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Guy,
But in the CCD case the sampling is integrated over the whole pixel area of
the sensor (assuming pixel coverage of 100% which is the case for EMCCDs
for exemple). There is no information about where did the photons arrive on
the square detector portion corresponding to a pixel. The value X you get
as intensity for this pixel is the integrated density over the whole
detector portion, so why would you take that value as a point-measurement
corresponding to the center of this area ? In that CCD case, isn't the mage
indeed an histogram with a bin that has (quite literally) a width equal to
the pixel size, not a sample at mathematical points corresponding to the
centers of the pixels ?
Christophe
On Mon, Apr 16, 2012 at 14:36, Guy Cox <[log in to unmask]> wrote:
> *****
> To join, leave or search the confocal microscopy listserv, go to:
> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> *****
>
> Mark,
>
> You are continuing to confuse the samples with the
> representation of those samples.
>
> Let's imagine we have a series of data points: 100 90 80 70 60 50 40
>
> We are mapping these on an image where each is separated by a defined
> distance. So we need to fill in this distance.
>
> You are saying that the 'correct' representation is:
> 100 100 100 100 100 90 90 90 90 90 80 80 80 80 80
> 70 70 70 70 60 60 60 60 60 50 50 50 50 50 40
>
> I am saying that this is a wildly implausible and totally unjustified
> interpretation, and the best representation we can derive from the data is:
> 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72
> 70 68 66 64 62 60 58 56 54 52 50 48 46 44 42
> 40
>
> EITHER way we are interpolating the sampled data - we have no option - so
> let's just get over this. Your proposed representation includes detail
> that we could not possibly detect, mine does not. Remember, these are
> SAMPLES. Neither representation changes our recorded data. End of story,
> IMHO.
>
> How did we get into this mess? Why does everyone then 'do it wrong'?
> Well, actually, everyone doesn't. Scanning probe microscopes always remap
> - because by the time they appeared the computing power to do it was
> available. When confocal microscopes first became widely available, in
> 1987, the data they produced completely overwhelmed available computing
> power (believe me, I was there and writing software). So we got used to
> the 'quick and dirty' approach. Consumer digital cameras do a sort of
> remap because the Bayer mosaic requires it, but modern sensors so far
> exceed the resolution of the camera optics that we never get to see any
> spurious frequencies anyway. Computer games consoles always remap. So do
> X-ray and EM tomography systems.
>
> Guy (arrogant bastard)
>
>
>
> ----Original Message-----
> From: Confocal Microscopy List [mailto:[log in to unmask]]
> On Behalf Of Mark Cannell
> Sent: Monday, 16 April 2012 7:37 PM
> To: [log in to unmask]
> Subject: Re: A pixel is not a little square
>
> *****
> To join, leave or search the confocal microscopy listserv, go to:
> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> *****
>
> 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:
>
> > *****
> > To join, leave or search the confocal microscopy listserv, go to:
> > http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> > *****
> >
> > " 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:[log in to unmask]]
> On Behalf Of Mark Cannell
> > Sent: Monday, 16 April 2012 5:27 PM
> > To: [log in to unmask]
> > Subject: Re: A pixel is not a little square
> >
> > *****
> > To join, leave or search the confocal microscopy listserv, go to:
> > http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> > *****
> >
> > 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:
> >
> >> *****
> >> To join, leave or search the confocal microscopy listserv, go to:
> >> http://lists.umn.edu/cgi-bin/wa?A0=confocalmicroscopy
> >> *****
> >>
> >> 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.
>
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