. For n = 2, the exact value of c is 3.5 while for the other values of n the approximated values of c are 2.6225, 2.3591 and 2.2346 respectively for n = 3, 4 and 5. For the examples presented we choose two common values of h_r working for all n=2,...,5 and two values specifically chosen for each n spreaded out between 2 and c.