快速小波轉換

快速小波轉換（英語：Fast wavelet transform）是利用數學的演算法則用來轉換在時域的波形或信號變成一系列的以正交基底（英语：Orthogonal basis）構成的小而有限的波、小波。當然，快速小波轉換本身可以很輕易地擴增它的維度以符合各種不同的需求，例如影像處理、壓縮、去除雜訊…等 $S_{n}^{(J)}:=2^{J}\left\langle {f(t),\phi (2^{J}t-n)}\right\rangle$

前項離散小波轉換

反離散小波轉換

利用 $S^{(M)}$ 且M<J的一系列常數集合，以及由 $d^{(K)}$ ，k=M,1,...,J-1的差分集合可以導出有遞迴關係式的數學式如下： $S_{n}^{(k+1)}:=\sum \limits _{k=-N}^{N}{a_{k}S_{2n-k}^{(k)}+\sum \limits _{k=-N}^{N}{b_{k}d_{2n-k}^{(k)}}}$

或是導入Z轉換，以k=J-1,J-2,...,M且 $n\in {\mathbb {Z} }$ 可改寫為

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

其中 $(\uparrow 2)$ 表示升高採樣操作子

同時參閱

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

參考資料

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