紧小波框架算子的python实现

紧小波框架（Tight Wavelet Frame）算子是一种用于信号分析和图像处理的数学工具，它可以分解信号并提取出其中的特征信息。以下是一些常见的紧小波框架算子的 Python 实现：

1. Haar 紧小波框架算子实现：

import numpy as np

def haar_matrix(N):
    H = np.zeros((N,N))
    n = int(np.log2(N))
    H[0,:] = 1/np.sqrt(N)
    for j in range(n):
        for k in range(2**j):
            p = 2**(n-j)
            q = 2*k*p
            for i in range(p):
                H[i+q,j+1] = 1/np.sqrt(p)
                H[i+p+q,j+1] = -1/np.sqrt(p)
    return H

def haar_wavelet(signal):
    n = signal.size
    H = haar_matrix(n)
    return H.dot(signal)

2. Daubechies 4 紧小波框架算子实现：

import numpy as np

def daub4_matrix(N):
    H = np.zeros((N,N))
    a = (1+np.sqrt(3))/(4*np.sqrt(2))
    b = (3+np.sqrt(3))/(4*np.sqrt(2))
    c = (3-np.sqrt(3))/(4*np.sqrt(2))
    d = (1-np.sqrt(3))/(4*np.sqrt(2))
    for i in range(N//2):
        H[2*i,2*i] = a
        H[2*i,2*i+1] = b
        H[2*i+1,2*i] = c
        H[2*i+1,2*i+1] = d
    return H

def daub4_wavelet(signal):
    n = signal.size
    H = daub4_matrix(n)
    return H.dot(signal)

3. Symlet 4 紧小波框架算子实现：

import numpy as np

def sym4_matrix(N):
    H = np.zeros((N,N))
    a = np.sqrt(3)/4
    b = (3+np.sqrt(3))/(4*np.sqrt(2))
    c = (3-np.sqrt(3))/(4*np.sqrt(2))
    d = np.sqrt(3)/4
    e = (1+np.sqrt(3))/(4*np.sqrt(2))
    f = -(1+np.sqrt(3))/(4*np.sqrt(2))
    g = (3-np.sqrt(3))/(4*np.sqrt(2))
    h = (3+np.sqrt(3))/(4*np.sqrt(2))
    for i in range(N//2):
        H[2*i,2*i] = a
        H[2*i,2*i+1] = b
        H[2*i,2*i+2] = c
        H[2*i,2*i+3] = d
        H[2*i+1,2*i] = e
        H[2*i+1,2*i+1] = f
        H[2*i+1,2*i+2] = g
        H[2*i+1,2*i+3] = h
    return H

def sym4_wavelet(signal):
    n = signal.size
    H = sym4_matrix(n)
    return H.dot(signal)

这些算子的实现可以通过 numpy 库中的矩阵运算实现，代码中使用的都是基本的矩阵操作。需要注意的是，不同的紧小波框架算子的矩阵结构不同，需要根据具体的算子进行实现。