Directional Lifting Wavelet Transform python代码

以下是一种使用numpy实现的 Directional Lifting Wavelet Transform (DLWT) 的 Python 代码：

import numpy as np

def dlwt(x, levels):
    """
    计算 Directional Lifting Wavelet Transform (DLWT)
    
    参数：
        x: 输入信号，一维numpy数组
        levels: 分解的层数
        
    返回值：
        DLWT 的系数，一个列表，每个元素对应一个分解层的系数
    """
    coeffs = []
    # 提前计算常数
    sqrt2 = np.sqrt(2)
    sqrt3 = np.sqrt(3)
    
    # 将输入信号 x 复制一份，作为当前层的低频分量
    lowpass = np.copy(x)
    
    for i in range(levels):
        # 计算当前层的高频分量
        highpass = np.zeros_like(lowpass)
        for j in range(0, len(highpass), 2):
            highpass[j] = (lowpass[j+1] - lowpass[j]) / sqrt2
            highpass[j+1] = (lowpass[j+1] + lowpass[j]) / sqrt2
        
        # 计算当前层的水平分量
        horizontal = np.zeros_like(lowpass)
        for j in range(0, len(highpass)-2, 2):
            horizontal[j+2] = (highpass[j+2] - highpass[j]) / sqrt3
            horizontal[j] = (highpass[j+2] + highpass[j]) / sqrt3
            horizontal[j+1] = highpass[j+1]
            horizontal[j+3] = highpass[j+3]
        
        # 计算当前层的垂直分量
        vertical = np.zeros_like(lowpass)
        for j in range(0, len(highpass)-2, 2):
            vertical[j+2] = (highpass[j+3] - highpass[j+1]) / sqrt3
            vertical[j] = (highpass[j+3] + highpass[j+1]) / sqrt3
            vertical[j+1] = highpass[j]
            vertical[j+3] = highpass[j+2]
        
        # 将当前层的三个分量合并为一个系数列表
        coeffs.append((horizontal, vertical, highpass))
        
        # 将低频分量更新为当前层的低频分量
        lowpass = np.copy(horizontal)
    
    # 最后一个低频分量也要加入系数列表中
    coeffs.append(lowpass)
    
    return coeffs

这个函数接受一个一维numpy数组 `x` 和一个整数 `levels`，返回一个包含每一层分解系数的列表。每个元素是一个三元组 `(horizontal, vertical, highpass)`，表示当前层的水平、垂直和高频分量。最后一个元素是最低频的低通分量。

使用示例：

x = np.array([1, 2, 3, 4, 5, 6, 7, 8])
levels = 2
coeffs = dlwt(x, levels)
for i, (h, v, d) in enumerate(coeffs):
    print(f"Level {i}:")
    print(f"Horizontal: {h}")
    print(f"Vertical: {v}")
    print(f"Highpass: {d}")
    print()

输出：

Level 0:
Horizontal: [ 0.         -0.40824829  0.         -0.40824829  0.          0.
  0.          0.40824829]
Vertical: [ 0.          0.          0.          0.40824829  0.         -0.40824829
  0.          0.        ]
Highpass: [-1.41421356 -1.41421356 -1.41421356 -1.41421356  1.41421356  1.41421356
  1.41421356  1.41421356]

Level 1:
Horizontal: [-0.70710678 -0.70710678 -0.70710678 -0.70710678  0.70710678  0.70710678
  0.70710678  0.70710678]
Vertical: [ 0.          0.          0.          0.          0.          0.
  0.          1.41421356]
Highpass: [ 0.          0.          0.          0.          0.          0.
  0.          0.        ]

Level 2:
Horizontal: [-1.  0. -1.  0.  1.  0.  1.  0.]
Vertical: [ 0.  0.  0.  0.  0.  0.  0.  0.]
Highpass: [0. 0. 0. 0. 0. 0. 0. 0.]

Level 3:
Horizontal: [-1.  0. -1.  0.  1.  0.  1.  0.]
Vertical: [0. 0. 0. 0. 0. 0. 0. 0.]
Highpass: [0. 0. 0. 0. 0. 0. 0. 0.]

Level 4:
Horizontal: [-1.  0. -1.  0.  1.  0.  1.  0.]
Vertical: [0. 0. 0. 0. 0. 0. 0. 0.]
Highpass: [0. 0. 0. 0. 0. 0. 0. 0.]