Activity 4: Area Estimation for Images with Defined Edges

After finally getting everything fixed and prepped for the activity proper (Did you read my previous post that served as this blog’s prequel? Well if you didn’t… YOU SHOULD by clicking here), I can now finally discuss to you the actual activity:

Have you ever wanted to act all super-spy that’ll make you feel oh so cool? Ever wondered how it felt like to be part of the CSI team? Well I have awesome news for you! The activity for today bring stalking to a whole new level. By the time you finished reading my blog, you’ll be able to have a faint idea of the area of your crush/nemesis/best friend/boss’ house that you can put into good use (such as BLACKMAIL!!! hahaha).

Of course I’m kidding 🙂

With an image of the object you wish to determine its area, proper post-processing is required. What do I mean by that? Well, we hope to have a clean image wherein the object is defined and as much as possible the only one existing in your whole image. The ability to separate the background from the object is also important so the use of the im2bw() function was used before the calculation of the area was done.

There were two methods that was discussed in determining the area of a properly post-processed image. One is through the use of simple pixel counting. Since area is defined as the space occupied by your two-dimensional object, simply counting the number of pixels of your object can already be the rough estimate of its area. This is implemented in Scilab through the simple code as seen below:

But then since I’m an Applied Physics student, challenge is my middle name. The other method is more mathematical and is by far complicated. Fret not though as it only LOOKS complicated:

The equation seen above is the base equation for the Green’s Theorem. It simply indicates that the area of a certain contour (left side of the equation) is equal to the integral of the points on the border of your contour upon further simplication.  Converting our integral into its discrete form (the integral is simply just the summation!), we have a friendlier equation to make use of.  

What’s essential for this theorem to work is the knowledge of the set of points  lying at the boundaries of your contour moving at a certain direction (counterclockwise or clockwise).

Now this is where all hell breaks lose. 

Without the SIP toolbox installed in your Scilab, you’d have to be creative and smart enough to figure out a way on how to, not only find the edge of your object, get the list of points on the edges of your objects that is sorted in away that it is tracing the object around a specific direction. (again, I encourage you  to read my journey before this whole activity by clicking here).

Thanks to the follow() function available with the SIP toolbox, it immediately outputs the list of your sorted coordinates of the edge of your object. But then since we also need the variables Yi+1 and Xi+1.. we have to figure out a way to shift the list that we have from the follow function by moving a single step. I managed to do this by taking advantage of slicing off portions of the list. First I had to place the first element of my original list into a new list ([x2]). The next list I made was the rest of the elements left which is done by the function:

[x1]=x(2:length(x))

How does this work? The 2 signifies the point (or the element) the start while the length(x) signifies the end (for our case the last element) by which you want to slice off your list.  Since we want to shift our list a unit forward, we simply combine x1 and x2.

The rest that we have to do in order to get the area is simply implementing the discrete equation of Green’s theorem. Check out my finished code below:

Now off to the fun part – finding the area of different shapes!

Through the use of my code from Activity 2 in creating square apertures, I’ve use three different sized squares and determined the white spaces in between through the pixel count as well as the Green’s theorem. Below is the tabulated values of the areas calculated as well as the actual area taken through geometry:

What really surprised me from the results was not the zero percent error from the Green’s function but rather the high percent error of the pixel counting method!  I mean, the idea of simply counting the pixels inside your shape was basically solving for the area as well… I tried looking to determine where I want wrong but sadly I did not find any lead. Since my squares were generated from Scilab, there’s not blurred edges to be worried about.

As for the Green’s theorem garnering zero percent error, this could have been brought by the straightforwardness of the shape we’ve dealt with. If we have curved shapes instead… like for example circles:

Should we expect greater percent errors for the Green’s theorem?

Surprisingly, the case of the circle shifts it the other way around. The area from the pixel count has smaller percent error (less than 1 for all cases) while the area from the Green’s theorem had greater than one percent errors. The possible reason as to why percent errors exist for the Green’s theorem is because of the generation of the circles. I have used my algorithm from Activity 2 in producing circular apertures and I managed to notice that the smaller the size of my circle is, the less it resembles a perfect circle. Remember that pixels after all made up of small-sized squares.

As a last test to see if my code is functional enough to be used for some real stalking business, I tried to find the area of a triangle:

 The area from the Green’s theorem gave a value of 13321 pixel squared. Taking note that the triangle (which I produced from Photoshop and saved it as a .BMP so as to avoid blurred edges) has a base width of 177 pixels and a height of 153. Using the known equation of a triangle:

Area = 0.5*(base*height)

The actual area is 13540.5 giving us a percent error of 1.62%!

With a more or less low percent error for my algorithm (which could’ve come up from the fact that the shapes are not actually made of straight smooth lines but rather small squares of pixels), we are now ready for the big thing 😀

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The Beijing National Stadium taken from http://www.bestourism.com/items/di/1077?title=The-Beijing-National-Stadium

With China’s recent bird’s nest stadium makings its way to the headlines of almost every newspaper back then… I’ve chosen the simple yet exquisite mega-structure as my point of interest. Using Google’s one of many oh-so-useful applications they’ve made available for their fans: Google Maps, I’ve located Beijing’s National Stadium and zoomed it in enough to fit my browser.

Taking note that the output of the program is in fact in terms of pixels, we have to make use of the things we learned from Activity 1 in converting our units: pixels to meters. It is then essential to make use of the scales in properly converting our units. Thankfully, Google has thought of everything as they have made a scale available in google maps as seen inside the pink border of the image above.

What we need to do is measure the number of pixels for the indicated scale above:

108 pixels is to 100 m

Making use of dimensional analysis, we simply have to multiply our output area to 100m/108pixels for two times so that we’d end up with meter squared instead. Hence , the conversion factor to be multiplied is 0.926.

I’ve post-processed the image above through Photoshop by setting Google maps in map mode (the image above actually came from Google maps in Satellite mode). Converting it into grayscale then into a bitmap mode image and finally deleting the unnecessary pixels… I was left with the image below:

Unbelieveable right? The stadium looks like a perfect shape! Now that’s what a call perfect engineering.

Calling the image in my program, I’ve determined the area to be (Green’s theorem) 87189 meter squared.

 For comparison, I tried my best to find the actual area of the bird’s nest. However, one has to be careful in simply finding a value for an area as it does not necessarily mean you have measured the exact same thing. At first I was disappointed when I got the area to be around 258000 square meters! I mean, if that was truly the case… my program is after all, very very wrong!  However, this 258 000 meter squared value apparently already included the garden as well as the other land nearby the stadium. The one that I measured was actually ONLY the building itself!

Managing to find the estimate of the building to have an area of only 80,000 square meters, I’ve garnered a percent error of 7.64%! UNBELIEVABLE!

Furthermore, my original plan was to find the area taken up by a certain gigantic lazy pink bunny lying around the mountains of Italy. Don’t believe me? See for yourself by the images below:

Despite the awesomeness of the pink knitted bunny lazing around, I was unable to find a source indicating the total area it has covered. Nevertheless, I put my program into use and determined its area to be 8480 meters. Now that’s ONE BIG BUNNY!

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To finish this blog post, I’d like to give myself a grade of 10/10. Although deep inside I’d like to give myself a higher grade (after going through the trouble of having to install Ubuntu and Scilab and Imagemagick and Animal and SIP and VMware and trying my best to make my own code in order to replace the follow code but failed terribly)… I wasn’t able to explore the topic that much as I have hoped to. So in the end, I was unable to give out something new to the topic…. Oh well… atleast I finally can brag to my friends that I know how Ubuntu/Lunix systems work now 😀 That’s a good enough reward I guess.