HartleyTransform

Computes the Hartley transform

SYNOPSIS

#include "dip_transform.h"

dip_Error dip_HartleyTransform ( in, out, trFlags, process )

DATA TYPES

binary, integer, float

FUNCTION

This function computes a Hartley transform on in and places the result in out.

Normalisation: 1/sqrt(dimension) for each dimension.

The main advantage of the Hartley transform over the Fourier transform is that is requires half the storage for real valued images. Note, that is also possible to directly reduce the storage requirements of the Fourier transform by just storing the right half plane, since for real valued images the left half plane can be derived from the right half using the symmetry properties of the Fourier transform.

Unfortunately there seem to be two definitions of the multi-dimensional Hartley transform (they are identical in the 1-D case). DIPlib implements the Bracewell (see below) variant, since this one is easy to implement and inherits the storage advantage from the 1-D case. The following are references which each use a different variant (all scaling factors have been dropped):

Bracewell, "Discrete Hartley Transform", J. Opt. Soc. Am, vol. 73, no. 12, December 1983 :

DHT(u,v) = Sum Sum I(x,y) cas( ux ) cas( vy )
            y   x

Kenneth R. Castleman, "Digital image processing", Prentice Hall, 1996 :

DHT(u,v) = Sum Sum I(x,y) cas( ux + vy )
            y   x

Using cas(a) = cos(a) + sin(a) :

cas(ux)cas(vy) = cos(ux)cos(vy)+cos(ux)sin(vy)+sin(ux)cos(vy)+sin(ux)sin(vy)
cas(ux+vy)     = cos(ux)cos(vy)+cos(ux)sin(vy)+sin(ux)cos(vy)-sin(ux)sin(vy)
                                                            ^^^

A subtle difference. The two definitions have very similar properties, for example the convolution property.

In implementation terms, Bracewell is equivalent to perform the one-dimensional Hartley transform along each dimension. The Castleman variant is equivalent to the definition: DHT = re(DFT) - im(DFT). On a final note, I've not noticed mention of the difference between the two variants, so the indications Bracewell's and Castleman's variant are not and should not be accepted "labels" to refer to the variants (For both variants I have selected the first reference I came across, not chronologically the first reference to use the variant).

Defaults: process may be zero, indicating that all dimensions should be processed.

ARGUMENTS

The dipf_FourierTransform enumeration consists of the following flags:

SEE ALSO

FourierTransform