The Hadamard Transform

The Hadamard transform and the Haar transform, to be considered in the next section, share a significant computational advantage over the previously considered DFT, DCT, and DST transforms. Their unitary matrices consist of and the transforms are computed via additions and subtractions only, with no multiplications being involved. Hence, for processors for which multiplication is a time-consuming operation a sustained saving is obtained.

The Hadamard unitary matrix of order n is the matrix , generated by the following iteration rule:

	(4.5.1)

where

	(4.5.2)

And denotes the Kronecker product of two matrices .

where A(i,j) is the (i,j) element of A,i,j=1,2...,N.. Thus, according to (4.5.1), (4.52) it is

And for

It is not difficult to show the orthogonality of , that is,

For a vector x of N samples and , the transform pair is

The 2-D Hadamard tansform is given by

The Hadamard transform has good to very good energy packing properties. Fast algorithms for its computation in subtractions and/or additions are also available.

Remark

Experimental results using the DCT, DST, and Hadamard transforms for texture discrimination have shown that the performance obtained was close to that of the optimal KL transform. At the same time, this near-optimal performance is obtained at substantially reduced complexity, due to the availability of fast computational schemes as reported before.