摘要
小波 {ψej,k,m(x ,y) |e=1,2 ,3,j,k,m∈Z}不仅可以构成L2 (R2 )空间的正交基 ,通过小波分解与重构 ,以及对Hj,Gj,H j ,G j 的行向量修改等 ,还可以产生N×N空间的正交基 .同时 ,N×N点阵信号 {Sl,r}( 0≤l,r,≤N - 1)的小波变换等价N×N于空间的正交变换 ,用我们的方法进行信号或图像压缩 ,不涉及对信号或图象进行周期延拓 ,可严格在N×N空间中进行 .首先研究了二维信号的小波分解与重构 ,给出了适用的二维张量积小波的分解与重构公式 .其次 ,给出了信号用分解公式进行小波分解与重构公式进行小波重构后完全恢复原信号的充要条件 ,并对完全重构充要条件的实现作了处理 .最后得到了N×N空间中由小波分解与重构滤波产生的正交基 .这样就推导出对N2 个数据进行小波分解后可精确重构的算法 ,该算法可避免信号做小波分解后在边界处不能精确重构 .
The wavelet {ψ~e_(j,k,m)(x,y)|e=1,2,3,j,k,m∈Z} can not only constitute an orthonormal basis for L^2(R^2), but also its decomposition and reconstruction filters can generate an orthonormal basis for N×N space. Moreover, a wavelet transform of a signal with N×N samples {S_(l,r)}_(0≤l,r≤N-1) is equivalent to an orthogonrmal transform in N×N space. Thus, by using the modified algorithm of wavelet transform proposed in the paper, the signal or image compression based on the wavelet transform can be operated in N×N space without the help of the periodic extension of signal or image .In detail, the decomposition and reconstruction of a two-dimensional signal are investigated. The proper decomposition and reconstruction formula of tensor product wavelet is offered. Then we discuss the sufficient and necessary conditions which can guarantee that the original signal can be completely reconstructed by using our decomposition and reconstruction formula of wavelet. In order to make the conditions hold true, we perform an orthonormalization process. Finally, the orthogonal basis for N×N space generated by the decomposition and reconstruction filters are obtained. Therefore, the algorithm that can exactly reconstruct the original signal after the decomposition of N^2 data is obtained, which avoid inexact reconstruction at the boundary after the decomposition of the signal.
出处
《南京大学学报（自然科学版）》
CAS
CSCD
北大核心
2005年第1期71-76,共6页
Journal of Nanjing University（Natural Science）
关键词
二维小波
小波分解与重构
信号压缩
two dimensional wavelet, signal compression, decomposition and reconstruction of wavelet
