Single Photon Finding Study

The effectiveness of single photon peak finders is analyzed in the text below. The V1 peak finder is used as the default peak finder along with the V4 and a square peak finder with which to compare.

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. Through the entirety of this study, the parameters were held constant unless otherwise stated. For the square peak finder, an initial threshold of 10 ADU was used. For the V1 and V4 algorithms, a rank of 1, a low threshold of 10 and a high threshold of 30 were used. The other parameters were set so as to not any noise subtraction in the calculation of the peak energies. The maximum peak size was set at 4 pixels since a single photon peak of greater than 4 pixels is unreasonable.

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 shifted 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 141.97 ± 0.05 and a sigma of 8.62 and the square peak finder having a mean of 146.80 ± 0.05 and a sigma of 8.61 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 V4R2 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. Using terminology borrowed from the section  Shape Effects for 3 Pixel Peaks, this algorithm rejects peaks such as Hockey Stick shapes. In this single photon case since the rank is 1, there isn't very much room for the V4R2 algorithm to give different results to those of the V1 algorithm. For example if the V4R2 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 V4R2 algorithms. Visually, there appears to be very little difference between the two algorithms (in fact, the overlapped plot is exactly the same). Although, the peak for the V4R2 distribution appears to be slightly taller. This algorithm resulted in a mean of 141.93 ± 0.05 with a sigma of 8.60. This sigma is very slightly smaller than that of the V1 algorithm. This change is better explained in the following section.

The Effect of Peak Size

Since the square peak finder only find 4 pixel peaks, only the data for the V1 and V4R2 peak finder is shown below.  This data consists of 30,837 peaks for the V1 algorithm and 30,988 peaks for the V4R2 algorithm found over 13 different sparse events. It can be seen for the 1 and 2 pixel peaks, the sigma is almost the same for each algorithm but for 3 and 4 pixel peaks, the difference is more significant. This is due to the filtering that the V4R2 algorithm does on the peaks as described before. So 3 and 4 pixel peaks for the V1 algorithm are instead 1 and 2 pixel peaks for the V4R2 algorithm and it's clear that this filtering is beneficial to the energy resolution although the benefit is slight. It'll be seen later that the number of peaks that are affected by this filtering is a very small percentage of the total number of peaks.

In principle, one can count photons by looking at individual pixel values but, in practice, this is difficult because it is a broad distribution. For event 656, an event with about 3000 peaks, the energy distribution of all pixels is shown below in a semilog plot. As can be seen, the distribution is quite broad and there is no clear one-photon peak.

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.

Shape 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). The effects of such cuts are shown below.

As expected, the Hockey Stick shapes peak does shift to the left and the peak is much sharper and narrower. The peak is now at about 137 ADU; this is less than the peak of the entire distribution, but it is closer. The L shape peaks were also affected but the result does not appear to be much better. These results can be seen in the table below.

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 as shown in the following plot.

One could argue that there is an improvement from cutting the Hockey Stick shapes, but that improvement is very slight.

One theory for the existence of the L shapes is masking due to bad pixels on the detector. These L shapes could actually be 4 pixel squares with a pixel masked which results in this odd shape. To test this theory, the bad pixel mask for the detector was overlayed with a plot of the max pixel for each L shape. If the theory is true, we should find bad pixels adjacent to the L shape max pixels. Of the 434 L shapes, this was true only 19 of these peaks which is about 4.4% of the L shapes. The follow plot shows a section of the entire detector. It is important to note that this number of 19 is an upper bound. Because the orientation of the L shape is unknown, it could be coincidental that there is a bad pixel adjacent to the L shape max pixel. So the actually number may be lower. Therefore this theory doesn't hold and so these L shapes seen are not caused by the bad pixel mask.