Each input Hermitian image represents a COMPLEX image in which certain symmetries exist. These symmetries allow all the information stored in the COMPLEX image to be compressed into a single REAL image. This routine effectively unpacks the two REAL images given as input into two COMPLEX images, multiplies them together pixel-by-pixel to produce another COMPLEX image, and then packs the COMPLEX image back into a single Hermitian REAL image. In fact it is not necessary to do the actual unpacking and packing of these Hermitian images; this algorithm generates the output Hermitian image directly and thus saves time.

See the Notes for more details of Hermitian images and how they are multiplied.

CALL KPG1_HMLTx( M, N, IN1, IN2, OUT, STATUS )

M = INTEGER (Given)

N = INTEGER (Given)

IN1( N, M ) = ? (Given)

IN2( N, M ) = ? (Given)

OUT( N, M ) = ? (Returned)

STATUS = INTEGER (Given and Returned)

• There is a routine for processing single- and double-precision arrays; replace " x" in the routine name by R or D as appropriate. The data type of the IN1, IN2, and OUT arguments must match the routine used.

• Fourier transforms supplied in Hermitian form can be thought of as being derived as follows (see the NAG manual, introduction to Chapter C06 for more information on the storage of Hermitian FFTs):

1) Take the one-dimensional FFT of each row of the original image. 2) Stack all these one-dimensional FFTs into a pair of two-dimensional images the same size as the original image. One two-dimensional image (" A" ) holds the real part and the other (" B" ) holds the imaginary part. Each row of image A will have symmetry such that A(J,row) = A(M-J,row) (J goes from 0 to M-1), while each row of image B will have symmetry such that B(J,row) = -B(M-J,row). 3) Take the one-dimensional FFT of each column of image A. 4) Stack all these one-dimensional FFTs into a pair of two-dimensional images, image AA holds the real part of each FFT, and image BA holds the imaginary part. Each column of AA will have symmetry such that AA(column,K) = AA(column,N-K) (K goes from 0 to N-1), each column of BA will have symmetry such that BA(column,K) =

• BA(column,N-K). 5) Take the one-dimensional FFT of each column of image B. 6) Stack all these one-dimensional FFTs into a pair of two-dimensional images, image AB holds the real part of each FFT, and image BB holds the imaginary part. Each column of AB will have symmetry such that AB(column,K) = AB(column,N-K) (K goes from 0 to N), each column of BB will have symmetry such that BB(column,K) = -BB(column,N-K). 7) The resulting four images all have either positive or negative symmetry in both axes. The Hermitian FFT images supplied to this routine are made up of one quadrant from each image. The bottom-left quadrant is taken from AA, the bottom-right quadrant is taken from AB, the top-left quadrant is taken from BA and the top-right quadrant is taken from BB.

The product of two Hermitian FFTs is itself Hermitian and so can be described in a similar manner. If the first input FFT corresponds to the four images AA1, AB1, BA1 and BB1, and the second input FFT corresponds to the four images AA2, AB2, BA2, BB2, then the output is described by the four images AA, AB, BA and BB where:

AA = AA1$\ast$AA2 $+$ BB1$\ast$BB2 - BA1$\ast$BA2 - AB1$\ast$AB2 BB = AA1$\ast$BB2 $+$ BB1$\ast$AA2 $+$ BA1$\ast$AB2 $+$ AB1$\ast$BA2 BA = BA1$\ast$AA2 - AB1$\ast$BB2 $+$ AA1$\ast$BA2 - BB1$\ast$AB2 AB = -BA1$\ast$BB2 $+$ AB1$\ast$AA2 $+$ AA1$\ast$AB2 - BB1$\ast$BA1