Single Photon Finding Study

The effectiveness of single photon peak finders is analyzed in the text below. Three different methods are compared: the V1, V4 and square peak finder methods which are described below.

The data from experiment xcs06016 and run 37 were used in this analysis. Only sparse events were used so multiple photon peaks or pixels shared by multiple photons are extremely rare; events with less than 3000 peaks found by each peak finder were used. A total of 13 events were used. The purpose of these analyses is to determine which (if any) of the following methods discussed below have an appreciable effect on the RMS of the distribution of the energy.

Comparing Peak Finders

Effect of Neighbor Thresholding

The V1 algorithm is a two-threshold algorithm. Once a pixel above the higher threshold is found, all pixels within a specific range of this pixel above the lower threshold are included as a peak. The range is determined by the rank used where a rank of n creates a box centered at the pixel above the high threshold hold with side length of 2n+1. In these analyses, the rank is set to 1 (i.e. a 3x3 box). The square peak finder has one threshold but lower energy pixels may be included in a peak. It first finds pixels above this threshold which are contained within a 2x2 area allowing for 1, 2, 3 or 4 pixel photons. Then the rest of the remaining square is included which contributes the most to the photon energy. For 1 pixel photons, there are 4 possibilities for the square peak. For 2 pixel photons, there are 2 possibilities. For 3 pixel photons, the pixel within the square that was not originally included is added on. And for 4 pixel photon, no extra energy is added. This is done under the assumption that single photon hits create square peaks on detectors.

The distribution of events for each peak finder is shown below. In the plot next to it, the square peak finder distribution is manually shifted by -4.8 to compare the shapes of the distribution which appear to be very similar.

One can see that the peaks roughly follow a Gaussian distribution but on the higher end, there is a very noticeable shoulder. At first, it appears to be a result of dense photons, a situation that was avoided. With the use of sparse events. A Gaussian curve was fitted to both distributions but to ignore the effects of the shoulder, only the data within 15 ADU of the bin with the maximum number of peaks was used so as to center the data used in the fit around the peak of the Gaussian. The result from this was the V1 peak finder having a mean of 141.93 ± 0.06 and an RMS of 8.37 and the square peak finder having a mean of 146.68 ± 0.060 and an RMS of 8.20 where the errors were obtained by σ/√N.

Just to confirm the suspicions, the sparsest event, event 634, was looked at separately. The V1 and square peak finders found only 545 and 584 peaks, respectively. It can be seen in the energy distribution of the peaks for event 634 that the high energy shoulder is still present so it cannot be due to multiple photon complications. Furthermore, by manually checking peaks, it can be seen that the high energy peaks do not neighbor other peaks and are merely just higher energy peaks.

One possible explanation is how the peaks are chosen. For the square peak finder, since the 2x2 region with the highest energy is chosen to complete the square for 1 and 2 pixel photons, it is possible that this includes noise that is higher than average thus shifting the total energy above the mean. Likewise for the V1 peak finder, large noise has a greater chance of being included since it may surpass the lower threshold again shifting the total energy of the peak above the mean.

It is possible that this shoulder is partly due to the K-beta emission of the material used in this particular experiment which happens to be copper (while the main peak is the K-alpha emission). After a quick loop-up, these values for copper are approximately 8040 keV for K-alpha and 8900 keV for K-beta which gives a 1.107 ratio of K-beta to K-alpha. If we look at the distribution of just the maximum energy pixel of each peak, as shown below, there is very visibly some type of peak on the higher energy side of the 1 pixel distribution. The main peak has a mean of 140.3 while the mean of the smaller peak is about 155. This gives a ratio of 1.104. The numbers used are approximations and the second peak is artificially shifted by the Gaussian from the first peak underlying it. So it is very likely that these two peaks are the K-alpha and K-beta lines of copper.

Alternative to the V1 Algorithm

The V4 algorithm was also explored as a possibly better alternative to the V1 algorithm. It is very similar to the V1 algorithm but does not include pixels above the lower threshold that are not adjacent to the peak on some side very similar to that of a flood-fill algorithm. This algorithm will not include in peaks pixels that are disconnected from the pixel above the high threshold. In this single photon case since the rank is 1, there isn't very much room for the V4 algorithm to give different results to those of the V1 algorithm. For example if the V4 algorithm finds a 1 pixel peak, there are only 4 possible other pixels that may be above the lower threshold that wouldn't be accepted. 

Below the first plot shows the energy distribution of the V1 and V4 algorithms while the second shows them overlapped. Visually, there appears to be very little difference between the two algorithms. This isn't too surprising since, when looking at single photons with a rank of 1, the cases which the V4 algorithm ignores would be very rare. The V1 data is the same as before, while a Gaussian fit to the V4 curve has a mean of 140.67 ± 0.06 with an RMS of 9.01. In the case shown below, the V4 algorithm was passed the parameters of r = 3 and dr = 3 which gives a large ring for noise. The opposite, no ring of noise, was tried but gave an even larger RMS.

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The Effect of Peak Size

Since the square peak finder only find 4 pixel peaks, only the data for the V1 peak finder is shown below.  This data consists of 29,342 peaks found from the 13 different events.

Lining Up and Recombining Pixel Distributions

As can be seen in the table above, there is a noticeable difference in the means of the energy distributions for each number of pixels which, ideally, shouldn't exist. Furthermore, the errors on the mean cannot account for this difference. One theory is that the larger peaks (the ones with more pixels) end up adding in more noise to the total energy of the photon thus shifting it to a slightly higher energy. If such is the case, one remedy would be to shift each distribution so that their peaks fall on the same bin. This was done by using the average of the four bins where the peaks existed as the new bin for the peaks. Below is the result compared to the normal distribution shown in the first graph on this page. As can be seen, the difference is very slight; the distribution is slightly sharper for the shifted data, but not by very much.

When fit with a Gaussian in the same manner as before, the results confirm this observation. Although there is a decrease in the width, it is very small. The noticeable change in the mean is most likely due to the choice of energy used to which the peaks were shifted.

Shapes Effects for 3 Pixel Peaks

Three pixel peaks are interesting peaks because they seem like they are extremely unlikely to happen, or at least less likely than what we see. To understand which shape is being discussed, the following names were chosen to represent the different possible shapes:

The distribution of 3 pixel peaks of these shapes is shown below out of a total of 4069 peaks for the V1 algorithm:

It makes sense that there are zero Idot peaks. This is because the rank is 1 and therefore it is impossible to have a line 3 pixels long that begins with the max peak; it would exceed the boundary from the rank. It is also understandable that there are no NaS because of the small number of possibilities to have no adjacent pixels to the max pixel. For the L shapes, some number of them had a pixel exceeding the high threshold on the pixel not adjacent to the max pixel which would represent two separate photons. A similar situation was also seen with the I shape (both ends of the I exceeded the high threshold). Their energy distribution is shown below.

The peak for the L shape at about 280 is a clear indication that there are two photon peaks that appear as the L shape. It is also interesting to note that the peak for the Hockey Sticks is at about 155 which is about 10 ADU higher than where the V1 algorithm normally peaks. This may be because the non-adjacent pixel is not part of the photon and instead this 3 pixel peak is actually only a 2 photon peaks. Furthermore, it may be beneficial to turn the L shapes into 2 pixel peaks when they aren't actually two photon peaks (i.e. when the non-max adjacent pixel is below the high threshold). This distribution is shown below.

As expected, the Hockey Stick shapes peak does shift to the left. The peak is now at about 137 ADU; this is less than the peak of the entire distribution, but it is closer. And, visually, the peak appears to be thinner than before. The L shape peaks were also affected but the result appears to be worse. Thus, cutting the Hockey Stick shapes but not cutting the L shapes would be the most beneficial. But because the Hockey Stick shape only make up about 2% of the total 3 pixel peaks, this change is not very significant.