# 穩定小波轉換

## 實現方式

${\displaystyle Z}$為一將${\displaystyle 0}$加入${\displaystyle x}$序列的運算，

${\displaystyle (Z_{x})_{k}}$${\displaystyle =x_{k/2},}$ for ${\displaystyle k=2,4,6,8...}$

${\displaystyle (Z_{x})_{k}}$${\displaystyle =0,}$ for ${\displaystyle k=1,3,5,7...}$

${\displaystyle h_{j}[n]=Zh_{j-1}[n]=Z^{2}h_{j-2}[n]=...=Z^{j-1}h_{1}[n]}$

${\displaystyle g_{j}[n]=Zg_{j-1}[n]=Z^{2}g_{j-2}[n]=...=Z^{j-1}g_{1}[n]}$

1. 原始信號與高通濾波器做旋積分之後會得到此信號中高頻的成分。此高頻的成分為第一個高頻的輸出。
2. 原始信號與低通濾波器做旋積分後會得到信號中低頻的成分，此低頻的成分再作為下一階濾波器的輸入。

${\displaystyle x_{j,L[n]}=x_{j-1,L[n]}*g_{j}[n]}$
${\displaystyle x_{j,H[n]}=x_{j-1,L[n]}*h_{j}[n]}$

## Matlab 使用範例

```[tmpAPP,tmpDET] =
dwt(A(j,ε1, ,ɛj),wname,'mode','per','shift',ɛj+1);
A(j+1,ɛ1, ,ɛj,ɛj+1) = wshift('1D',tmpAPP,ɛj+1);
D(j+1,ɛ1, ,ɛj,ɛj+1) = wshift('1D',tmpDET,ɛj+1);
```

## 同義轉換

• 穩定小波轉換 (Stationary Wavelet Transform)
• 冗餘小波轉換 (Redundant Wavelet Transform)
• à trous演算法 (Algorithme à trous)
• 準連續小波轉換 (Quasi-continuous wavelet transform)
• 平移不變量小波轉換 (Translation invariant wavelet transform)
• 轉移不變量小波轉換 (Shift invariant wavelet transform)
• 循環平移演算法 (Cycle spinning)
• 最大重複離散小波轉換 (Maximal overlap discrete wavelet transform, MODWT)
• 非抽樣小波轉換 (Undecimated wavelet transform, UWT)

## 參考文獻

• P. P. Viadyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993, ISBN 0-13-605718-7
• G. P. Nason and B. W. Silverman, The stationary wavelet transform and some statistical applications, Lecture Notes in Statistics
• M.V. Tazebay and A.N. Akansu, Progressive Optimality in Hierarchical Filter Banks, Proc. IEEE International Conference on Image Processing (ICIP), Vol 1, pp. 825-829, Nov. 1994
• P. Dutilleux, An implementation of the algorithme à trous to compute the wavelet transform, in Wavelets: Time-Frequency Methods and Phase Space, J.-M. Combes, A. Grossman, and P. Tchamitchian,Eds. Berlin, Germany: Springer-Verlag, 1989, pp. 298–304.