Coiflet小波

Coiflet小波是英格麗·多貝西（Ingrid Daubechies）應Ronald Coifman的要求所設計的一種離散小波。Coiflet小波的調整函式（scaling function）及小波函式（wavelet function）能同時擁有高消失動量，且其波形接近對稱，常被用於數位訊號處理。[1]

Coiflet with two vanishing moments

性質

定義

一個正交小波（orthogonal wavelet）系統若符合以下條件，則稱為Generalized Coiflet Wavelet(GOC)。

{\begin{array}{lcl}\\{\mathcal {M_{\tilde {\psi }}}}(0,l]=0&{\mbox{for }}l{\mbox{ =0,1,...,L-1}}\\{\mathcal {M_{\tilde {\phi }}}}(t_{0},l]=\delta [l]&{\mbox{for }}l{\mbox{ =0,1,...,L-1}}\\\end{array}}

其中

{\psi }

為小波函式，

{\phi }

為調整函式

濾波器長度與消失動量

若

N

為濾波器長度，而

L

為系統消失動量，則最小的

N

為：

N={\begin{cases}3L-1,&{\mbox{if }}L{\mbox{ is odd}}\\3L,&{\mbox{if }}L{\mbox{ is even}}\\\end{cases}}

近線性相位（Near-linear phase）濾波器

當

|w|

夠小時，GOC的低通濾波器的頻率響應擁有漸進型態（asymptotic form）：

\angle {H(w)}=-t_{0}w+C\cdot {w^{L'}}+O(w^{L'+2})

其中

L'=2\left\lfloor L/2\right\rfloor +1

，

C

則為常數項。因此

h

具有漸近線性相位（asymptotic linear phase）的特性：

\lim _{L\to \infty }\angle {H(w)}=-t_{0}w

應用

Coiflet小波除了被使用在影像壓縮外，目前也有被使用在電力系統訊號監測上[2]

係數

下表為coiflet小波調整函式C6~30的係數，小波函式的係數可以藉由每兩個係數變換一次符號來推導(例如C6 wavelet = {−0.022140543057, 0.102859456942, 0.544281086116, −1.205718913884, 0.477859456942, 0.102859456942})

**Coiflets係數**
k	C6	C12	C18	C24	C30
-10					-0.0002999290456692
-9					0.0005071055047161
-8				0.0012619224228619	0.0030805734519904
-7				-0.0023044502875399	-0.0058821563280714
-6			-0.0053648373418441	-0.0103890503269406	-0.0143282246988201
-5			0.0110062534156628	0.0227249229665297	0.0331043666129858
-4		0.0231751934774337	0.0331671209583407	0.0377344771391261	0.0398380343959686
-3		-0.0586402759669371	-0.0930155289574539	-0.1149284838038540	-0.1299967565094460
-2	-0.1028594569415370	-0.0952791806220162	-0.0864415271204239	-0.0793053059248983	-0.0736051069489375
-1	0.4778594569415370	0.5460420930695330	0.5730066705472950	0.5873348100322010	0.5961918029174380
0	1.2057189138830700	1.1493647877137300	1.1225705137406600	1.1062529100791000	1.0950165427080700
1	0.5442810861169260	0.5897343873912380	0.6059671435456480	0.6143146193357710	0.6194005181568410
2	-0.1028594569415370	-0.1081712141834230	-0.1015402815097780	-0.0942254750477914	-0.0877346296564723
3	-0.0221405430584631	-0.0840529609215432	-0.1163925015231710	-0.1360762293560410	-0.1492888402656790
4		0.0334888203265590	0.0488681886423339	0.0556272739169390	0.0583893855505615
5		0.0079357672259240	0.0224584819240757	0.0354716628454062	0.0462091445541337
6		-0.0025784067122813	-0.0127392020220977	-0.0215126323101745	-0.0279425853727641
7		-0.0010190107982153	-0.0036409178311325	-0.0080020216899011	-0.0129534995030117
8			0.0015804102019152	0.0053053298270610	0.0095622335982613
9			0.0006593303475864	0.0017911878553906	0.0034387669687710
10			-0.0001003855491065	-0.0008330003901883	-0.0023498958688271
11			-0.0000489314685106	-0.0003676592334273	-0.0009016444801393
12				0.0000881604532320	0.0004268915950172
13				0.0000441656938246	0.0001984938227975
14				-0.0000046098383254	-0.0000582936877724
15				-0.0000025243583600	-0.0000300806359640
16					0.0000052336193200
17					0.0000029150058427
18					-0.0000002296399300
19					-0.0000001358212135

Matlab程式

F = coifwavf(W)會回傳N=str2num(W)的coiflet小波的調整函式，其中N只能為1~5的整數。[3]

雙正交Coiflet小波

定義

一個order為

(L,{\tilde {L}})

的雙正交Coiflet小波需符合以下條件：

{\begin{array}{lcl}\\{\mathcal {M_{\tilde {\phi }}}}(0,l]=\delta [l]&{\mbox{for }}l{\mbox{ =0,1,...,L-1}}\\{\mathcal {M_{\tilde {\psi }}}}(0,l]=0&{\mbox{for }}l{\mbox{ =0,1,...,L-1}}\\{\mathcal {M_{\phi }}}(0,l]=\delta [l]&{\mbox{for }}l{\mbox{ =0,1,...,L-1}}\\{\mathcal {M_{\psi }}}(0,l]=0&{\mbox{for }}l{\mbox{ =0,1,...,}}{\tilde {L}}{\mbox{-1}}\\\end{array}}

性質

當

L

為偶數時，

h

會對稱於原點：

h[n]=h[-n]

。這讓雙正交coiflet在圖片壓縮方面能有較好的峰值信噪比（PSNR）。

参考资料

(PDF). [2016-01-25]. （原始内容 (PDF)存档于2016-03-05）.
. Shyh-Jier Huang, and Cheng-Tao Hsieh. 2002. 缺少或|url=为空 (帮助)
. http://www.mathworks.com/. [22 January 2015]. （原始内容存档于2016-05-22）.

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[1] (PDF). [2016-01-25]. （原始内容 (PDF)存档于2016-03-05）.

[2] . Shyh-Jier Huang, and Cheng-Tao Hsieh. 2002. 缺少或|url=为空 (帮助)

[3] . http://www.mathworks.com/. [22 January 2015]. （原始内容存档于2016-05-22）.