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Hi Johannes

I can see we are talking at crossed purposes, I was referring to the article "Pixels are not square". If you have to display your data, then the square display pixel representing the integrated intensity over that area in the camera pixel is _perfectly_ correct -it has nothing to do with trying to 'undo' an instrument PSF. I recall publishing a paper some time ago where the detector pixels were visible in the image. The production editor complained that the image was too low a resolution and insisted that it be retaken ay a higher resolution. I pointed out that the square pixels he could see were actually made up of many smaller pixels (as required to meet their 600 dpi printing requirement) and that it was not appropriate to reprocess the image to hide the actual detector/instrument resolution  (I eventually got the image accepted by him but it was a protracted email fight).  I have also seen 'images' representing 2D histograms that have been smoothed and displayed at much higher pixel resolution than the underlying histogram data without rescaling the intensity calibration. This is, in my opinion, a form of scientific fraud, it leads the reader to assume that there was much more data obtained than was actually present and a false sense of reliability in the underlying data set (but I bet the production editor was happy)... 

As far as assigning a data value to a spatial point within the detector pixel, that is quite problematic as the pixel integrates the signal over its area -I think we agree on that. Unless the pixel were very small compared to the width of the instrument PSF then it's value does not 'correctly' assign the intensity recorded to the center of the pixel (or anywhere else over the area of the pixel). Perhaps one way of thinking about this is to imagine a narrow spike being detected by a single camera pixel (yes, the sampling frequency is slightly greater than twice the bandwidth) the camera pixel reports the local integrated amplitude/energy of the spike function correctly but the phase information (needed for reconstruction) is uncertain due to the width of that single pixel. The problem disappears if you know you are looking at a single Airy disk (or other known bandlimited waveform with 2B < Fs) as only one solution for the position of the disk centre will fit the intensities recorded over the width of  several pixels (I'm ignoring the uncertainty due to noise).  It is the extra pixels around the pixel of interest that supply the extra information needed to assign an intensity to a _position_ within that pixel from the integrated signal over the pixels. This is why super resolution microscopy becomes possible. BUT to display _raw_ camera data, I still think you should use square pixels for square pixel cameras to show that you don't know how to assign (or have not yet determined the assignment (it's raw data)) the intensity to a point...

Perhaps a non-mathematical, intuitive way of describing the situation is this: You need to pieces of information for determining a sinusoid: It's amplitude and phase. So you need 2 measures to solve the equation and that is given by the values of 2 adjacent pixels for a sinusoid just under Fs/2. If you move the phase of highest frequency sinusoid both pixel intensities change (note that both pixels always have different intensity).  It's the ability to solve the phase component that makes super resolution microscopy possible. BUT to display _raw_ camera data, I still think you should use square pixels for square pixel cameras to show that you don't know how to assign (or have not yet determined the assignment (it's raw data)) the intensity to a point...

Hope this clears up the confusion between our POVs for others reading this.

For those interested in how to display super resolution data when you only have intensities at points and not pixels, we published a nice (IMO) method that may not corrupt spatial information as much as assigning a Gaussian or square to each point:

Baddeley D, Cannell MB, Soeller C. Visualization of Localization Microscopy Data. Microsc. Microanal. 2010 Jan. 18;16(01):64.  

Cheers Mark


On 15/04/2012, at 6:24 AM, Johannes Schindelin wrote:

> Hi Mark,
> 
> On Sun, 15 Apr 2012, Mark Cannell wrote:
> 
>> The point is that the pixels integrate over their entire face which is
>> square so that this may be represented by square pixels perfectly
>> faithfully.
> 
> Point is: at the resolution we're operating, PSFs play a role. It does not
> matter whether the detector is square. The non-square part happens earlier
> in the process, the detector cannot undo it.
> 
> All you can safely assume of a pixel is that it has one or more values
> (even referring to the value as an "intensity" need not be correct) and
> that it has a center. The rest depends on the setup.
> 
> Ciao,
> Johannes