This activity is designed to experience how measurement of the actual dimension of the object is being done given only the image of the object. By knowing the relationship between the pixel location in the image and the physical variable, one can already tell the object's actual length. This method is not easy and erroneous but I guess it is the simplest technique for a rough estimation.
I got the graph I used from a paper entitled Electrolytic Determination and Separations with the Use of a Rotating Anode from The Journal of American Chemical Society Volume XXIX of 1907 (January 25 1907). Thanks to the author of the paper, Julia Langness, for this graph.
Thanks to Mr. Luis C. Buno III for helping me in rotating the image using MATLAB and answering some of my questions. The image had to be rotated since the scanned image is tilted.
Hand-drawn graphs are indeed difficult to reconstruct. The gridlines of the graph are unevenly thick and not aligned, thus, unevenly spaced. Due to this problem, the determination of the the equivalent pixel location in the image of a point in the graph is very much prone to error. The table below shows the pixel location of the ticks in the X-axis (Minutes) and the Y-axis (Grams). It can be seen that the difference between two consecutive ticks in the X-axis greatly varies ranging from 103 to 115, a very large deviation by 12. The same deviation in the Y-axis can be observed. To minimize the error due to the large deviation in reconstructing the gridlines of the graph, the mean of the differences was used.
By the way, the pixel location of the ticks in both axes was determined using the Paint program and then tabulated using Microsoft Excel. Instructions for determining the pixel location can be found in the manual for this activity.
Equations relating the pixel location to the physical variable in both axes were established. The proportionality factor was obtained by getting the ratio of the increment per division to the mean difference. It must be noted that the origin of the graph does not correspond to the (0,0) pixel location of the image. To account for this offset, the (X,Y) pixel location of the origin was subtracted from/to the pixel location of the points in the graph. Hence, the equations are the following:
X-axis: minutes = 2*(1355-x)/113
Y-axis: grams = 0.05*(y-67)/109
Several points to represent the best fit line, including the data points, were selected to be able to create a smooth reconstructed curve. The pixel location of each of these points were determined. The reconstruction of the graph was done by converting the pixel locations back into their equivalent physical values using the generated equations.
Note that the best fit line is also unevenly thick, so this introduces additional error.
The fun part in this activity was determining the pixel locations in the image of the points in the graph. Wait, was it really fun? hhhhmmmm... Yeah, I had fun especially because this is the first activity. But, it was also tiresome.. :( and painful to my eyes since I had to stare for so long at the computer. The frustrating part, on the other hand, was the next thing to be done. That is, overlaying the image of the graph on the reconstructed graph It wasn't supposed to be frustrating but I wasn't able to find out immediately how to overlay the image. And my seatmate was done already. Time was past 11am. Our class is until 11:30am only. So, I became a 'lil bit' fidgety. Thanks to the power of Google, I figured it out how to overlay an image as a background on a chart.
To overlay an image in a chart using Excel, the following must be done:
Go to Format Plot Area. Probably by pointing the mouse to the Plot Area and then right click.
Then, in the Fill tab, choose the option Picture or texture fill. Insert the image you want to overlay.
Below is the figure of the reconstructed graph with the original graph overlaid on it. The gridlines, as well as the reconstructed curve, were made thicker to compensate the errors (uneven thickness of the lines in the original graph) that were introduced above. It must be noted that the original graph has to be cropped properly so that it fits, more or less, the reconstructed graph.
Looking at the figure above, it can be seen that the graph is not perfectly reconstructed. Some of the reconstructed and original gridlines do not coincide. Same also with the curve and the data points. The reasons for this are already explained above. It is because the gridlines are unevenly thick and unevenly spaced.
The author of this paper is probably better in drawing horizontal lines than vertical lines. :) As you can see above, more vertical gridlines are inclined.
Although I was not able to perfectly reconstruct the graph, which isn't my fault (hehehe), I think I still deserve a grade of 9. :) Why? I didn't finish it on time but I believe I did this activity correctly as I have explained what I did above.
Guys, please comment!!! :) Tell me what you think about what I did and what I said above.
I got the graph I used from a paper entitled Electrolytic Determination and Separations with the Use of a Rotating Anode from The Journal of American Chemical Society Volume XXIX of 1907 (January 25 1907). Thanks to the author of the paper, Julia Langness, for this graph.
Thanks to Mr. Luis C. Buno III for helping me in rotating the image using MATLAB and answering some of my questions. The image had to be rotated since the scanned image is tilted.
Hand-drawn graphs are indeed difficult to reconstruct. The gridlines of the graph are unevenly thick and not aligned, thus, unevenly spaced. Due to this problem, the determination of the the equivalent pixel location in the image of a point in the graph is very much prone to error. The table below shows the pixel location of the ticks in the X-axis (Minutes) and the Y-axis (Grams). It can be seen that the difference between two consecutive ticks in the X-axis greatly varies ranging from 103 to 115, a very large deviation by 12. The same deviation in the Y-axis can be observed. To minimize the error due to the large deviation in reconstructing the gridlines of the graph, the mean of the differences was used.
By the way, the pixel location of the ticks in both axes was determined using the Paint program and then tabulated using Microsoft Excel. Instructions for determining the pixel location can be found in the manual for this activity.
Equations relating the pixel location to the physical variable in both axes were established. The proportionality factor was obtained by getting the ratio of the increment per division to the mean difference. It must be noted that the origin of the graph does not correspond to the (0,0) pixel location of the image. To account for this offset, the (X,Y) pixel location of the origin was subtracted from/to the pixel location of the points in the graph. Hence, the equations are the following:X-axis: minutes = 2*(1355-x)/113
Y-axis: grams = 0.05*(y-67)/109
Several points to represent the best fit line, including the data points, were selected to be able to create a smooth reconstructed curve. The pixel location of each of these points were determined. The reconstruction of the graph was done by converting the pixel locations back into their equivalent physical values using the generated equations.
Note that the best fit line is also unevenly thick, so this introduces additional error.
The fun part in this activity was determining the pixel locations in the image of the points in the graph. Wait, was it really fun? hhhhmmmm... Yeah, I had fun especially because this is the first activity. But, it was also tiresome.. :( and painful to my eyes since I had to stare for so long at the computer. The frustrating part, on the other hand, was the next thing to be done. That is, overlaying the image of the graph on the reconstructed graph It wasn't supposed to be frustrating but I wasn't able to find out immediately how to overlay the image. And my seatmate was done already. Time was past 11am. Our class is until 11:30am only. So, I became a 'lil bit' fidgety. Thanks to the power of Google, I figured it out how to overlay an image as a background on a chart.
To overlay an image in a chart using Excel, the following must be done:
Go to Format Plot Area. Probably by pointing the mouse to the Plot Area and then right click.
Then, in the Fill tab, choose the option Picture or texture fill. Insert the image you want to overlay.
Below is the figure of the reconstructed graph with the original graph overlaid on it. The gridlines, as well as the reconstructed curve, were made thicker to compensate the errors (uneven thickness of the lines in the original graph) that were introduced above. It must be noted that the original graph has to be cropped properly so that it fits, more or less, the reconstructed graph.
Looking at the figure above, it can be seen that the graph is not perfectly reconstructed. Some of the reconstructed and original gridlines do not coincide. Same also with the curve and the data points. The reasons for this are already explained above. It is because the gridlines are unevenly thick and unevenly spaced.The author of this paper is probably better in drawing horizontal lines than vertical lines. :) As you can see above, more vertical gridlines are inclined.
Although I was not able to perfectly reconstruct the graph, which isn't my fault (hehehe), I think I still deserve a grade of 9. :) Why? I didn't finish it on time but I believe I did this activity correctly as I have explained what I did above.
Guys, please comment!!! :) Tell me what you think about what I did and what I said above.
I disagree, your score is 10, not 9. Great work! That's really the problem with hand-drawn graphs. They're not accurate.
ReplyDeleteBy the way, I like your blog name.