The Spectral Radius Of Bi-orthogonal Wavelets And Its Application

With the requirements of image quality is increasing and multimedia ap-plications is surging, The JPEG2000was born, which is a new generation of static image compression technology with higher compression rate.The most core part of the international standard is the wavelet transform.According to the theory of Cohen and Daubechies, the famous wavelet (CDF9-7)can be constructed.The length of its filters are odd numbers,and the number of the vanishing moments are four.Actully, It is considered to be the most excellent wavelet transform. With the same theory, we can structure the even filters of biorthogonal wavelet. But the wavelet may not have good mathematical property,there is also no good performance in image compression.This con-tradiction phenomenon explains the theory of biorthogonal wavelet need to improve.So we hope to define a new concept for biorthogonal wavelet, and un-der the guidance of it, we can construct the more biorthogonal wavelet with excellent properties.By the error analysis of transform of bi-orthogonal wavelets, a new concept of spectral radius is defined to measure the orthogonal degree for bi-orthogonal wavelets. Effecting wavelet properties of the important factors are the length of filters and the vanishing moment of high pass filter in the decomposition.In fact, the spectral radius this paper found is one of the most important factors. Based on this new concept, we have offered two class of wavelets:One is based on the model of minimizing spectral radius of the bi-orthogonal wavelets. The other is based on the balance of vanishing moments and the spectral radius of the bi-orthogonal wavelets.The high vanishing moments is one of the advantages of biorthogonal wavelets. Under this condition, minimizing spectral radius can make the or-thogonal degree as high as possible. Based on this idea, the excellent bi-orthogonal wavelets with the fiters of even length is designed. We use the SPIHT algorithm to test the wavelet performance in image compression. In the SPIHT algorithm, we use the Huffman entropy coding algorithm, the de- composition times is6, and the boundary is processed by period extension.The compressed Image is measured for good or bad with peak signal-to-noise ra-tio (PSNR). Experimental results has shown that the new bi-orthogonal wavelet(Opl6-8) designed performs better than the famous9-7wavelet on the ability of compression coding. To bi-orthogonal wavelets with the filters of even length, it is very rare.If only to minimize spectral radius for the purpose, and the length of filters is restricted, the performance of the obtained wavelet may not be op-timal. Reducing the vanishing moments, we can seek a balance between the spectral radius and the vanishing moment.It is beneficial for improving the performance of wavelets. Comparing with minimizing the spectral radius of Op12-8, the Ba12-8had a big enhancement in image compression capacity under this ideology. The performance of Ba12-8is as well as CDF9-7. By the way, under the guidance of the thought, we can find excellent bi-orthogonal wavelets in the odd or even length of the filters.