import matplotlib.pyplot as plt
import numpy as np

Read and plot the image

img = plt.imread("imagedata.jpg")
plt.axis('off') 
plt.imshow(img)
plt.show()

If you do not use the plot function then note that img is a three dimensional (RGB) object.

print(img.shape)
print(img)

(564, 400, 3)
[[[179 173 177]
  [180 174 178]
  [180 174 178]
  ...
  [181 172 175]
  [178 172 174]
  [176 170 172]]

 [[181 175 179]
  [181 175 179]
  [181 175 179]
  ...
  [179 170 173]
  [179 170 173]
  [177 171 173]]

 [[183 177 181]
  [182 176 180]
  [181 175 179]
  ...
  [178 169 174]
  [179 170 173]
  [181 172 175]]

 ...

 [[160 124 102]
  [148 113  91]
  [148 117  96]
  ...
  [111  78  59]
  [121  86  67]
  [110  75  55]]

 [[168 132 110]
  [152 120  99]
  [146 115  94]
  ...
  [115  82  63]
  [123  88  69]
  [114  79  60]]

 [[163 127 105]
  [155 123 102]
  [148 117  97]
  ...
  [109  76  57]
  [123  87  71]
  [122  86  70]]]

Reshape img into a matrix

#The array has 564 rows each of pixel 400x3. 
#Reshape it into the form of a matrix that PCA can understand. # 1200 = 400 * 3

img_reshaped = np.reshape(img, (564, 1200))
print(img_reshaped.shape)
print(img_reshaped)

(564, 1200)
[[179 173 177 ... 176 170 172]
 [181 175 179 ... 177 171 173]
 [183 177 181 ... 181 172 175]
 ...
 [160 124 102 ... 110  75  55]
 [168 132 110 ... 114  79  60]
 [163 127 105 ... 122  86  70]]

Make the data centred at origin

img_mean = img_reshaped.mean(axis=0)
img_reshaped = img_reshaped-img_mean

Apply Principal Component Analysis

from sklearn.decomposition import PCA

components=40
pca = PCA(n_components=components)
pca.fit(img_reshaped)

PCA(copy=True, iterated_power='auto', n_components=40, random_state=None,
  svd_solver='auto', tol=0.0, whiten=False)

Note that T=XW, where W is the weight matrix with each column representing the eigen vector. Also WW'=I because W is an orthogonal coordinate system and eigen vectors are normalized to unit length.

To get back from T coordinate system to W coordinate system one needs to do as follows, X=TW'. Note that if W is the complete matrix then X is the exact representation, otherwise not. Also remember to do the de-normalization of X to get to the original values.

#Following represents the values in the new coordinate system T=XW
img_transformed_coordinate = pca.transform(img_reshaped)
print(img_transformed_coordinate.shape)

#To go back to the old coordinate systesm, either use the inbuilt command or do the operations manually X=TW'
#img_original_coordinate = pca.inverse_transform(img_transformed_coordinate)
img_original_coordinate = img_transformed_coordinate.dot(pca.components_)

#Shifting the mean to original values
img_original_coordinate = img_original_coordinate+img_mean

#Reshaping the matrix to 564*400*3
img_final = np.reshape(img_original_coordinate, (564,400,3))
img_final = img_final.astype('int')

img_final[img_final<0] = 0
img_final[img_final>255] = 255

plt.axis('off') 
plt.imshow(img_final)
plt.savefig('imagedata_new.jpg')
plt.show()

(564, 40)

Data Compression Achieved

#Number of values required to store the original image
original_number_of_values = 564*400*3
#Number of values required to store the new image
#Values in transformed coordinate + principal component values + mean values
new_number_of_values = 564*components+1200*components+components

space_required_in_percentage = new_number_of_values/original_number_of_values*100
print("%.2f" % space_required_in_percentage)

10.43