查看“︁快速小波轉換”︁的源代码

快速小波轉換（{{lang-en|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[[序列|系列]]的以{{tsl|en|Orthogonal basis|正交基底}}構成的小而有限的波、[[小波分析|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[[图像处理|影像處理]]、壓縮、去除[[雜訊]]…等
<math>S_n^{(J)} : = 2^J \left\langle {f(t),\phi (2^J t - n)} \right\rangle</math>

== 前項離散小波轉換 ==

== 反離散小波轉換 ==
利用<math>S^{(M)}</math>且M<J的一系列常數集合，以及由<math>d^{(K)}</math>，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：
<math>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)} } } </math> 

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

<math>S^{(k + 1)} (Z) = a(Z) \cdot ( \uparrow 2)(s^{(k)} (Z)) + b(Z) \cdot ( \uparrow 2)(d^{(k)} (Z))
</math>

其中<math>( \uparrow 2)</math>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 ==
{{reflist}}

* 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
[[Category:小波分析]]
[[Category:图像处理]]
[[Category:離散轉換]]