Image compression via SVD¶

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ASE7030: Convex optimization, Inha University.

Jong-Han Kim (jonghank@inha.ac.kr)

In [1]:

import numpy as np
import matplotlib.pyplot as plt#
from PIL import Image
import requests
inha = Image.open(requests.get("https://jonghank.github.io/ase7030/files/SymbolMark.jpg", stream=True).raw)
inha = np.array(inha)/255.

In [2]:

plt.figure(figsize=(8,8), dpi=100)
plt.imshow(inha)
plt.axis('off')
plt.title('Raw color image')
plt.show()

In [3]:

R_channel = inha[:,:,0]
G_channel = inha[:,:,1]
B_channel = inha[:,:,2]
X = 0.299*R_channel + 0.587*G_channel + 0.114*B_channel

m, n = X.shape
print(m,n)

plt.figure(figsize=(8,8), dpi=100)
plt.imshow(X, cmap='gray', vmin=0, vmax=1)
plt.title(f'Grayscale image ({m*n} real numbers)')
plt.axis('off')
plt.show()

420 390

In [4]:

U,S,Vt = np.linalg.svd(X)

plt.figure(figsize=(14,8), dpi=100)
plt.subplot(121)
plt.plot(S)
plt.grid()
plt.xlabel('Index')
plt.ylabel('Singular values')
plt.subplot(122)
plt.semilogy(S)
plt.grid()
plt.xlabel('Index')
plt.ylabel('Singular values')
plt.show()

In [5]:

plt.figure(figsize=(8,8), dpi=100)
plt.imshow(X, cmap='gray', vmin=0, vmax=1)
plt.axis('off')
plt.title(f'Original image ({m*n} real numbers)')
plt.show()

for k in [2,5,10,20,40,60,80]:
  plt.figure(figsize=(8,8), dpi=100)
  X_compressed = U[:,:k]@np.diag(S[:k])@Vt[:k,:]
  plt.imshow(X_compressed, cmap='gray', vmin=0, vmax=1)
  plt.title(f'Compressed image ({(m+n+1)*k} real numbers), rank {k}')
  plt.axis('off')
  plt.show()

In [5]: