Let's start by loading the olivetti face dataset from scikit-learn; it contains 400 face images, each with a dimension of 64 x 64 pixels.

The following code block shows a few random faces from the dataset:

from sklearn.datasets import fetch_olivetti_faces 
faces = fetch_olivetti_faces().data
print(faces.shape) # there are 400 faces each of them is of 64x64=4096 pixels
fig = plt.figure(figsize=(5,5)) 
fig.subplots_adjust(left=0, right=1, bottom=0, top=1, hspace=0.05, wspace=0.05) 
# plot 25 random faces
j = 1
np.random.seed(0)
for i in np.random.choice(range(faces.shape[0]), 25): 
    ax = fig.add_subplot(5, 5, j, xticks=[], yticks=[]) 
    ax.imshow(np.reshape(faces[i,:],(64,64)), cmap=plt.cm.bone, interpolation='nearest') 
    j += 1
plt.show()

The following screenshot shows the output of the preceding code—25 face images randomly selected from the dataset:

Next, let's pre-process the dataset to perform a z-score normalization (subtracting the mean face from all the faces and dividing by the standard deviation); this is a necessary step before we apply PCA on the images. Next, the principal components are computed with PCA() and only the 64 (instead of 4,096) dominant principal components are chosen; the dataset is projected onto these PCs. The following code block shows how to do it and visualizes how much variance of the image dataset is captured by choosing more and more principal components:

from sklearn.preprocessing import StandardScaler
from sklearn.decomposition import PCA
from sklearn.pipeline import Pipeline
n_comp =64
pipeline = Pipeline([('scaling', StandardScaler()), ('pca', PCA(n_components=n_comp))])
faces_proj = pipeline.fit_transform(faces)
print(faces_proj.shape)
# (400, 64)
mean_face = np.reshape(pipeline.named_steps['scaling'].mean_, (64,64))
sd_face = np.reshape(np.sqrt(pipeline.named_steps['scaling'].var_), (64,64))
pylab.figure(figsize=(8, 6))
pylab.plot(np.cumsum(pipeline.named_steps['pca'].explained_variance_ratio_), linewidth=2)
pylab.grid(), pylab.axis('tight'), pylab.xlabel('n_components'), pylab.ylabel('cumulative explained_variance_ratio_')
pylab.show()
pylab.figure(figsize=(10,5))
pylab.subplot(121), pylab.imshow(mean_face, cmap=pylab.cm.bone), pylab.axis('off'), pylab.title('Mean face')
pylab.subplot(122), pylab.imshow(sd_face, cmap=pylab.cm.bone), pylab.axis('off'), pylab.title('SD face')
pylab.show()

The following diagram shows the output. As can be seen, around 90% of the variances are explained by only the first 64 principal components:

The following screenshot shows the mean and the standard deviation face images computed from the dataset, respectively: