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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.

Image RemovedImage Removedsquare_v1_compare    square_v1_shift_compare

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 a sigma of 8.37 and the square peak finder having a mean of 146.68 ± 0.060 and an RMS a sigma of 8.20 where the errors were obtained by σ/√N.

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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.

Image Removedv1_max_pix_dist_per_pix

Alternative to the V1 Algorithm (WIP)

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.

Image RemovedImage Removedv4_v1_compare    v4_v1_shift_compare

The Effect of Peak Size

Since the square peak finder only find 4 pixel peaks, only the data for the V1  and V4 peak finder is shown below.  This data consists of 29,342 peaks found from the 13 different events.

v1_pix_distributionImage Removed

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.

Image Removedv1_shifted_pix_compare

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

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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.

Image Removedv1_3pix_shape_uncut

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.

Image Removedv1_Lcut_vs_Luncut   v1_HScut_vs_uncut

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. 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.

v1_HScut_vs_uncut

In the following table, the stats for these cuts are shown. These stats are only for the 3 pixel distribution, so the effect on the entire distribution will be less.

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