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Gonzalez and Woods, Ch. 3.3.1--3.3.7
Here are some Fourier Transforms for some common functions:
For this section, assume that 
is the Fourier Transform of some
function 
.
A function 
is said to be linear if 
.
The Fourier Transform is linear:
This means
- Multiplying an image by a constant multiplies the images's Fourier Transform
by the same constant.
- Adding two images togeter adds the two Fourier transforms together.
The Fourier Transform of a two-dimensional function is the Fourier
Transform in one dimension of the Fourier Transform in the other direction:
if
then
What this means is that we can compute the two-dimensional Fourier
Transform by doing a one-dimensional Fourier Transform of the rows and
then taking a one-dimensional Fourier Transform of the columns of the result.
This can be extended to any arbitrary dimension.
Rotating the an image rotates its Fourier Transform.
Translating an image doesn't change the magnitude of the Fourier
Transform, but does change its phase:
Changing the spatial unit of distance changes the Fourier Transform as
follows:
There is an inverse relationship between spatial distance and frequency,
resulting from 
where P is the spatial period of the sinusoid.
Expanding a signal shrinks its Fourier Transform (contracts it to lower
frequencies with longer periods),
and contracting a signal expands its Fourier Transform (stretches it to
higher frequencies with shorter periods).
- linearity
- rotational invariance
- separability
- scaling property
© Bryan S. Morse, 1995