编写程序实现矩阵的SVD分解，编写一个综合程序，实现基于SVD分解的图片压缩、降噪。并根据图像质量评价指标PSNR、SSIM评估压缩、降噪结果。画出PSNR、SSIM随奇异值保留数量的变化关系曲线，以及其他结果展示

首先，我们需要实现矩阵的SVD分解。下面是一个示例代码：

import numpy as np

def svd(matrix):
    U, s, V = np.linalg.svd(matrix, full_matrices=True)
    return U, s, V

# 示例代码
matrix = np.array([[1, 2], [3, 4], [5, 6]])
U, s, V = svd(matrix)
print(U)
print(s)
print(V)

接下来，我们可以使用SVD分解来实现图片压缩和降噪。下面是一个示例代码：

import cv2
import numpy as np

def compress_image(image, k):
    U, s, V = svd(image)
    compressed_s = np.zeros_like(s)
    compressed_s[:k] = s[:k]
    compressed_image = U.dot(np.diag(compressed_s)).dot(V)
    return compressed_image.astype(np.uint8)

def denoise_image(image, k):
    U, s, V = svd(image)
    denoised_s = np.zeros_like(s)
    denoised_s[k:] = s[k:]
    denoised_image = U.dot(np.diag(denoised_s)).dot(V)
    return denoised_image.astype(np.uint8)

# 示例代码
image = cv2.imread('lena.png', cv2.IMREAD_GRAYSCALE)
compressed_image = compress_image(image, 100)
denoised_image = denoise_image(image, 100)
cv2.imwrite('compressed.png', compressed_image)
cv2.imwrite('denoised.png', denoised_image)

最后，我们可以使用PSNR和SSIM评估压缩和降噪结果。下面是一个示例代码：

import cv2
import numpy as np
from skimage.metrics import peak_signal_noise_ratio, structural_similarity

def evaluate(image1, image2):
    psnr = peak_signal_noise_ratio(image1, image2)
    ssim = structural_similarity(image1, image2)
    return psnr, ssim

# 示例代码
original_image = cv2.imread('lena.png', cv2.IMREAD_GRAYSCALE)
compressed_image = cv2.imread('compressed.png', cv2.IMREAD_GRAYSCALE)
denoised_image = cv2.imread('denoised.png', cv2.IMREAD_GRAYSCALE)
psnr1, ssim1 = evaluate(original_image, compressed_image)
psnr2, ssim2 = evaluate(original_image, denoised_image)
print('Compressed PSNR:', psnr1, 'SSIM:', ssim1)
print('Denoised PSNR:', psnr2, 'SSIM:', ssim2)

最后，我们可以画出PSNR和SSIM随奇异值保留数量的变化关系曲线。下面是一个示例代码：

import cv2
import numpy as np
from skimage.metrics import peak_signal_noise_ratio, structural_similarity
import matplotlib.pyplot as plt

def evaluate(image1, image2):
    psnr = peak_signal_noise_ratio(image1, image2)
    ssim = structural_similarity(image1, image2)
    return psnr, ssim

def compress_image(image, k):
    U, s, V = svd(image)
    compressed_s = np.zeros_like(s)
    compressed_s[:k] = s[:k]
    compressed_image = U.dot(np.diag(compressed_s)).dot(V)
    return compressed_image.astype(np.uint8)

def denoise_image(image, k):
    U, s, V = svd(image)
    denoised_s = np.zeros_like(s)
    denoised_s[k:] = s[k:]
    denoised_image = U.dot(np.diag(denoised_s)).dot(V)
    return denoised_image.astype(np.uint8)

# 示例代码
original_image = cv2.imread('lena.png', cv2.IMREAD_GRAYSCALE)
psnrs1, ssims1 = [], []
psnrs2, ssims2 = [], []
for k in range(1, 500, 10):
    compressed_image = compress_image(original_image, k)
    denoised_image = denoise_image(original_image, k)
    psnr1, ssim1 = evaluate(original_image, compressed_image)
    psnr2, ssim2 = evaluate(original_image, denoised_image)
    psnrs1.append(psnr1)
    ssims1.append(ssim1)
    psnrs2.append(psnr2)
    ssims2.append(ssim2)
    
plt.figure(figsize=(10, 5))
plt.subplot(121)
plt.plot(psnrs1, label='Compressed')
plt.plot(psnrs2, label='Denoised')
plt.title('PSNR')
plt.legend()
plt.subplot(122)
plt.plot(ssims1, label='Compressed')
plt.plot(ssims2, label='Denoised')
plt.title('SSIM')
plt.legend()
plt.show()

这里我们使用了matplotlib库来画出随奇异值保留数量变化的PSNR和SSIM曲线。