WHT of uniform distribution.
After filling an array with random numbers from the uniform distibution go pair-wise
through the array to get an x and y coordinate to plot. Do the same for the
WHT transformed random data. Notice that the WHT transforms the data into the
Gaussian distribution due to the central limit theorem, which applies not only
to sums of random numbers but also sums and differences as found in the WHT.
WHT of image.
Natural images when transformed by the WHT have a small number of high magnitude
coefficents and a large number of low magnitude coefficents. Randomly sign flipping
the data in the natural image make all the WHT outputs have a Gaussian distibution.
This makes a very fast random projection algorithm with many uses (eg. locality
sensitive hashing, neural networks and associative memory.)
You can improve the quality of the random projections by repeating the sign flip, WHT
process a number of times or by including random permutations for even higher
random projection entropy.