Walsh functions and lossy compression.

Walsh functions are best introduced via Hadamard matrices.

Hadamard matrices :

Walsh-1D functions :

Walsh functions are given by the rows (equivalently the columns) of the Hadamard matrices.

Two examples : s = 3 (8 functions) and s = 4 (16 functions) :

Walsh functions are orthonormed with respect to the following scalar product :

An arbitrary binary 1D-function (say {1,1,1,1,1,1,-1,-1} if s = 3) is expandable in the basis of 1D-Walsh functions :

Walsh-2D functions :

The following character, R, is pixellized in a 64x64 grid :

The walsh coefficients for the letter R :

The walsh coefficients for the letter R, sorted in decreasing order (absolute value) :

An arbitrary binary 2D-function (say Rpixels) is expandable without loss in the basis of 2D-Walsh functions :

Light truncation :

Light truncation with greylevels erased :

Medium truncation :

Medium truncation with greylevels erased :

Strong truncation :

Strong truncation with greylevels erased :