【正文】 ral characteristics: Compactly supported wavelets with extremal phase and highest number of vanishing moments for a given support width. Associated scaling filters are minimumphase filters. Family Daubechies Short name db Order N N strictly positive integer Examples db1 or haar, db4, db15 db系列小波 Orthogonal yes Biorthogonal yes Compact support yes DWT possible CWT possible Support width 2N1 Filters length 2N Regularity about N for large N Symmetry far from Number of vanishing moments for psi N 圖 : Biorthogonal()小波系 主要特點體現(xiàn)在具有線性相位型，主要應用于信 號和圖象的重構中。) HAARINFO Information on Haar wavelet. Haar Wavelet General characteristics: Compactly supported wavelet, the oldest and the simplest wavelet. scaling function phi = 1 on [0 1] and 0 otherwise. wavelet function psi = 1 on [0 [, = 1 on [ 1] and 0 otherwise. Family Haar Short name haar Examples haar is the same as db1 其他012/112/101?????? xxH? Orthogonal yes Biorthogonal yes Compact support yes DWT possible CWT possible Support width 1 Filters length 2 Regularity haar is not continuous Symmetry yes Number of vanishing moments for psi 1 圖 : 在命令窗口輸入 waveinfo(39。 ^^????、 、 、??和 Haar 小波 waveinfo(39。 (4)正則性。 (3) (若存在 )的消失矩階數(shù)。即當時間或頻率趨于無窮大時，上述各量從有限值收斂到 0的速度； (2) 對稱型。 目前主要是通過用小波分析方法處理信號的結果與理論分析結果的誤差萊判定小波基的好壞 ， 并由此選定小波基 。常用小波函數(shù)及 Matlab常用指令 ● 一 、常用小波函數(shù) 與標準傅立葉變換相比 ， 小波分析中用到的小波函數(shù)沒有唯一性 ， 小波函數(shù) 具有多樣性 。由此而帶來的問題是使用不同的小波基分析同一個問題會產生不同的結果 ， 沒有一個選擇最優(yōu)小波基的統(tǒng)一方法 。 ()x?常用的指導性選擇標準有： (1) 的支撐長度。它在圖象處理中對于避免移相非常有用 。對于壓縮非常有用 。對信號或圖象的重構獲得較好的平滑效果非常有用。haar39。haar39。通常表示為 。 Nr=2 Nd=2,4,6,8 Nr=3 Nd=1,3,5,7,9。 Nr=6 Nd=8 General characteristics: Compactly supported biorthogonal spline wavelets for which symmetry and exact reconstruction are possible with FIR filters (in orthogonal case it is impossible except for Haar). Family Biorthogonal Short name bior Order Nr,Nd Nr = 1 , Nd = 1, 3, 5 r for reconstruction Nr = 2 , Nd = 2, 4, 6, 8 d for deposition Nr = 3 , Nd = 1, 3, 5, 7, 9 Nr = 4 , Nd = 4 Nr = 5 , Nd = 5 Nr = 6 , Nd = 8 Examples , Orthogonal（正交） no Biorthogonal(雙正交的 ) yes Compact support yes DWT possible CWT possible Support width 2Nr+1 for rec., 2Nd+1 for dec. Filters length max(2Nr,2Nd)+2 but essentially bior ld lr effective length effective length of Lo_D of Hi_D bior 2 2 bior 6 2 bior 10 2 bior 5 3 bior 9 3 bior 13 3 bior 17 3 bior 4 4 bior 8 4 bior 12 4 bior 16 4 bior 20 4 bior 9 7 bior 9 11 bior 17 11 Regularity for psi rec. Nr1 and Nr2 at the knots Symmetry yes Number of vanishing moments for psi dec. Nr Remark: bior , and are such that reconstruction and deposition functions and filters are close in value. 圖 : Coiflet(coifN)小波系 由 Daubechies構造 ,N=1,2,3,4, dbN更好的 對稱性。 圖 : General characteristics: Compactly supported wavelets with highest number of vanishing moments for both phi and psi for a given support width. Family Coiflets Short name coif Order N N = 1, 2, ..., 5 Examples coif2, coif4 Orthogonal