1 import numpy as np  2 from PIL import Image  3   4   5 x1 = np.arange(9).reshape((3,3))  6 x2 = np.arange(10,19,1).reshape((3,3))  7   8 y2 = np.stack((x1,x2),axis=1)  9 # -*- coding: utf-8 -*- 10 """ 11 arr =  12     [[0, 0, 0, 2, 2],  13      [0, 0, 0, 3, 3],  14      [0, 0, 0, 1, 1], 15      [1, 1, 1, 0, 0],  16      [2, 2, 2, 0, 0],  17      [5, 5, 5, 0, 0],  18      [1, 1, 1, 0, 0]] 19      20 u = 7 * 7 21 sigma = 7 * 5, 只返回了对角元素, 其余0元素被省略 22 V = 5 * 5 23 """ 24 from PIL import Image 25 import numpy as np 26   27 import matplotlib as mpl 28 import matplotlib.pyplot as plt 29 arr = np.array([[0, 0, 0, 2, 2], [0, 0, 0, 3, 3], [0, 0, 0, 1, 1], 30                 [1, 1, 1, 0, 0], [2, 2, 2, 0, 0], [5, 5, 5, 0, 0], [1, 1, 1, 0, 0]]) 31   32 # 1. 分解 33 u, sigma, v = np.linalg.svd(arr) 34   35 # 2. 重构 36 new_arr = np.mat(u[:, 0:2]) * np.mat(np.diag(sigma[0:2])) * np.mat(v[0:2, :]) 37  38      39 def xprinf(x): 40     #f = open("out.txt", "w")# 打开文件以便写入 41     print("\n**************************\n") 42     #np.set_printoptions(threshold = 1e6)#设置打印数量的阈值 43     np.savetxt("result.txt", x) 44     #f.close() 45     print(x) 46     print("\n**************************\n") 47  48 def restore_1(u, sigma, v, k): 49     new_arr = np.mat(u[:, 0:k]) * np.mat(np.diag(sigma[0:k])) * np.mat(v[0:k, :]) 50     new_arr[new_arr < 0] = 0 51     new_arr[new_arr > 255] = 255    52     return  np.rint(new_arr).astype('uint8') 53  54 def restore_x(u, sigma, v, k): 55     return  np.array(restore_1(u, sigma, v, k)) 56     57 def restore_2(u, sigma, v, k): 58     m = len(u) 59     n = len(v) 60     a = np.zeros((m, n)) 61     # 重构图像 62     #a = np.dot(u[:, :k], np.diag(sigma[:k])).dot(v[:k, :]) 63     # 上述语句等价于： 64     for i in range(k): 65         ui = u[:, i].reshape(m, 1) 66         vi = v[i].reshape(1, n) 67         test= np.dot(ui, vi) 68         a += sigma[i] * np.dot(ui, vi) 69  70     a[a < 0] = 0 71     a[a > 255] = 255 72     return np.rint(a).astype('uint8')     73      74 A = Image.open('lu.jpg') 75 a = np.array(A)  # 转换成矩阵 76  77 #print(a.shape) 78 #xprinf(a) 79  80 def percentage2n(eigVals,percentage): 81     sortArray=np.sort(eigVals)   #升序 82     sortArray=sortArray[-1::-1]  #逆转，即降序 83     arraySum=sum(sortArray) 84     tmpSum=0 85     num=0 86     for i in sortArray: 87         tmpSum+=i 88         num+=1 89         if tmpSum>=arraySum*percentage: 90             return num 91  92 u_r, sigma_r, v_r = np.linalg.svd(a[:, :, 0]) 93 u_g, sigma_g, v_g = np.linalg.svd(a[:, :, 1]) 94 u_b, sigma_b, v_b = np.linalg.svd(a[:, :, 2]) 95  96  97 k=percentage2n(sigma_r,0.8) 98  99 100 101 R_1=restore_x(u_r, sigma_r, v_r,k)102 G_1=restore_x(u_g, sigma_g, v_g,k)103 B_1=restore_x(u_b, sigma_b, v_b,k)104 105 106 I_1 = np.stack((R_1, G_1, B_1), axis=2)107 108 109 110 #plt.imshow(I_1)111 #plt.show()112 113 Image.fromarray(I_1.astype(np.uint8)).save('new.jpg')114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 # plt.imshow(R)131 132 # plt.show()133 # plt.imshow(G)134 135 # plt.show()136 # plt.imshow(B)137 # plt.show()138 #I = np.stack((R, G, B), axis=2)139 # plt.imshow(R)140 # plt.show()141 # new_arr = np.mat(u[:, 0:2]) * np.mat(np.diag(sigma[0:2])) * np.mat(v[0:2, :])142 143 # print(np.diag(sigma[0:2]))144 # print("\n\n\n")145 # print(sigma[0:2])146 # print(np.stack((x1,x2),axis=0))147 # print("\n\n----------------------------------\n\n")148 149 # print(np.stack((x1,x2),axis=1))150 # print("\n\n----------------------------------\n\n")151 152 # print(np.stack((x1,x2),axis=2))153 #print(np.arange(10,28,2).reshape((3,3)))

 

对称矩阵可以分解成 特征向量+特征值，根据特征值的比重，构造主成分矩阵。

对于一般的矩阵（不对称），就要使用奇异值分解了，不失一般性。

u_r, sigma_r, v_r = np.linalg.svd(a[:, :, 0])    一个矩阵可以分解成u_r, sigma_r, v_r 这三个矩阵，m*m , m*n, n*n 的矩阵。 选取K值，可以压缩矩阵 m*k , k*k, k*n。   与主成分分析类似