Quantifying Observational Error in FWHM Measurements of Optical Signals

[TadyZ]

Hello,

i have a optical signal(x is length in pixels and y is amplitude ) and i have found a number of FWHM values of peaks. The width is measured in pixels, let's say the average width of 50 peaks is 8,7 pixels, standard deviation is 0,6 and the average amplitude of peaks is 38 and standard deviation is 1,2. I want to find what's the error for the average value if i say that accuracy is 1 pixel.

I want to plot a graph with all the peaks, average value and to show if deviations may be caused by observational errors or something else.

I think that my attempt to find it is too simple. I just take 1/8,7 and find that the error is 11,5%.

 

[mfb]

All your peaks are supposed to be the same (repeated measurements of the same thing)? How did you get FWHM and amplitude? If that was a fit result, the fit result should also include the mean value and its uncertainty for every peak.

Without background, the uncertainty on the mean will follow the FWHM divided by the square root of the number of events used to determine it - with some prefactor depending on what exactly you do.

 

[TadyZ]

No, it's not repeated measurements, in theory values suppose to be the same, but they have deviations. Amplitude is brightness of a pixel, i got FWHM using MATLAB function, this one to be exact.

If you want background:
i'm measuring width of a white space(if it's telling you something, it's picket fence test for QA of LINAC MLC) in the picture that I've attached. I need to measure it on every horizontal line and find which line has the highest deviation with respect to average value. I already did that. But i also need to find out if the deviations are not caused by observational error. In practice i know that it's not, because measurements with slightly different setting show similar results, but i need to prove it in theory.

 

[mfb]

Matlab can do a fit to determine the central value of the peak.

Forget "number of events", sorry - I'm too used to have those instead of direct amplitude measurements.
Hmm... it might be possible to derive the expected variation of the peak positions from more fundamental principles, but you could just take the observed variance and look for outliers.

 

[gleem]

@Choppy

What are the MLC manufacturers specs?

 

[Choppy]

I don't have specs easily accessible right now - I'm out of my home country for a couple of weeks.

In a picket fence test you should be looking for two things really:
(1) a deviation in the gap width that's significantly larger than the other gaps, and
(2) a deviation in the peak location that's significantly outside of the rest of the locations in that column. (This second one needs to assume that the mean peak location for the column correct, so it helps to have a means of verifying that.)

So really, as I believe what Gleem is getting at, is you first have to look a the specifications for the test - either from your MLC manufacturer or the literature. In principle, you could use a set of images to determine what a distribution of gap FWHMs and peak locations should look like under proper operating conditions and then use a basic statistical test for each MLC leave at each bar of the fence to see if you have a significant deviation from those, but this won't necessarily give you what you're looking for, since a statistically significant deviation does not necessarily correlate with a clinically significant difference. Instead, look to the literature for tolerances that are clinically significant.

You could start by searching LoSasso, as I believe that name is one of the big ones as far as picket fence testing goes.

 

[TadyZ]

Yes, i know what to look for when you are making Picket Fence test, usually the deviations are visible by eye and it's really easy to spot it. It's even easier to spot when i have numerical data. I my data there are no deviations that are so big that could be considered as something out of Quality Assurance boundaries. MLC is working fine.

But my case is different, there are small deviations and i need to be really specific that I'm measuring actual positioning inaccuracy but not getting observational errors. I need it for my master thesis. So, is there a way to calculate error if I'm using FWHM method to find impulse width?

 

[gleem]

So you want to find the "intrinsic" uncertainty ( better word than error) in the position of a peak having measured the width of a the peak?

 

[TadyZ]

Yes, Gleem, you can put it like that.

 

[gleem]

I want to plot a graph with all the peaks, average value and to show if deviations may be caused by observational errors or something else.

That is you want to graphically show the deviation of each peak's FWHM with respect to the mean of the data set using the this intrinsic uncertainty as a criterion for judging whether or not a peak might be deviating too much from expectation?

 

[Khashishi]

When fitting line center, you can often have much better precision than 1 pixel by modeling the line shape properly (Gaussian, multiple Gaussians, Lorentzian, Voigt, etc.)
I think the best way to estimate the fitting error is by generating a million random lines with Poisson noise, and fitting them, and taking the standard deviation of the difference between your generated centers and your fitted centers. Make sure each generated line has a random center which can be between two pixels.

 

[TadyZ]

That is you want to graphically show the deviation of each peak's FWHM with respect to the mean of the data set using the this intrinsic uncertainty as a criterion for judging whether or not a peak might be deviating too much from expectation?

Yes! :D final graph should look like this:

Each leaf pair(line in image below) is average width of 5 measurements(5 white lines)

Each line pixel intensity profile looks like this:

I just don't know how to calculate uncertainty for FWHM.

 

[Khashishi]

What is the meaning of the y-axis? Is that photon counts?
I would write a simple program that takes your curve and replaces the value at each x location with a new random value, Poisson distributed (assuming those are photon counts), with mean equal to your original curve. Then fit that new curve. Repeat many times. Each trial gives you new fit parameters. The standard deviation among the trials gives you the uncertainty of the fit due to shot noise.