Construction Of Compactly Supported Multi-wavelet

Wavelet analysis has been a new branch of mathematics since 1980s. As a time-frequency analysis method, it has been widely applied in image compression and denoising. Comparing with uni-wavelet, multi-wavelet can simultaneously has properties like compactly supported, orthogonality and symmetry etc, which decide that multi-wavelet can get broad application and research in signal processing. Image transmission and storage have a bright future in visual telephone, digital television, remote medical treatment and satellite remote sensing, but the massive image data limits the communication bandwidth and storage media. To adapt the communication bandwidth and storage media in use, image data must be compressed by coding properly and redundant information must be removed.This paper makes some systematic researches on the theory and application of single and multi-wavelet in image compressing. The main work is summarized as follows:We describe the theory of uni-wavelet and multi-wavelet in detail, and apply the decomposition and the reconstruction algorithm of wavelet transform to two-dimensional picture decomposition and the reconstruction algorithm, then give the principles of image compression, the evaluation criteria of image compression and restoration and some common image coding methods. The image compression flow is given when the wavelet transform is applied to image compression. Finally, the decomposition algorithm and the reconstruction algorithm are demonstrated through my images and Barbara image at different levels of wavelet transform.We introduce the multi-resolution analysis theory, construct two dimensional compactly supported symmetrical multi-wavelet through translating a compactly supported orthogonal uni-wavelet to more than one wavelet function. Multiplying selected appropriate orthogonal matrix by two dimensional compactly supported symmetrical multi-wavelet, we can construct two dimensional compactly supported symmetrical and anti-symmetrical multi-wavelet with the corresponding properties and subsequently give the matrix coefficient relation between original uni-wavelet and the multi-wavelet. In the end of this part, the two dimensional compactly supported symmetrical and anti-symmetrical multi-wavelet is constructed by Haar wavelet and selected original matrix. This method is simple, easy to implement and can construct compactly supported wavelet, so it can be used in practice.The compactly supported symmetric orthogonal multi-wavelet can be constructed by multiplying proper original matrix by given compactly supported symmetric orthogonal multi-wavelet, and the DGHM compactly supported symmetric orthogonal multi-wavelet is taken as an example to show how to this process.