快速小波转换

快速小波转换（英语：Fast wavelet transform）是利用数学的演算法则用来转换在时域的波形或信号变成一系列的以正交基底（英语：Orthogonal basis）构成的小而有限的波、小波。当然，快速小波转换本身可以很轻易地扩增它的维度以符合各种不同的需求，例如影像处理、压缩、去除杂讯…等 $S_{n}^{(J)}:=2^{J}\left\langle {f(t),\phi (2^{J}t-n)}\right\rangle$

前项离散小波转换

反离散小波转换

利用 $S^{(M)}$ 且M<J的一系列常数集合，以及由 $d^{(K)}$ ，k=M,1,...,J-1的差分集合可以导出有递回关系式的数学式如下： $S_{n}^{(k+1)}:=\sum \limits _{k=-N}^{N}{a_{k}S_{2n-k}^{(k)}+\sum \limits _{k=-N}^{N}{b_{k}d_{2n-k}^{(k)}}}$

或是导入Z转换，以k=J-1,J-2,...,M且 $n\in {\mathbb {Z} }$ 可改写为

$S^{(k+1)}(Z)=a(Z)\cdot (\uparrow 2)(s^{(k)}(Z))+b(Z)\cdot (\uparrow 2)(d^{(k)}(Z))$

其中 $(\uparrow 2)$ 表示升高采样操作子

同时参阅

G. Beylkin, R. Coifman, V. Rokhlin, "Fast wavelet transforms and numerical algorithms" Comm. Pure Appl. Math., 44 (1991) pp. 141–183

参考资料

A.N. Akansu Multiplierless Suboptimal PR-QMF Design Proc. SPIE 1818, Visual Communications and Image Processing, p. 723, November, 1992
A.N. Akansu Multiplierless 2-band Perfect Reconstruction Quadrature Mirror Filter (PR-QMF) Banks US Patent 5,420,891, 1995
A.N. Akansu Multiplierless PR Quadrature Mirror Filters for Subband Image Coding IEEE Trans. Image Processing, p. 1359, September 1996
M.J. Mohlenkamp, M.C. Pereyra Wavelets, Their Friends, and What They Can Do for You (2008 EMS) p. 38
B.B. Hubbard The World According to Wavelets: The Story of a Mathematical Technique in the Making (1998 Peters) p. 184
S.G. Mallat A Wavelet Tour of Signal Processing (1999 Academic Press) p. 255
A. Teolis Computational Signal Processing with Wavelets (1998 Birkhäuser) p. 116
Y. Nievergelt Wavelets Made Easy (1999 Springer) p. 95