We will first define the DFT for two dimensional signals.
And then we will look at the amount of information that we can extract from
the magnitude and the phase of the DFT of an image.
Fourier analysis of two dimensional signals can be developed exactly
as we did for the one dimensional case.
Since here we're concerned mostly with digital images,
namely finite support two dimensional images.
We will only review the definition of the two dimensional DFT.
So let's consider a two dimensional finite support
signal of support big N1 times big N2.
The DFT is defined as the double sum for
the first index that goes from 0 to big N1 minus 1.
And the second index that goes from 0 to big N2 minus 1,
of the values of the dimensional signal over the support.
Times the product of two complex exponentials.
Whose frequencies are 2 pi over N1 and
1k1, and 2 pi over N2 and 2K2.
The products of these two complex exponentials represents a basis
function for the space of images of size N1 times N2.
The DFT can be easily inverted just like we did in the one dimensional case.
We take basically complex exponentials with the sign reversed and
we repeat the sum.
Taking now the DFT coefficients in to the sum.
Normalization is by convention applied to the inverse formula.
And in this case we have to divide the sum by the product of N1 times N2.
It is certainly instructive to look in more detail at the basis functions for
the space of N1 times N2 images.
These have the form that we have shown before in the DFT sum.
And we can easily prove like in the one dimensional case,
that they are orthogonal.
There are N1 times N2 basis functions for an image of that size.
And so it would be very hard to look at each one of them even for
images of moderate size.
But we can try and
plot some key representative basic functions to give you an idea.
Of what the building blocks are for an image in Fourier space.
We will show these basis functions by plotting the real part only as
a grayscale image.
Where the value of zero is indicated by black and
the value of one is indicated by white.