Irregular Filters for Semi-regular Dubuc-Deslauriers Wavelet Tight Frames

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.

本论文提出了一套适用于半正则Dubuc-Deslauriers 2n点小波紧框架的非规则框架滤波器，其网格参数为h_ell = 1，h_r > 0。具体包括以下情况：

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

这些滤波器是通过《半正则Dubuc-Deslauriers小波紧框架》一文中描述的方法计算得出，该文已提交至《计算与应用数学杂志》SMART 2017特刊。

滤波器为矩阵Q_irr的列，其中R_irr = Q_irr * transpose(Q_irr)。为了避免数值波动，Q_irr通过奇异值分解计算，奇异值阈值设置为10^-8。

滤波器仅依赖于h_ell与h_r的比率，当此比率倒置时，只需翻转滤波器即可。因此，在考虑h_ell = 1且h_r大于或等于1时，不失一般性。

此外，对于任何固定的自然数n和h_ell = 1，h_r的可用区间为(1/c, c)，其中h_r = 1退化为常规情况。对于n = 2，c的精确值为3.5，而对于n = 3，4，5的其他值，c的近似值分别为2.6225，2.3591和2.2346。对于所展示的例子，我们选择了适用于所有n=2,...,5的两个常见h_r值，以及针对每个n特别选择的两个值，这些值分布在2和c之间。