Activity 4 - Area Estimation for Images with Defined Edges

In this activity, what we need to do is to find the area of a certain region of concern in an image by (1) counting the pixels enclosed by the said region and (2) by  using Green's theorem whose discrete form is given by:

1. Pixel counting 

In pixel counting, we need our image to be binarized such that the area of interest is white and the background is black. Then to find the area, we just count the number of pixels which are white (or 1). 

2. Green's Theorem

To able to use Green's theorem we first need to learn how to take the points which consist the contour of the said region. This is fairly easy to do if your image has defined bounds by using the edge() command in SciLab and finding the indices of the array elements which are white. But another requirement of Green's theorem is for the contour to be taken either in the clockwise or counterclockwise direction. This is where the challenge lies. One technique in sorting the points is by taking the angles between the x-axis and the line which connects the points to the center of the image which acts as the origin. 

3. Code Snippet
The code snippet below shows the implementation of the technique in SciLab.

//Load image as array

I = imread("C:\Users\Tin\Desktop\AP_186\Act4-Area Estimation\square_250.bmp");

//Actual areas

area1 = 250*250

area2 = %pi*125^2

//Estimate area by pixelcount

pixels = size(find(I > 0));

pixelcount = pixels(2);

err_p = (pixelcount - area1)/area1;

//For Green's theorem

//Isolate edge

E = edge(I, 'canny',0)

E = bool2s(E);

//find points on edge

[i,j]= find(E);

ind = [i;j]';

//plot(ind(:,1),ind(:,2),"bo")

s = size(I); // size of image

n = size(ind); // size of index list

//set origin at center of image 

c_x = s(1)/2;

c_y = s(2)/2;

//Calculate angles

t = n(1);//for limit of for loop

angles = zeros(n(1));

for i =1:t

    x = -(ind(i,1)-c_x);

    y = ind(i,2)-c_y;

    r = (x^2 + y^2)^0.5

    if x==0 & y>=0 then

        angles(i) = 90;

    elseif x==0 & y<0 then

        angles(i) = 270;

    elseif x<0 & y<=0 then

        angles(i) = atan(y/x)*180/%pi + 180;

    elseif x>0 & y<=0 then

        angles(i) = 360 + atan(y/x)*180/%pi;

    elseif x>0 & y>0 then

        angles(i) = atan(y/x)*180/%pi;

    elseif x<0 & y>0 then

        angles(i) = atan(y/x)*180/%pi + 180;

    end

end 

//save angles (x,y) pairs in another array then sort 

B = zeros(n(1),3);

B(:,2:3) = ind;

B(:,1) = angles;

C = lex_sort(B);

//Estimate area using Green's theorem

sum_ = 0

for j = 1:t-1

    sum_ = sum_ + (C(j,3)*C(j+1,2)- C(j,2)*C(j+1,3));

end

A = 0.5*sum_;

err_g = (A-area1)/area1;

4. Results for black and white regular geometric shapes

I used both techniques for different black and white images with known area. Note that we consider that the actual areas of the regions we are interested in are exactly the same as the regions in white. That is, we consider the edges to be completely smooth instead of being discretized. This way, we will be able to include the errors that may result when we delineate our regions of interest in a real-world scene.  The results are as follows (in square pixels):

250 pixel by 250 pixel square

Actual Area: 62500

Green's theorem: 62559.5 (0.095% error)

Pixel counting: 62500 (0% error)

100 pixel by 100 pixel square

Actual Area: 10000

Green's theorem: 10079  (0.97% error)
Pixel counting: 10 000 ( 0% error) 

Circle with radius of 125 pixels

Actual Area: 49087.4

Green's theorem: 48995.5 (0.19% error)

Pixel counting: 48936 (0.31% error)

Circle with radius of 50 pixels

Actual Area:7854

Green's theorem: 7917  (0.8% error)

Pixel counting: 7820 (0.4% error)

Triangle with base = 401pixels and height = 264pixels

Actual Area: 52932 

Green's theorem:  53049 (0.22% error)

Pixel counting: 53167 (0.44% error)




From the estimates obtained using Green's method and pixel counting, the techniques may be considered accurate, yielding percent errors which are less than 1%. Let us now apply the technique to a real-world object or region. 


5. real-world application
The following figure shows the City Commercial Center (C3) in our city (Pagadian City) in map view and in its binarized form made using Gimpv.2.8. 

Using the code above, the following edge was detected.

The contour is obtained using the code above is shown below:

The theoretical area of the region in sq. pixels was determined by manually measuring each side and taking areas of the rectangles that make up the entire region. The areas calculated in square pixels are as follows :

Theoretical: 66830 sq. pixels

Green's theorem : 67622 sq. pixels (1.14% error)

Pixel counting: 67595 sq. pixels (1.18% error)

By counting the number of pixels which correspond to 20m in the scale bar, I obtained a scaling factor of  3.4 pixels/m which I can use to convert my computed area in sq. meters.


Green's theorem: 5849 sq. m. 
Pixel counting: 5847 sq. m 


These values for the area are good estimates with percent errors less than 2%. 


I would like to thank Ms. Maria Isabel Saludares for giving me an idea on the use of angles when my first idea did not work. Also, I thank Mr. Gino Borja and Ms. Eloisa Ventura for helpful discussions on the commands and syntax in SciLab.


Finally, I give my self a grade of 10/10 for being able to take the estimates of areas using Green's theorem and pixel counting. I have gained much skill while doing this activity.


Reference:
M. Soriano, AP 186 Activity 4 - Area estimation for images with defined edges instruction sheet, 2012