Applying PCA Analysis on Images

Apply PCA analysis on Images¶

To extract the main characteristics for a group of images¶

Based on Programming Computer Vision with Python by Jan Erik Solem¶

Directly apply PCA to raw image is interesting but does not yield much useful information because of pixel location plays an important role¶

Better apply PCA to extracted features¶

In [29]:

import os
import glob
import math
from PIL import Image
import numpy as np
import matplotlib.pyplot as plt

# Ensure plots embeded in notebook
%matplotlib inline

In [2]:

# Prepare data
# In this case, font images for "a"
imgSrcDir = 'a_font_thumbs'
a_font_files = [os.path.join(imgSrcDir, f) for f in os.listdir(imgSrcDir) if f.endswith(".jpg")]

In [40]:

# Define PCA function
def pca(X):
    """
    Principal Component Analysis
    input: X, matrix with trainnig data stored as flattened arrays in rows
    return: projection matrix (with important dimensions first), variance and mean.
    
    SVD factorization:  A = U * Sigma * V.T
                        A.T * A = V * Sigma^2 * V.T  (V is eigenvectors of A.T*A)
                        A * A.T = U * Sigma^2 * U.T  (U is eigenvectors of A * A.T)
                        A.T * U = V * Sigma
                        
    """
    
    # get matrix dimensions
    num_data, dim = X.shape
    
    # center data
    mean_X = X.mean(axis=0)
    X = X - mean_X
    
    if dim > num_data:
        # PCA compact trick
        M = np.dot(X, X.T) # covariance matrix
        e, U = np.linalg.eigh(M) # calculate eigenvalues an deigenvectors
        tmp = np.dot(X.T, U).T
        V = tmp[::-1] # reverse since the last eigenvectors are the ones we want
        S = np.sqrt(e)[::-1] #reverse since the last eigenvalues are in increasing order
        for i in range(V.shape[1]):
            V[:,i] /= S
    else:
        # normal PCA, SVD method
        U,S,V = np.linalg.svd(X)
        V = V[:num_data] # only makes sense to return the first num_data
    return V, S, mean_X
        

In [4]:

# load images into matrix
immatrix = np.array([np.array(Image.open(im, 'r')).flatten()
                 for im in a_font_files], 'f')

In [37]:

# Perform PCA
V, S, immean = pca(immatrix)

Path b

In [17]:

# Show Results
# First one is the mean image
# Rest 7 are the top 7 features extracted for font 'a'
tmp_img = np.array(Image.open(a_font_files[0], 'r'))
m,n = tmp_img.shape
fig = plt.figure()
plt.gray()

plt.subplot(3,4,1)
plt.imshow(immean.reshape(m,n))
plt.axis('off')

for i in range(11):
    plt.subplot(3,4,i+2)
    plt.imshow(V[i].reshape(m,n))
    plt.axis('off')

plt.show()

Apply PCA to Face Images :-)¶

In [54]:

## List of Bill Clinton Face Images
lwf_src_dir = "lfw/Bill_Clinton"
bclinton_files = [os.path.join(lwf_src_dir, f) for f in os.listdir(lwf_src_dir) if f.endswith(".jpg")]
## Display first 7 images
fig_org = plt.figure()

tmp_img = np.array(Image.open(bclinton_files[0], 'r').convert('L'),'f')
m,n = tmp_img.shape


# load images into matrix
lwf_immatrix = np.array([np.array(Image.open(im, 'r').convert('L'),'f').flatten()
                 for im in bclinton_files], 'f')

for i in range(8):
    plt.subplot(2,4,i+1)
    plt.imshow(lwf_immatrix[i].reshape(m,n))
    plt.axis('off')
    

In [50]:

# PCA on face images
bV, bS, bimmean = pca(lwf_immatrix)

In [51]:

# Show Results
# First one is the mean image
# Rest 11 are the top 11 features extracted for pictures of Bill Clinton

fig = plt.figure()
#plt.gray()

plt.subplot(3,4,1)
plt.imshow(bimmean.reshape(m,n))
plt.axis('off')

for i in range(11):
    plt.subplot(3,4,i+2)
    plt.imshow(bV[i].reshape(m,n))
    plt.axis('off')

plt.show()

In [ ]: