Inverse formulas of parameterized orthogonal wavelets (Q1616583)

According to the orthogonality conditions, the design of orthogonal wavelets can be viewed as a low-pass filter design problem. The parameterization of filter coefficients of orthogonal wavelets has been researched before, and it could be used to construct personalized wavelets for some applications such as data compression. One of the main drawbacks of personalizing the filters is the slowness to calculate them. In this report, for the known parameterization of length 4, 6, 8, and 10 filters, their inverse formulas to get the parameter values from a set of valid low-pass coefficients are presented, and a repair technique is proposed to deal with the problem of mismatching between the domain and the range of the parameters for which the filter parameterizations and their inverse formulas are defined. The authors validate these inverse formulas when parameters are restricted to $[0, 2\pi)$ for practical applications through image compression application where parameters are optimized to maximize the number of negligible wavelet coefficients. The research work has potential in the optimization of larger filters for some applications such as digital watermarking, classification of neurological signals and image compression. Of course, constructing personalized wavelets needs more computational cost than using standard wavelets.