If anyone wants to visualize how a variety of basis transforms (including Fourier/DFT, Eigenvectors/PCA, Haar Wavelets, Random basis, and Identity basis) can be applied to images, I created this a few years ago which seems to have become the #2 google result for "basis transform":

http://www.cc.gatech.edu/~phlosoft/transforms/

It's a bit dated by being a Java applet, but source code is all there, written in Processing. (I should probably update it now that processing.js is a thing. Years pass quickly.)