Name of this window

Tim Wescott wrote:
> 
> spasmous@yahoo.com wrote:
> 
> > I've started using a simple windowing function to low pass filter a 2D
> > image. Following the FFT, I multiply in the frequency domain with a
> > function that increases with distance from the origin (allowing for
> > wrapping). Then I IFFT back to the image domain. Basically the window
> > is this:
> >
> > f = sqrt(alpha / (alpha + x^2 + y^2))
> >
> > where x,y range from 1-n/2 to n/2 and alpha is a scalar.
> > Anyone know what this filter is called?
> >
> If the "windowing" is done in the frequency domain it's called a filter,
> not a window.
> 
> Now who's the idiot?

Well he did make the mistake of supplying more information then the
question required. Had he asked simply "I have a 2d window. Basically
the window is this:

	f = sqrt(alpha / (alpha + x^2 + y^2))

Does anybody have a name for this shape"

Or better yet he should have asked  no more than this:
"what is this window called?-> sqrt(1/(1+x^2))"

Don't know the answer myself, it is very similar to a gaussian in shape.

-jim


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jim wrote:

  ...

>   .... Had he asked simply "I have a 2d window. Basically
> the window is this:
> 
> 	f = sqrt(alpha / (alpha + x^2 + y^2))
> 
> Does anybody have a name for this shape"
> 
> Or better yet he should have asked  no more than this:
> "what is this window called?-> sqrt(1/(1+x^2))"

  ...

I took x and y to be the pixel coordinates, and therefore different at
different differences from the origin. I would have written
                  sqrt(h�/(h� + r�)).
I see that as a (mysterious) way to fade to black away from the origin.

The problem with that view is that it's neither a filter nor a shape, so
it's probably wrong. To have asked the question without defining any of
the quantities is wronger.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry Avins wrote:
> 
> jim wrote:
> 
>   ...
> 
> >   .... Had he asked simply "I have a 2d window. Basically
> > the window is this:
> >
> >       f = sqrt(alpha / (alpha + x^2 + y^2))
> >
> > Does anybody have a name for this shape"
> >
> > Or better yet he should have asked  no more than this:
> > "what is this window called?-> sqrt(1/(1+x^2))"
> 
>   ...
> 
> I took x and y to be the pixel coordinates, and therefore different at
> different differences from the origin. I would have written
>                   sqrt(h�/(h� + r�)).

Yes, but for purposes of this discusion we only need to consider the
cross-section of the implied surface:
  wn = sqrt(h�/(h� + x�)) 
and h is a constant which might as well be 1 as it's essentially a scale
factor for the width of the central hump.

> I see that as a (mysterious) way to fade to black away from the origin.
> 
> The problem with that view is that it's neither a filter nor a shape, so
> it's probably wrong. To have asked the question without defining any of
> the quantities is wronger.
>

It's clearly both a filter and a shape. The OP already has explained
he's using it as a filter. He knows what it does in that context. The
question was if the frequency domain shape or spacial domain filter had
been given a name. 

-jim


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jim wrote:

> 
> Jerry Avins wrote:

  ...

>>The problem with that view is that it's neither a filter nor a shape, so
>>it's probably wrong. To have asked the question without defining any of
>>the quantities is wronger.
>>
> 
> 
> It's clearly both a filter and a shape. The OP already has explained
> he's using it as a filter. He knows what it does in that context. The
> question was if the frequency domain shape or spacial domain filter had
> been given a name. 

If I knew what alpha, x, and y represented, I could better understand
the question.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry Avins wrote:
>
> If I knew what alpha, x, and y represented, I could better understand
> the question.
>

I was going to write this thread of as a lost cause but it's becoming
hilarious. Jerry I thought you plonked me, now here you are again
asking questions! If you study the OP it says "x,y range from 1-n/2 to
n/2 and alpha is a scalar".

Although if you meant, what does "x" *really* mean you could try
alt.philosophy. Or maybe if you don't know the name, you could just
stay silent? Just a thought.

jim wrote:

"
Or better yet he should have asked no more than this:
"what is this window called?-> sqrt(1/(1+x^2))"

Don't know the answer myself, it is very similar to a gaussian in
shape.
"

Although it does have that hump-type of look to it, it is, in a way,
about as far away from Gaussian as possible.

>From a probability point of view, it is the density function of a
Cauchy distributed random variable. You can view the Cauchy
distribution as the edge member of a one-parameter family of
probability distributions (the t_n-distributions where the parameter n
is the degrees of freedom - Cauchy has n=1). As n goes to infinity,
this family converges (in the weak sense) to the Gaussian distribution.

Cauchy distribution is notorious and gives nice counter-examples to
almost any convergence theorem in probability theory (for example, the
renormalized sum of a number of Cauchy distributed rv's is again Cauchy
distributed, thus stays away from Gaussian as fas as possible, in the
above sense, mocking the CLT).

The characteristic function (Fourier transform) of the Cauchy
distribution is of the form exp(-c |f|), where c is some constant
related to the variance (interestingly, it decays slower than the
Gaussian in both time and frequency domain).

Regards,
Andor