http://confocal-microscopy-list.275.s1.nabble.com/A-pixel-is-not-a-little-square-tp7468992p7470099.html
sampling. But while sampling provides the data needed to reconstruct
reconstruction envisaged by sampling theory. This requires a filtering
of the raw data. Look at a sine wave sampled at 2.3 times its
source is stable. Poisson noise is unavoidable and can only be minized
by increasing the the number of photons detected. Poisson noise
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> Sorry Guy, I still think you don't see the point I'm trying to make.
> The camera actually says "The mean signal from x to x+dx is ..."
> (where dx is the sensor pixel size). It does NOT say the signal at x
> is 'K' and that is where I think the confusion lies. The camera
> output is a 2D 'histogram' and showing little boxes with the same
> intensity is (I say again) a perfectly accurate representation of
> the data ( i.e. F(x) for x -> x + dx = K). With respect, it is not,
> as you say inaccurate -even if it is unaesthetic. If you fit a
> sinusoid you have just carried out a fitting exercise... That is not
> a "more accurate" presentation of the data despite what your Smith
> says (even if it may be a more accurate representation of the object
> which has been discretized). One should not loose sight of the fact
> that you have made some (possibly large) assumptions in the fitting
> process.
>
> Put mathematically, if you smooth out the displayed pixel edges you
> extend the actual sampling frequency (note how you are putting new
> unrecorded samples between recorded data values -which is what
> drawing a line between points actually does) -you are adding
> information to the data that was NOT present in the RAW data. It may
> be that your additional information is correct and adds value (e.g.
> the band limit of the microscope is...) but one should not loose
> sight of distinction between the addition of data/information by the
> experimenter (which may or may not be wrong) and that reported by
> the instrument (the closest to truth the experimenter can get).
>
> At the risk of boring some readers on this list, let me emphasize my
> point : The camera actually says "The mean signal from x to x+dx is
> ..." (where dx is the sensor pixel size). It does NOT say the signal
> at x is 'K' . This can be portrayed as a square with constant color
> and I can think of no other truer portrayal of the measured data.
> Hopefully dx is less than the resolution of the viewer at final
> display resolution but if it is not, then the only choice (IMHO) is
> between aesthetics (or some other goal) and truthfully displaying
> the recorded data -there is no middle ground.
>
> Cheers Mark
>
> PS My CD player can't output square waves because the detector etc.
> has a rather finite bandwidth... Even if it could, my ears are too
> many dB down at 44 kHz to sample it correctly and hear the artifacts
> introduced by digital sampling ... :-)
>
> On 16/04/2012, at 9:18 AM, Guy Cox wrote:
>
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>>
>> " There are _no_ 'higher harmonics' present in the data, only in
>> ones 'artistic' interpretation for display purposes." That is
>> exactly what I said!
>>
>> I also never said that the data is a continuous function, I said it
>> is a series of discrete samples of a continuous function. So when
>> you choose to display it you have to do something. Drawing little
>> boxes is NOT 'doing nothing' and neither is it 'displaying the raw
>> data'. On the contrary - it is corrupting the data with
>> frequencies which shouldn't be there AND confusing the human eye
>> (for which, presumably, we are doing the drawing). The raw numbers
>> are useful - indeed essential - for the computer but fundamentally
>> cannot just 'be displayed' to the human eye as an image. Our
>> sampling rationale is based on sine-wave frequencies and therefore,
>> as Alvy Ray Smith said, sinusoidal mapping is the truest (not the
>> most aesthetic, though this is also true) way of displaying the
>> data. It doesn't add any spurious higher harmonics, it presents
>> the data as accurately as our sampling permits. Drawing little
>> boxes may be easier, but it is just as much mapping the measured
>> samples to a displayed image - the difference is that this method
>> is both inaccurate and un-aesthetic.
>>
>> If your CD player spat out square waves to the speakers, you'd take
>> it back to the shop pretty promptly!
>>
>>
>>
>> Guy
>>
>> -----Original Message-----
>> From: Confocal Microscopy List
>> [mailto:
[hidden email]] On Behalf Of Mark Cannell
>> Sent: Monday, 16 April 2012 5:27 PM
>> To:
[hidden email]
>> Subject: Re: A pixel is not a little square
>>
>> *****
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>>
>> I think I see the problem, the spurious frequencies arise from your
>> thinking the _data_ is a continuous function and treating it as
>> such (by "drawing a line ..."), but it is not, it is discrete and
>> can be faithfully represented by a _discrete_ Fourier transform
>> (which folds at Fs/2). The hiighest frequency in the DFT is Fs, but
>> we know we shouldn't look at that right? There are _no_ 'higher
>> harmonics' present in the data, only in ones 'artistic'
>> interpretation for display purposes.
>>
>> If it looks jagged, that is because in reality sampled data really
>> is! The problem really arises because you do not know how to fill
>> in the space between data samples. You can interpolate (or not).
>> If you interpolate you are making a statement about the model
>> underlying the data and have just carried out a fitting exercise.
>> Fitting is NOT raw data presentation. If you just plot data values
>> you make no assumption about what should join the data, no model
>> has been fit to the data. Every scientist should know the
>> difference between a histogram and a continuous distribution and
>> not be fooled by the vertical lines at the histogram boundaries
>> (which is what you show in a pixel image).
>>
>> The choice is yours, in one case you faithfully show unadulterated
>> sampled data (the histogram looks less 'pretty' than a curve) or
>> you fit a model and interpolate. The trouble with the latter is
>> that the model is probably wrong and you hide the defects in the
>> data (e.g. camera pixel size) from the keen eyed reviewer... Of
>> course if the data points are really close together, the myopic
>> reviewer can't see defects in you data :-) ! From Guy's
>> reasoning, it would be impossible to represent any digitally
>> sampled data because you are always pixelating a continuous
>> function (all pictures get mad up of little squares -the printer
>> dumps blobs of ink etc). So, where does the pixelation become
>> acceptable? This is now aesthetic and has nothing to do with
>> science or mathematics (those with perfect vision will always see
>> discretization 'artifacts' more easily) .
>>
>> Cheers Mark
>>
>> On 16/04/2012, at 3:31 AM, Guy Cox wrote:
>>
>>> *****
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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.
>>>
>>> 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.
>>>
>>> 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.
>>>
>>> "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.
>>>
>>> *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.
>>>
>>>
>>> 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.
>