import copy
import japanize_matplotlib
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import scipy
from matplotlib.pyplot import imshow
from numpy.random import randn
from scipy import stats
Z = np.array([0, 5, -1, -2, -4, 1]).reshape(-1, 3).T
print(Z)
u, s, vh = np.linalg.svd(Z)
print("u:", u, "\n" "s:", s, "\n" "vh:", vh)
u2, s2, vh2 = np.linalg.svd(Z.T)
print("u2:", u2, "\n" "s2:", s2, "\n" "vh2:", vh2)
Z = np.array([0, 5, 5, -1]).reshape(-1, 2).T; print(Z)
u3, s3, vh3 = np.linalg.svd(Z.T)
print("u3:", u3, "\n" "s3:", s3, "\n" "vh3:", vh3)
val, vec = np.linalg.eig(Z)
print("values:", val, "\n" "vectors:", vec)
def svd_r(z, r):
u, s, vh = np.linalg.svd(z)
sigma = np.zeros((z.shape[0], z.shape[1]))
for i in range(r):
sigma[i, i] = s[i]
tt = np.dot(np.dot(u, sigma), vh)
return tt
m = 100; n = 80; z = np.random.normal(size=m*n).reshape(m, -1)
F_norm = list()
for r in range(n):
m = svd_r(z, r)
F_norm.append(np.linalg.norm(z-m, ord="fro") ** 2)
print(np.array(F_norm).shape)
plt.plot(list(range(n)), F_norm)
plt.xlabel("rank")
plt.ylabel("squared Frobenius norm")
plt.show()
from PIL import Image
image = np.array(Image.open("lion.jpg"))
rank_seq = [2, 5, 10, 20, 50, 100]
mat = np.zeros((image.shape[0], image.shape[1], 3))
fig = plt.figure(figsize=(40, 35)); k = 1
for j in rank_seq:
for i in range(3): mat[:, :, i] = svd_r(image[:, :, i], j)
Image.fromarray(np.uint8(mat)).save("compressed/lion_compressed_mat_rank_%d.jpg" % j)
Im = Image.open("compressed/lion_compressed_mat_rank_%d.jpg" % j)
fig.add_subplot(3, 2, k)
imshow(Im)
k = k + 1
def mat_r(z, mask, r):
min = np.inf
m = z.shape[0]; n = z.shape[1]
for j in range(5):
guess = np.random.normal(size=m*n).reshape(m, -1)
for i in range(10):
guess = svd_r(mask * z + (1 - mask) * guess, r)
value = np.linalg.norm(mask * (z - guess), ord="fro")
if value < min: min_mat = guess; min = value
return min_mat
image = np.array(Image.open("lion.jpg"))
m = image.shape[0]; n = image.shape[1]
mask = np.random.binomial(1, 0.5, size=m*n).reshape(m, -1)
rank_seq = [2, 5, 10, 20, 50, 100]
fig = plt.figure(figsize=(40, 35)); k = 1
mat = np.zeros((image.shape[0], image.shape[1], 3))
for j in rank_seq:
for i in range(3): mat[:, :, i] = mat_r(image[:, :, i], mask, j)
Image.fromarray(np.uint8(mat)).save("compressed/lion2_compressed_mat_rank_%d.jpg" % j)
Im = Image.open("compressed/lion2_compressed_mat_rank_%d.jpg" % j)
fig.add_subplot(3, 2, k)
imshow(Im)
k = k + 1
def soft_svd(lambd, z):
u, s, vh = np.linalg.svd(z)
sigma = np.zeros((z.shape[0], z.shape[1]))
for i in range(r):
sigma[i, i] = s[i]
return np.dot(np.dot(u, sigma), vh)
def mat_lasso(lambd, z, mask):
m = z.shape[0]; n = z.shape[1]
guess = np.random.normal(size=m*n).reshape(m, -1)
for i in range(20):
guess = soft_svd(lambd, mask * z + (1-mask) * guess)
return guess
image = np.array(Image.open("lion.jpg"))
m = image[:, :, 1].shape[0]; n = image[:, :, 1].shape[1]
p = 0.5; lambd = 0.5
mat = np.zeros((m, n, 3))
mask = np.random.binomial(1, p, size=m*n).reshape(-1, n)
for i in range(3):
mat[:, :, i] = mat_lasso(lambd, image[:, :, i], mask)
Image.fromarray(np.uint8(mat)).save("compressed/lion3_compressed_mat_soft.jpg")
i = Image.open("compressed/lion3_compressed_mat_soft.jpg")
imshow(i)