even/odd symmetry in 2D FT of real-valued image

Hi folks,

If I have a 2D image whose values are purely real, and I take the 2D
FT, then I expect the 2D FT to have even symmetry. I think this applies
to a reflection about one axis - that is, half the resulting 2D FT
image is a reflection of the other half. So all the information is in
kept if I use just one half of the 2D FT image.

But does it apply in some circumstances to each quadrant being a
reflection of each other quadrant? So that the information is kept even
if I use just one quadrant of the 2D FT image?

Thanks,

Chris

Chris Bore skrev:
> Hi folks,
>
> If I have a 2D image whose values are purely real, and I take the 2D
> FT, then I expect the 2D FT to have even symmetry. I think this applies
> to a reflection about one axis - that is, half the resulting 2D FT
> image is a reflection of the other half. So all the information is in
> kept if I use just one half of the 2D FT image.
>
> But does it apply in some circumstances to each quadrant being a
> reflection of each other quadrant? So that the information is kept even
> if I use just one quadrant of the 2D FT image?

In general -- no.

The symmetry properties in the 2D FT are a bit different than
one might expect. The symmetry is around origo, not the axes.

To see why, think of the 2D FT of an Nx by Ny image as two
sequencial 1D FTs, the first one applied in the x direction and
the second in the y direction. After the first 1D FTs, the partially
transformed image contains complex-valued samples and is
conjugate symmetrical around Ny/2, as expected.

The second set of FTs, in the y direction, take *complex-valued*
data as input. As you know, no general symmetry properties can
be expected for FTs on complex-valued data,sou you will need
to keep half the tranformed image.

Rune

Thank you. That is most helpful.

Chris

On 2006-04-11 05:25:41 -0300, "Chris Bore" <chris.bore@gmail.com> said:

> Hi folks,
> 
> If I have a 2D image whose values are purely real, and I take the 2D
> FT, then I expect the 2D FT to have even symmetry. I think this applies
> to a reflection about one axis - that is, half the resulting 2D FT
> image is a reflection of the other half. So all the information is in
> kept if I use just one half of the 2D FT image.
> 
> But does it apply in some circumstances to each quadrant being a
> reflection of each other quadrant? So that the information is kept even
> if I use just one quadrant of the 2D FT image?
> 
> Thanks,
> 
> Chris

The word you are looking for is "inversion" as in "a 2-d FT of a real function
has inversion symmetry". Inversion in 1-d is the same as reflection. Inversion
is a "reflection" through the origin. So you will see the reflection you
were expecting along either axis but it is an inversion in the interior.

For much more than you ever wanted to know about symmmetry and FTs take a
look at the crystallographic literature. They mix FTs with spacial symmetry
groups in both 2-d and 3-d.