Research On Second-Generation Wavelet Based On Interpolation And Application In Image Compression

In recent years, wavelet analysis becomes an interesting research area. It is a breakthrough development on mathematics after Fourier analysis. Many mathematical theories are included in wavelet analysis, and it provides a powerful tool for application area and has been paid great attention. Wavelet filter bank is the bridge that applying wavelet to engineering technology, and the design of wavelet filter bank is the critical issue on the development and application of wavelet theory. The second-generation wavelet theory based on lifting scheme has an essential improvement compared with the first-generation wavelet. It provides many improved properties and does not depended on the Fourier analysis theory. The design of second-generation wavelet filter bank based on lifting scheme becomes the focus problem of wavelet analysis. Wavelet image compression represents the state of the art in image compression theory and technology. The features of wavelet image compression are the higher compression radio and the better quality of reconstruction image. Therefore, the two parts of the system of image compression using wavelet, the design of second-generation wavelet filter bank and the image compression algorithm using wavelet, are discussed in this thesis.The second-generation wavelet can be obtained by decomposing the first-generation wavelet into some lifting steps and dual lifting steps. The lifting theory built by Daubechies and Sweldens decomposes the synthesis polyphase matrix E ( z ), using the Euclidean algorithm of Laurent polynomial, into the second-generation wavelet filter banks based on lifting scheme. The decomposition method based on synthesis polyphase matrix E ( z )has some problems when using it to compose the Haar wavelet and Daubechies 9/7 wavelet into lifting steps. For example, the decomposition using synthesis polyphase matrix of Haar wavelet dose not meet the lifting theorems built by Daubechies and Sweldens. Moreover, the lifting decomposition of Daubechies 9/7 given by them is implemented using analysis polyphase matrix E ( z ), but they do not present the lifting theorems based on analysis polyphase matrix. These problems are investigated in this thesis, and we find that they can be resolved by decomposing the analysis polyphase matrix into lifting steps. The relationship between polyphase representation and lifting representation of wavelet filter bank is introduced in this thesis, and the lifting theorem and dual lifting theorem are built according to the analysis polyphase matrix. The prediction filters and update filters of lifting wavelet are given using the new lifting method. The lifting based on analysis polyphase matrix overcomes the disadvantage of the traditional lifting theory, and presents the prediction filters and update filters of lifting wavelet filter bank. Therefore, the lifting theory based on the decomposition of analysis polyphase matrix plays a vital role in the development and application of the second-generation wavelet theory.The second-generation wavelet can be obtained by decomposing the first-generation wavelet using lifting theory. It is difficult to build the second-generation wavelet filter bank using lifting theory directly. The Neville filter theory, which built by Kova?evi? and Sweldens, discusses the relationship among the prediction filter, the update filter and the Neville filter according to vanishing moments. As soon as the prediction filter fit for Neville filter theory is built, the corresponding update filter is also obtained using Neville filter theory. Therefore, Neville filter theory is suitable to construct the second-generation wavelet filter banks via lifting theory. In this thesis, the Neville filter theory and Lagrange interpolation are combined to construct the new lifting wavelet filter banks, and they are named as Neville–Lagrange lifting wavelet filter banks. Furthermore, the computational complexity, the normalization, the waveform, and the experiments of Neville–Lagrange lifting wavelet filter banks are discussed in this thesis.The embedded zerotree wavelet (EZW) is a classic algorithm of wavelet image compression, and we find that it can be improved. The subordinate pass of EZW encodes the data in subordinate list by constructing the quantizer layer by layer. As the increasing of decomposition level, the construction of quantizer becoming more and more complicated. Therefore, the coding of subordinate pass needs a lot of logic judgements. Moreover, the number of quantization intervals increases exponentially ( 2 k ? 1) according to the decomposition level k. Our research shows that the construction of quantizer is not necessary, and the subordinate pass can be implement by outputing the binary-bit of the data in subordinate pass list directly. Therefore, the simplified coding of EZW algorithm using bit successive approximation quantization for the subordinate pass is presented in this thesis.Furthermore, there are some problems about dominant pass of EZW algorithm are not proposed yet. (1) Each insignificant coefficient in the three highest frequency subbands (LH1, HL1, and HH1) is encoded using two bits in dominant pass of traditional EZW, however one bit can be used to achieve this purpose. (2) The four sub-coefficients of a significant coefficient must be coded even if all their descendant coefficients are insignificant, and this will waste the coding resource. (3) The dominant pass of EZW algorithm adopts the scheme"The significant coefficients under previous threshold do not be encoded in current dominant pass". It seems always better than the scheme"The significant coefficients under previous threshold encoded in current dominant pass", but this is not correct according to our research. The more zerotrees will be generated using the scheme"The significant coefficients under previous threshold encoded in current dominant pass"in most cases. To resolve the above three problems, the adaptive EZW algorithm with extended coding symbol is presented in this thesis. The previous two problems can be resolved using the EZW with extended coding symbol, and the third problem can be resolved using the adaptive EZW.