互为Hilbert变换对的双正交小波构造

The Design of Hilbert Transform Pairs of Biorthogonal Wavelet Bases

证明了两个双正交小波滤波器组构成Hilbert变换对的充要条件,并从理论上说明了两个线性相位双正交小波系统构成Hilbert变换对的必要条件是它们的长度分别为奇数和偶数.在此基础上通过选择合适的小波消失矩和优化过程中的目标函数,提出了一种构造这类Hilbert变换对的新算法.采用该算法不但可以得到系数对称的线性相位小波滤波器组,而且在性能基本相当的条件下,滤波器长度较已有算法大幅度减小(以13/19和12/16小波为例,可以降到约为原来的1/2).通过适当调整设计参数,还可以得到全为有理系数的小波滤波器,从而进一步减少计算代价.实验表明上述构造得到的Hilbert变换在用于复数小波进行图像去噪时,处理时间可以降低为原来的2/3左右.

The Hilbert transform pairs of biorthogonal wavelet bases are studied in this paper. A sufficient and necessary condition is given and proved, and a new algorithm for the design of Hilbert transform pairs is proposed. Being different from the existing algorithms, two linear phase wavelet filter banks are obtained with all symmetric coefficients here, by letting their length be odd and even separately and improving the goal function at the same time, Results of numerical experiments show that the above algorithm can shorten the length of FB efficiently in the similar approximation degree（Take 13/19 tap and 12/16 tap wavelets as an example, the lengths are reduced about 1/2）. Especially, we can get a group of rational coefficients by adjusting the parameters in construction. Authors＇ work is valuable for the rapid and efficient application of many new wavelet theories such as complex wavelet and phaselet.