FAST COMPUTATION OF THE 2-DIMENSIONAL GENERALIZED HARTLEY TRANSFORMS

The two-dimensional generalised Hartley transforms (2-D GDHTs) are various half-sample generalised DHTs, and are used for computing 2-D DHT and 2-D various convolutions. Fast computation of 2-D GDHTs is achieved by solving (n(1) + (n(01)/2))k(1) + (n(2) + (n(02)/2))k(2) = (n + (1/2))k mod N, n(01), n(02) = 1 or 0. The kernel indexes on the left-hand side and on the right-hand side belong to the 2-D GDHTs and the 1-D H-3, respectively. This equation categorises N x N-point input into N groups which are the inputs of a 1-D N-point H-3. By decomposing to 2-D GDHTs, an N x N-point DHT requires a 3N/2(i) 1-D N/2(i)-point H-3, i = 1,..., log(2) N-2. Thus, it has not only the same number of multiplications as that of the discrete Radon transform (DRT) and linear congruence, but also has fewer additions than the DRT. The distinct H-3 transforms are independent, and hence parallel computation is feasible. The mapping is very regular, and can be extended to an n-dimensional GDHT or GDFT easily.