图像去雾的原始对偶算法的程序

以下是图像去雾的原始对偶算法的Python程序：

import numpy as np
from scipy import misc, fftpack
from PIL import Image

def im2double(im):
    return np.float64(im) / np.max(im)

def double2im(im):
    return np.uint8(im*255)

def wthresh(x, T):
    return np.sign(x) * np.maximum(np.abs(x) - T, 0)

def shrink(x, tau):
    return np.maximum(np.abs(x) - tau, 0) * np.sign(x)

def dual_filtering(Y, lambda_, sigma, iter_=100):
    # initialization
    h, w = Y.shape
    u1 = np.zeros((h, w))
    u2 = np.zeros((h, w))
    u3 = np.zeros((h, w))
    p = np.zeros((h, w, 2))
    div_p = np.zeros((h, w))

    # Fourier transform matrix
    F = fftpack.fft2(np.eye(h, w))

    # dual step size
    sigma1 = 1 / (sigma * lambda_ * np.sqrt(2))
    sigma2 = 1 / (sigma * lambda_ * np.sqrt(2))

    # main iteration
    for i in range(iter_):
        # update u1, u2, u3
        grad_u = np.gradient(u3)
        u1 = wthresh(u1 + sigma1 * grad_u[0], lambda_ * sigma1)
        u2 = wthresh(u2 + sigma1 * grad_u[1], lambda_ * sigma1)
        u3 = shrink(Y + div_p, sigma2)

        # update p
        grad_u = np.gradient(u3)
        p[:, :, 0] = p[:, :, 0] + sigma * grad_u[0]
        p[:, :, 1] = p[:, :, 1] + sigma * grad_u[1]

        # update divergence of p
        div_p = - np.gradient(p[:, :, 0], axis=0) - np.gradient(p[:, :, 1], axis=1)

    return u3

def dehaze(image_path, lambda_=5, sigma=0.05, iter_=100):
    # read image
    im = np.array(Image.open(image_path).convert('L'))

    # normalize image
    I = im2double(im)

    # apply the algorithm
    J = dual_filtering(I, lambda_, sigma, iter_)

    # denormalize the image
    J = double2im(J)

    return J

其中，`im2double`函数将图像像素值归一化到[0, 1]范围内，`double2im`函数将像素值恢复到[0, 255]范围内；`wthresh`函数是一个软阈值函数，用于实现L1范数正则化；`shrink`函数是一个硬阈值函数，用于实现L1范数正则化；`dual_filtering`函数是原始对偶算法的实现，其中`Y`是输入图像，`lambda_`是正则化参数，`sigma`是步长参数，`iter_`是迭代次数；`dehaze`函数是整个算法的接口函数，用于读取图像、调用`dual_filtering`函数、并返回去雾后的图像。