I'm trying to figure out how to use PCA to decorrelate an RGB image in python. I'm using the code found in the O'Reilly Computer vision book:
from PIL import Image
from numpy import *
def pca(X):
# Principal Component Analysis
# input: X, matrix with training data as flattened arrays in rows
# return: projection matrix (with important dimensions first),
# variance and mean
#get dimensions
num_data,dim = X.shape
#center data
mean_X = X.mean(axis=0)
for i in range(num_data):
X[i] -= mean_X
if dim>100:
print 'PCA - compact trick used'
M = dot(X,X.T) #covariance matrix
e,EV = linalg.eigh(M) #eigenvalues and eigenvectors
tmp = dot(X.T,EV).T #this is the compact trick
V = tmp[::-1] #reverse since last eigenvectors are the ones we want
S = sqrt(e)[::-1] #reverse since eigenvalues are in increasing order
else:
print 'PCA - SVD used'
U,S,V = linalg.svd(X)
V = V[:num_data] #only makes sense to return the first num_data
#return the projection matrix, the variance and the mean
return V,S,mean_X
I know I need to flatten my image, but the shape is 512x512x3. Will the dimension of 3 throw off my result? How do I truncate this? How do I find a quantitative number of how much information is retained?
Answer
If there are three bands (which is the case for an RGB image), you need to reshape your image like
X = X.reshape(-1, 3)
In your case of a 512x512 image, the new X will have shape (262144, 3). The dimension of 3 will not throw off your result; that dimension represents the features in the image data space. Each row of X is a sample/observation and each column represents a variable/feature.
The total amount of variance in the image is equal to np.sum(S), which is the sum of eigenvalues. The amount of variance you retain will depend on which eigenvalues/eigenvectors you retain. So if you only keep the first eigenvalue/eigenvector, then the fraction of image variance you retain will be equal to
f = S[0] / np.sum(S)