From: Stephen Montgomery-Smith <stephen@math.missouri.edu>
Subject: Re: Fourier analysis of Images
Date: Mon, 15 Nov 1999 21:58:20 -0600
Newsgroups: sci.math
Keywords: Gibb's phenomenon

"David C. Ullrich" wrote:
> 
>     Huh? If you _modify_ the FT and then take the inverse transform of
> course you're going to get a modified image back, which means precisely
> that some pixels will be different colors from the original.
> 
>     Could be that you expected smaller changes than what you got.
> Try using a _smoother_ mask. (Don't just nuke some frequencies and
> leave the rest the same, instead multiply all the frequencies by a nice
> continuous mask function that equals 0 for the frequencies you want
> to lose and 1 for most of the frequencies you want to keep, but
> which makes the transition from 1 to 0 in a smooth way instead of
> just jumping from 1 to 0 suddenly.)
> 

I think that David has a good point.  You may be experiencing some
kind of Gibb's phenomenon.  Suppose one considers the Fourier series
for the square wave.  Graphing the partial sums - maybe something like

sum_{n=-N}^N a_n exp(inx)

where the a_n are whatever you calculate from the formula you will see that
near the discontinuities that you get wild oscillations.

Now translate that into your example.  If you Fourier transform your
image, then cut off the high frequencies, and then Fourier transform
back again, then near discontinuities, for example, near boundaries
between a region of one color and another, you are going to get wild
oscillations in your function, that is, spots of very different colors.

The standard way to lessen or remove Gibb's phenomena is instead of
considering the partial sums, to consider something like

sum_{n=-N}^N phi(n) a_n exp(inx)

where phi(n) is a function so that phi(0)=1, and phi(n)=0 for |n|>N, but
for 0<|n|<N, the function changes in a smooth fashion.  A common example
is

phi(n) = max(0,1-|n|/N)

Another example is something like

phi(n) = 1-2(n/N)^2   |n|<N/2
phi(n) = 2(1-n/N)^2   N/2<|n|/N
phi(n) = 0 otherwise

which is better than the previous function in that it is not only 
continuous, but so is its derivative.  In fact, the more derivatives
of phi(n) you get continuous, the more you will remove the Gibb's
phenomenon.

-- 

Stephen Montgomery-Smith              stephen@math.missouri.edu
307 Math Science Building             stephen@showme.missouri.edu
Department of Mathematics             stephen@missouri.edu
University of Missouri-Columbia
Columbia, MO 65211
USA

Phone (573) 882 4540
Fax   (573) 882 1869

http://www.math.missouri.edu/~stephen