边界镜像对称延拓双正交小波变换矩阵的构造

计算小波变换的Mallat算法需要进行逐级分解和重构,对于有限长信号的小波变换来说,为了保证其完全重构,有必要对其进行边界延拓。基于边界周期延拓的小波变换算法极易实现,也常见于文献,而边界对称延拓较周期延拓则更适合用于信号和图像的处理,但基于边界对称延拓的小波变换矩阵实现方法却很少出现在文献中。为了用矩阵-向量乘积实现信号的小波变换,给出了一种在信号镜像对称延拓方式下,任意深度小波变换矩阵的构造方法,并证明了该延拓方式下实现Mallat算法的完全重构条件。作为实例,绘出了B ior3.3小波的分解和重构矩阵的基向量及波形图。将构造的变换矩阵用于基于小波的图像处理中,不仅可以避免逐级迭代,大大简化运算量,而且边界效应也明显减少。

Construction of Biorthogonal Wavelet Transform Matrices with Mirror-symmetric Boundary-extension

Iterative decomposition and reconstruction are needed in Mallat algorithm.In order to realize perfect reconstruction,finite-length signals must be extended to some extent before they can be transformed.The algorithm based on periodic boundary-extension always can be seen in the literature.Symmetric boundary-extension has better performance than periodic method in image processing,whereas the matrix transform method based on symmetric boundary-extension is seldom mentioned in the literature.A method of constructing decomposition and reconstruction matrices with arbitrary wavelet transform depth in mirror-symmetric boundary-extension is proposed for wavelet transform in matrix-vector multiplication,and the condition for perfect reconstruction of Mallat algorithm is proved.As an example,the base vectors and base graphs of Bior3.3 wavelet were given.The application of wavelet transform matrices in the wavelet-based image processing can avoid iterative operation,simplify the calculation and meanwhile reduce the edge effect evidently.