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Hi Guy,

I feel better knowing you are an even slower typist (or explainer?) than 
I am. Thanks, George

On 4/15/2012 10:31 PM, Guy Cox wrote:
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> To join, leave or search the confocal microscopy listserv, go to:
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>
> OK, having slept on it, I now feel that just maybe I can explain what this is all about.  If only the list would let us include pictures it would be much easier!
>
> Let's assume we have a digital image, from any source, consisting of pixels with a spacing s.  The smallest spacing we can resolve in this image is 2s, and this will correspond, in frequency space, with a frequency f.  f represents the bandpass limit of this system,  no higher frequencies can be passed.  Now imagine we have a row of pixels containing the following values:
>
> 255  0  255  0  255  0  255  0  255
>
> If we represent these pixels by little squares, we'll have something like a chessboard.  Taking a line along this chessboard will give us a square wave.  Now this square wave cannot be represented within the bandpass limit of the system, defined by the frequency f.  To represent a square wave we need an infinite series of sine waves f + 3f + 5f +7f .....    To get even a crude approximation to a square wave we need f + 3f - that is a frequency three times higher than the image can contain.
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> In other words, we've introduced a whole series of spurious frequencies into our image that not only were not there to start with, they could not possibly have been there.   Does this matter?  After all, we know they can't be real.  It does matter, because we are talking about a visual representation of our data - that's why we drew the little boxes in the first place.  Our eyes are very sensitive to edges* and the edges will take over if we let these frequencies come within the bandwidth of our eyes.   We will find it very hard to actually see the finest detail in our picture (defined by 2s, remember) because if we enlarge it enough to see this easily we'll also get the edges created by these spurious frequencies.  In everyday terms, the pixellation takes over from the picture.
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> Note that in all this discussion I have  not mentioned microscopes, cameras or anything - we are just talking about a digital image from any source.  It applies to confocal, widefield, and electron microscopes, telescopes, X-ray images and your holiday snaps.  Coming back to the microscopic world, if we oversample to the point where r, our minimum resolved distance, is substantially greater than 2s, we may not need to enlarge to the point where we see the spurious frequencies.  This is probably why some contributors to this discussion have advocated considerable levels of oversampling (though they probably didn't realise this, they just knew they got good pictures that way).  But oversampling in fluorescence can be very hard on our specimens.
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> "But I'm using a CCD detector so my image is made up of little squares".  Yes, you can produce a 'coloured in' picture of your detector that way.  I'm assuming the image is actually what you want to see, though, not the detector.
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> *Amusingly, the human eye does the same thing to emphasize edges as computer image processing does - it makes the dark side of the edge darker than it is and the light side lighter.
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>                                                                                                                                     Guy
>
> PS.  This has doubtless confirmed my reputation among some people as an arrogant bastard.  They are probably right, but at least I'm an arrogant bastard who tries to help.  It's taken me two hours to write this.
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>