Digital Scanning

You know how it would really be convenient to know the values of points in a plot that you want to recreate? When sometimes your professor or adviser asks you to recreate a plot or graph from a book or a research paper that somehow lacks the details like initial conditions, it would really be neat if you are able to know the values of points in a plot without asking the original author for them.


The first thing you can do is to have the plot scanned. Any plot would do, whether it's digitally made or hand-drawn. For this one, I used a hand-drawn plot back from 80s when computers weren't that popular yet. You have to make sure that the plot is scanned uniformly, with the paper lying totally flat on the scanbed. Any warps or distortion on the scanned plot will make things more complicated. Let's do the basics first.

The hand-drawn plot is now converted to a digital image, with points which we can measure in pixels. Microsoft Paint or any other image-editing software can give you the pixel values of any point on the graph. These points would of course correspond to a real physical value as indicated by the labels on the axes of the plot. The next problem is to figure out how the pixels are related to the real physical values. Once we've found this, we will be able to convert the pixel values of any point on the plot to its real physical value.


The digitally scanned plot is the similar to the hand-drawn plot, magnified or minified only by a factor. Assuming that warps in the scanning are negligible, then we can say that the real values are related to the pixel values linearly. Thus we can use the linear equation y = mx + b, x being the pixel value, y the real physical value, m the slope or ratio of pixel to real value, and b being the offset. Using points from the axis with known real physical values, m and b can be easily calculated. That's it! Using this conversion, one need only to pick a point, take note of the pixel values and convert.

Oh but wait. I accidentally picked a bit complicated plot. Notice that the x-axis of this plot is not uniformly scaled  (boxed in red). To resolve this, I had to cut the graph into four partitions, with the tick marks on the x-axis as the boundaries. I chose not to include the parts of the plot after x = 200 since there's no known upper boundary for it. It's not that hard really. It's as easy as what I described above, only that for each partition, there should be a different conversion for the horizontal pixel values.

The points I chose are in red. I picked points that include the values of the tick marks so I can easily check if my conversions are wrong. The real values of the points I picked were:

x	y
10.00	2.86
12.09	3.07
14.96	3.32
17.52	3.60
20.00	3.87
23.88	4.12
28.82	4.42
33.94	4.75
38.88	5.09
44.35	5.48
50.00	5.92
56.20	6.13
67.05	6.56
77.91	7.00
88.76	7.48
100.00	7.99
122.66	8.52
144.53	9.05
166.41	9.58
185.94	10.06
200.00	10.45
10.00	2.86
12.09	3.07
14.96	3.32
17.52	3.60
20.00	3.87
23.88	4.12
28.82	4.42
33.94	4.75
38.88	5.09
44.35	5.48
50.00	5.92
50.00	5.92
56.20	6.13
67.05	6.56
77.91	7.00
88.76	7.48
100.00	7.99
122.66	8.52
144.53	9.05
166.41	9.58
185.94	10.06
200.00	10.45

To better visualize how awesome that is, I plotted these in Excel for each partition, using the scanned image of the hand-drawn plot as the background.

 

As you can very well see, the recreated points in first and second partitions did not coincide that well with the hand-drawn plot. I guess this is somehow due to my lack of expertise then in pinpointing exact points in the bold lines of the original plot. The tick marks on 2 and 4 on the y-axis are also a bit off. Perhaps this is due to some small deviations in the intervals between tick marks. If I also did partitions on the y-axis, this error may be minimized.


That was kinda fun. :D


I'm giving myself a perfect score, yay! 10/10! :D
But I'm not really sure of 10, I left my rubric at school. Anyway, I'm still giving myself a 10 because I was able to do everything pretty well. The plots are almost the same, and I'm pleased about my explanation. There's a 10. Hope you understand everything. ;)






Hand-drawn figure taken from:


Y.Afek, G. Berlad, A. Dar, G. Eillam. The Collective Tube Model for High Energy Particle-Nucleus and Nucleus-Nucleus Collisions. Multiparticle Production on Nuclei At Very High Energies - Proceedings of the Topical Meeting in Trieste, 10-15 June 1976. International Atomic Energy Agency. Austria, July 1977.