Hello guys! I'm back again. This time, we are tasked to play around with a method for computing the area of closed curves with defined edges in images. This exercise will make way for us to understand the capability and flexibility of image processing in useful applications like measuring the area of a lot defined in Google map. Applying the computational method used in this activity, for example, in evaluating the extent of a catastrophic event such as forest fire using its aerial images; we can save a lot of resources as compared to actual evaluation of the place.
FAQ
What method can be used in evaluating the area of a figure with defined edge?
- The Green's Theorem is the most utilized method in finding the area in this activity.
What is Green's Theorem?
- Green's Theorem is a mathematical construct which relates the line integral of the contour of a figure to the area of the figure enclosed by that specific contour. The Green's Theorem is mathematically represented by
(1)How can we use Green's Theorem to derive the area of a closed figure with defined edge?
- From the Green's Theorem, the area of a continuous curve can be evaluated by the following equation
(2) In computational approach, this formula will be helpful after applying discretization method. The discretized form of (2) is given by
(3)This equation can now be used to create a programming algorithm to derive the area of a closed figure with defined edge.
Are the numerical calculations of the area accurate using the Green's Theorem method?
Applying the method to a circle wherein we know the actual analytic area, we can estimate the error produced by the method.
First Test:
- Circle: r = 70 px
- Numerically computed area = 15, 071 px^2
- Analytically computed area = 15, 394 px^2
- Error = 2.10%
Second Test:
- Circle: r = 140 px
- Numerically computed area = 60, 865 px^2
- Analytically computed area = 61, 575 px^2
- Error = 1.15%
Third Test:
- Circle: r = 280 px
- Numerically computed area = 244, 905 px^2
- Analytically computed area = 246, 301 px^2
- Error = 0.57%
From the three tests, we can observe that as the pixel dimension of the object whose area is to be calculated is larger, then the error of the computed area using Green's Theorem decreases relative to its analytic area.
How can you apply the computation of area using Green's Theorem in daily life?
One of its vast applications include the determination of area of a certain region in a map. As an example, I chose to measure the area of the College of Science Administration Building using this method.
Map Calculation (College of Science Area):
Computed Area in pixels = 20852 px^2
Scale:
0.934579439 ft/px
0.285714285 m/px
Using the scaling factor taken from the scale accompanying the image of the map, we can approximate the area of the CS by simply multiplying the scaling factor with the calculated value returned by the Green's method.
Approximated Land Area of CS = 5, 957.71 m^2
Actual Land Area = ?
Image of the CS admin bulding taken from Google Maps!..
Summing up everything discussed in this How-To++, we can say that the Green's method offers a convenient way of calculating the area of a region shown in a map. This is quite a resourceful and less rigorous approach to measuring area of regions with defined edges.
It must also be noted that the larger the size of the region in pixels the more accurate the result of the area measurement. Thus, the converse of the event must be taken with high caution. Take into consideration that if you dealt with a region with a small pixel area, the resulting measurement has a high margin of error which may lead to wrong conclusions.
In finality, I want to consider treating myself a 12/10 for this activity for the effort I expended in putting things together and tried to be creative (A simple FAQ outline) in presenting this blog... :)
Source:
Activity Sheet for Activity 4 - Dr. Maricor Soriano
Google Maps
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