Re: 2D FFT

Richard Owlett <rowlett@xxxxxxxxxxxxx> wrote:
Rune Allnor wrote:
On 2 Feb, 15:47, Richard Owlett <rowl...@xxxxxxxxxxxxx> wrote:
(snip)

For example I have a gut feeling that a 2D transform of elevation data
in the region of the San Andreas Fault or the Great Rift Valley would
show a prominent feature. But what vector space would it be in?

The vector space would be the space of MxN arrays, where M and N
are the dimensions of the image. But don't go there.

The first thing to know is that in rectangular coordinates
the FFT is separable. You take the FFT of all the rows, then
all the columns, or the other way around, and get the same result:
the 2D FFT of the data.

In 1D we know that, for example, f(x)=sin(3x) one the solution
for the amplitude of vibration for a violin string as a function
of position on the string.

In equation form, consider f(x,y)=sin(3x)sin(3y), a function
with frequency 3 in x and y. Using the sine addition angle
identities in reverse, sin(3x)sin(3y)=(cos(3x-3y)-cos(3x+3y))/2
(Consider it over the range 0 to 1 for both x and y, the
frequencies being 3 radians/unit length in x and y.)

The solution is the sum (difference) of two cosine waves, one going in
the (x+y) direction, the other in the (x-y) direction.
In electrical terms, they are standing waves in two dimensions.
(The frequency in (x+y) and (x-y) directions is 3/sqrt(2)).

Among others, sin(Ax)sin(By) are the solutions to the vibrating
modes of a square drum head with uniform tension. (Circular
drum heads require Bessel functions for the modes.)

In electrical terms, they are the solutions for the current
in a square plate, going to zero at the boundary.

Thank you. *THANK YOU* ;)

Instead,
try and build some intuition by playing with the 2D FFTs of
binary B&W images that contain simple shapes.

Your suggestion would answer a different "Why" - "why/how does 2D FT work?"
My question is more "why would you want to do a 2D transform?"

They come up often in optics. One of my favorite optics lab
experiments generates 2D fourier transforms using lenses:

Start with a uniform monochromatic light source and illuminate
a transparent object. (A black and white slide, for example, or
a piece of metal with holes cut out of it.) After the light goes
through the object place a lens of focal length f. Put an image
sensor (TV camera with lens removed) a distance f from the lens.
The image will be the 2D fourier transform of the object.

If you rotate the object the transform will rotate the same way.

In the same way that you need higher frequencies to resolve waveforms
with sharp transitions, you need larger lenses to resolve higher
spatial frequencies (in addition to collecting more light).

I don't have any application, _that I *know* of_ , for a 2D transform.
But I see it regularly discussed by people with a "signals" orientation.
I have an interest in one class of signals (speech). My question might
be phrased "what gain would I have for effort invested in studying 2D
transforms?" I could investigate the Carnot cycle, but would it improve
my driving?

As far as I know, speech signals are 1D. Studying optics
might improve your driving, though, if you can't see very well.

-- glen
.

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