Implementing Hough transform for (phi_0,q/p_T) feature space

[noemina]

• [x] Define the transformation to move from the (x,y) parameter space to the (phi_0,q/p_T) feature space. A bit of math is posted below.

• [x] Plot the tracks in the (phi_0,q/p_T) and produce the binned version of the accumulation histogram, where the crossing point can be evaluated.

• [x] Make heat map with different colour scheme (e.g. inverting the colour palette).
• [x] Evaluate efficiency of the Hough transform as a function of p_T for the precise and approximated transformation.
• [x] Produce heat map for the longitudinal transformation.
• [x] Start combining the information in the longitudinal and transverse planes.

About the efficiency plots:

• [x] Correct the p_T used for the efficiency plots: it has to use the truth p_T
  • Make two plots:
  • [x] Number of generated particles vs generated p_T
  • [x] Number of generated found particles vs generated p_T (for the precise and approximated transformation)
• [x] Add efficiency vs eta (again for the truth particle)

[AndrewSpano]

Started the hough transformation. The formula I ended up using is:

The results (for one-particle files) are this:

I had some issues with seaborn and heatmaps, I will upload tomorrow the code with the requested plots.

[AndrewSpano]

I implemented both equations for the Hough transform (with/without sinus). We will take a look at the sinus first and then check the other.

• Had some bugs in the code (wasn't using enough samples, and one '-' sign was missing). Now the correct plots looks like this:

  

  By applying the Hough transform, we get the following hits per bin (small bin size):

  

  Let's see if the estimated emittance angle is legit by checking the ground truth:

  

  It's spot on! Regarding the heatmap, we can see that the bin-separation is not clearly visible. This is due to the small bin size. The reason I chose a small bin size was to get a more accurate emittance angle. If we run the pipeline again, using a bigger bin size (0.3, 0.3) we get:

  

  which is way clearer at explaining which regions (bins) have more hits. Of course, this drops the accuracy of the predicted parameters:

  

  This happens because phi is actually 1.016, hence it falls in the bin that covers points in the range 0.9 - 1.2. We can easily deduct now that decreasing the bin size might give more accurate results (in some cases, like this one).

  Now regarding the many-particle files, the Hough transform looks like a cluster of colored rubber bands:

  

  The heatmap is this:

  

  The "crowded" region is along the central horizontal line, which makes total sense according to the tracks in the Hough Space.

• Now let's see the results when we simplify sin(x - phi) to x - phi:

  For single-particle files:

  

  

  

  For many particle files:

  

  

[AndrewSpano]

• Turns out the heatmap didn't look bad because of the colors. The problem was that I wasn't "zooming in" in the RoI. I refactored the script so that now we can choose which range of values for q/p_T we want to zoom in: The new heatmaps are these:

  • using sin() in the formula:

  • single-particle

    

    

  • many-particles

    

    

  • simplifying sin(x) to (x) in the formula:

  • single-particle

    

    

  • many-particles

    

    

• Regarding the efficiency of the 2 transformations pet p_T values: Initially I thought that I had to group the tracks per q/p_T values. This managed to waste quite some time as I wasn't getting very good results. Luckily, I later realized that I had to group the tracks per p_T values intead:

  • For the formula using the sin() function:

  

  • For the simplified formula (without sin), I tried the same hyper-parameters, and got these results:

  

  Makes sense that the same parameters (bin-size, least-number-of-hits-per-bin) won't give similar results, so I tried fine-tuning those parameters by hand a bit:

  

  Still the results are nowhere near as good as with the sin().

  One key takeaway though is that the sin() version had many duplicates. They can be avoided by finding a good bin size (which I didn't invest time in doing it).

• Regarding the heatmap in the r-z plane, the tracks in the Hough space and the heatmap look like this:

  

  

• Regarding the combination of the transformations, I thought of taking an intersection of sets: That is, pick the tracks that appear in both transformation along with the same tracks (again, in both transformations). This is ongoing.

[AndrewSpano]

• Regarding fixing the transverse momentum issue:

  First I did the count plots for 1 event:

  

  Then I computed the counts for all the events (separately) and aggregated the results to get these counts per p_T and eta:

  

  

  The efficiency looks like this:

  

  Still, the results don't look as expected. I will have to do some more analysis to find out what's wrong, though I have gone through the code for hours over and over again.

• Regarding the combination of the two approaches, I have finished it for one-particle events. I will complete it tomorrow for many-particle events and then I will push the code. For the evaluation, I will make a plot: efficiency vs x-y vs r-z vs combination, to see if it actually improves the results.

• Also, I finally booked a ticket to Switzerland. I will be flying August 3 to Zurich. Then I will take a train to Geneva. I hope the city and the views are at least as entertaining as this one :D

  

[AndrewSpano]

Finally found the problem with the high pT tracks. I wasn't considering the whole range of values for the approximated transformation, which is (-π, π). I was considering only (0, π), which is a valid assumption for the precise transformation since the sinus() is a periodic and even function. Here are the results I get:

• 1 data file, 25 tracks:

  

  

  

• all the data files:

  

  

  

  

  

They get closer as p_T gets higher.

[noemina]

This is great to see!!! All makes sense now :)

Give the results you get we can continue using the approximated transformation for this project.