haar, an Octave code which computes the Haar transform of data.
In the simplest case, one is given a vector X whose length N is a power of 2. We now consider consecutive pairs of entries of X, and for I from 0 to (N/2)-1 we define:
S[I] = ( X[2*I] + X[2*I+1] ) / sqrt ( 2 )
D[I] = ( X[2*I] - X[2*I+1] ) / sqrt ( 2 )
We now replace X by the vector S concatenated with D. Assuming that (N/2)
is greater than 1, we repeat the operation on the (N/2) entries of S, and
so on, until we have reached a stage where our resultant S and D each contain
one entry.
For data in the form of a 2D array, the transform is applied to the columns and then the rows.
Thanks to comments by Stephen Becker, the code has been modified so that the haar_1d() and haar_1d_inverse(), and haar_2d() and haar_2d_inverse() functions will be proper inverses of each other even in the case when the vector or matrix dimensions are NOT powers of 2.
The computer code and data files described and made available on this web page are distributed under the MIT license
haar is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.
cosine_transform, an Octave code which demonstrates some simple properties of the discrete cosine transform (DCT).
sftpack, an Octave code which implements the "slow" Fourier transform, intended as a teaching tool and comparison with the fast Fourier transform.
walsh, an Octave code which implements versions of the Walsh and Haar transforms.