A pixel is Not a Little Square!, and a Voxel is Not a Little Cube (Alvy Ray Smitth 1995)
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
No tengo ni puñetera idea lo que quiere decir todo esto pero queda bien, a que sí. Si estoy de acuerdo conque un Voxel is Not a Little Cube y la afirmación anterior, por toda su explicación matemática posterior.
Pero algo supera todo esto: LA IMAGEN que te embriaga en si misma sin tanta palabreria.
A pixel is Not a Little Square!,
ResponderEliminarand a Voxel is Not a Little Cube
(Alvy Ray Smitth 1995)
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
No tengo ni puñetera idea lo que quiere decir todo esto pero queda bien, a que sí.
Si estoy de acuerdo conque un Voxel is Not a Little Cube y la afirmación anterior, por toda su explicación matemática posterior.
Pero algo supera todo esto: LA IMAGEN que te embriaga en si misma sin tanta palabreria.
Foto Preciosa.