!pip install scikit-image

# start by importing some necessary packages %matplotlib inline import numpy as np import matplotlib.pyplot as plt from skimage.io import imread

im = imread("hidden_message.png") im = im.astype(np.float)/255. print("Image dimensions:",im.shape) plt.imshow(im) plt.axis('off') plt.show()

T = 0.004 m0 = np.zeros((240,180)) m1 = np.zeros((240,180)) m2 = np.zeros((240,180)) for i in range(len(m0)): for j in range(len(m0[0,:])): s0 = np.array(im[4*i:4*(i+1), 4*j:4*(j+1),0]) _, s0, _ = np.linalg.svd(s0) s0 = np.sort(s0)[::-1] s1 = np.array(im[4*i:4*(i+1), 4*j:4*(j+1),1]) _, s1, _ = np.linalg.svd(s1) s1 = np.sort(s1)[::-1] s2 = np.array(im[4*i:4*(i+1), 4*j:4*(j+1),2]) _, s2, _ = np.linalg.svd(s2) s2 = np.sort(s2)[::-1] if s0[1] - s0[2] > T: m0[i][j] = 1 if s1[1] - s1[2] > T: m1[i][j] = 1 if s2[1] - s2[2] > T: m2[i][j] = 1 M = (m0 + m1 + m2) plt.imshow(M, cmap = 'gray') plt.axis('off') plt.show()

A = np.loadtxt('recon.txt') U, s, Vt = np.linalg.svd(A, full_matrices=False) print(s[0:5]) print('Recon.txt has an effective rank of 3 with values of approximately 0 following the third term')

u_hat = U[:,0:3] Vt_hat = Vt[0:3, :] s_hat = np.diag(s[0:3]) U_tot = [] S = [] for i in range(10): u1 = u_hat[i*2][0] * u_hat[i*2][0] u2 = u_hat[i*2][0] * u_hat[i*2][1] + u_hat[i*2][1] * u_hat[i*2][0] u3 = u_hat[i*2][0] * u_hat[i*2][2] + u_hat[i*2][2] * u_hat[i*2][0] u4 = u_hat[i*2][1] * u_hat[i*2][1] u5 = u_hat[i*2][1] * u_hat[i*2][2] + u_hat[i*2][2] * u_hat[i*2][1] u6 = u_hat[i*2][2] * u_hat[i*2][2] U_tot.append([u1, u2, u3, u4, u5, u6]) S.append(1) u1 = u_hat[i*2+1][0] * u_hat[i*2+1][0] u2 = u_hat[i*2+1][0] * u_hat[i*2+1][1] + u_hat[i*2+1][1] * u_hat[i*2+1][0] u3 = u_hat[i*2+1][0] * u_hat[i*2+1][2] + u_hat[i*2+1][2] * u_hat[i*2+1][0] u4 = u_hat[i*2+1][1] * u_hat[i*2+1][1] u5 = u_hat[i*2+1][1] * u_hat[i*2+1][2] + u_hat[i*2+1][2] * u_hat[i*2+1][1] u6 = u_hat[i*2+1][2] * u_hat[i*2+1][2] U_tot.append([u1, u2, u3, u4, u5, u6]) S.append(1) u1 = u_hat[i*2][0] * u_hat[i*2+1][0] u2 = u_hat[i*2][0] * u_hat[i*2+1][1] + u_hat[i*2][1] * u_hat[i*2+1][0] u3 = u_hat[i*2][0] * u_hat[i*2+1][2] + u_hat[i*2][2] * u_hat[i*2+1][0] u4 = u_hat[i*2][1] * u_hat[i*2+1][1] u5 = u_hat[i*2][1] * u_hat[i*2+1][2] + u_hat[i*2][2] * u_hat[i*2+1][1] u6 = u_hat[i*2][2] * u_hat[i*2+1][2] U_tot.append([u1, u2, u3, u4, u5, u6]) S.append(0) U_tot = np.array(U_tot) S = np.array(S) X, resid, rank, singular = np.linalg.lstsq(U_tot, S) S = np.array([ [X[0], X[1], X[2]], [X[1], X[3], X[4]], [X[2], X[4], X[5]] ]) print("S:\n", S)

eigs, Q = np.linalg.eigh(S) D = np.diag(eigs) B = np.dot(Q, np.sqrt(D)) R = np.array([ [0.7175, 0.1907, 0.6699], [0.6799, 0.0173, -0.7331], [0.1513, -0.9815, 0.1172] ]) Br = np.dot(np.dot(Q, np.sqrt(D)), R)

M = np.dot(u_hat, Br) O = np.dot(np.linalg.inv(Br), np.dot(s_hat, Vt_hat))

print("Q:\n", Q, "\n") print("Delta:\n", D, "\n") print("B:\n", Q, "\n") print("Rotated B:\n", Br, "\n") print("M:\n", M, "\n") print("O:\n", O)

import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D fig = plt.figure(figsize=(4,4)) ax = fig.add_subplot(111, projection='3d') ax.scatter(O[0,:],O[1,:],O[2,:], alpha=0.1, s=0.5) # plot the point (2,3,4) on the figure plt.show()