AP186 Activity 10 Blog Report - Enhancement by Histogram Manipulation

In this activity, we will be using histogram manipulation to enhance dark-looking images.

Take this image, for example:

Fig. 1. Image to be enhanced in this activity.

The grayscale of the above image was obtained using I = rgb2gray(I_rgb). The result is:

Fig 2. Grayscale equivalent of the image in Fig. 1.

The grayscale histogram of the image is then obtainable using plot(imhist(I)), which gives:

Fig. 3. Grayscale histogram of the image.

The maximum and minimum grayscale values were obtained to be 245 (with 1 pixel having that value) and 0 (with 15 pixels having that value). Meanwhile, the maximum and minimum number of pixels having a particular grayscale value between 0 and 245 is 5257 pixels and 1 pixel. The obtained cumulative distribution function (CDF) using plot(y) where y = cumsum(nhist,1)) and nhist = imhist(I) is:

Fig. 4. Cumulative distribution function of the image.

The maximum cumulative number of pixels in Fig. 4 is 76800, and this can be used for the desired CDF's maximum as well.

In order to create the desired CDF, I created an array of integers from 0 to 76800 (max cumulative number of pixels) to act as the x-axis of the desired CDF. Then, to act as the y-axis, I divided each element of the x-axis array by 76800 then multiplied it by 245 (max grayscale value) to get the y-axis array. The desired CDF when plotting the x-axis array and the y-axis array then looks like:

Fig. 5. Desired cumulative distribution function of the image.

The enhanced image is then obtained by plugging in the grayscale image matrix I (plus 1, since the indices in Scilab start at 1) to the original CDF, and plugging in the result of that to the desired CDF. The resulting array is then reshaped into a matrix with the dimensions of the original image, giving the enhanced image. In code form, this is:

I2 = matrix(desired_y(y(I+1))/255,240,320);

where desired_y is the y-axis array of the desired CDF, y is the original CDF, I is the grayscale image matrix, and 240 by 320 is the original image pixel dimensions. The values are divided by 255 to get a range of 0 to 1 in the new image.

The resulting enhanced image, known as a histogram equalized image, looks like:

Fig. 6. Resulting histogram equalized image for the grayscale of the original image.

Upon saving the image in Fig. 6, I immediately read it using imread() to get the histogram of the image using imhist() as well as the CDF using cumsum(). These ended up looking like:

Fig. 7. Histogram of the histogram equalized image in Fig. 6.

The above looks like a more equalized histogram apart from the darker parts, which may have been a result of the 0 to 1 fractional values settling for integers from 0 to 255. Looking at the histogram with just 10 bins, the plot looks very equalized. The CDF of the histogram equalized image is:

Fig. 8. Cumulative distribution function of the histogram equalized image in Fig. 6.

The above CDF seems to imitate the desired CDF, which is expected.

For a hopefully better enhanced image, we have to consider that the human eye does not have a linear response. Thus, we can try using a sigmoid function as the desired CDF.

To create a sigmoid function, I used the second answer in this Stack Overflow question as reference:
http://stackoverflow.com/questions/3741063/scilab-x-6-6-y-1-1e-x-why-it-doesnt-work

Since I need a desired CDF that has an x-axis array from 0 to 76800, similar to Fig. 5, and a y-axis array from 0 to 245 (the desired CDF has to return a grayscale value after histogram manipulation enhancement), the correct modification to the sigmoid function from the source above is:

desired_y = -6:(12/76799):6; desired_x = 76800*ones(desired_y)./(1+%e.^-desired_y);
desired_y2 = 245*(desired_y+6)/12;
plot(desired_x,desired_y2); //desired CDF

The desired CDF looks like any old sigmoid function but with the x-axis and y-axis swapped:

Fig. 9. Desired cumulative distribution function following a sigmoid.

I think this is the correct form, since the fourth page of the Activity 10 manual gives a sample desired CDF with a regular looking sigmoid, and the x-axis there has a range of 0 to 255. Thus, the sigmoid must have the x-axis and the y-axis flipped if the 0 to 255 range should actually be on the y-axis in our desired CDF.

The enhanced image using the sigmoid CDF turned out to be:

Fig. 10. Enhanced image for the grayscale of the original image.

Using this technique for a colored image, we reuse some of the code from Activity 7:

I = double(I);
R = I(:,:,1); G = I(:,:,2); B = I(:,:,3);
Int = R + G + B;

We then perform histogram manipulation on the "Int" matrix. I checked the maximum value in the "Int" matrix and obtained 731. I then checked the minimum value, and it is 0, as expected (those are in the grayscale, after all). Thus, I replaced the 0 to 245 range from the previous code with the 0 to 731 range more applicable to the "Int" matrix.

In getting the histogram of the "Int" matrix, to be used for obtaining the original CDF, imhist() cannot be used. Thus, we replace the nhist = imhist(I) command with the [nhist,nind] = histc(732,Int) command. And then, in order to use cumsum(), we turn nhist into 732 rows by 1 column instead of 1 row by 732 columns.

[nhist,nind] = histc(732,Int,normalization=%f);
nhist = matrix(nhist,732,1);
y = cumsum(nhist,1);

Anyway, after making those small changes, the code with the sigmoid function works for the "Int" matrix. I named the resulting matrix as the "Int2" matrix. The new RGB channels are then calculated by first getting the r, g, and b as follows:

Int(find(Int==0))=100000; //avoids dividing by zero
r = R./ Int; g = G./Int; b = B./Int

Then, the new RGB channels are calculated using the "Int2" matrix by multiplying the matrix to the obtained r, g, and b:

R2 = r.*Int2; G2 = g.*Int2; B2 = b.*Int2;
I_enhanced(:,:,1) = R2; I_enhanced(:,:,2) = G2; I_enhanced(:,:,3) = B2;

I then used 255 to normalize all of the values of the multimatrix "I_enhanced" corresponding to the colored enhanced version of the original image. The resulting image is as follows:

Fig. 11. Enhanced colored image using histogram manipulation.

Compared with Fig. 1, Fig. 11 has much better colors on the trees, the background buildings, and even the details of the clothes of some of the spectators. In common with Fig. 6 and Fig. 10, the enhanced images have these weird horizontal scan lines on the very dark parts of the picture. I think this can be attributed to the treatment of my cellphone to those parts when it saves the picture as a relatively low-quality .jpg image.

Although I did this activity in a hurry while still sick, I enjoyed it for how it produced cool resulting images.

Self-Evaluation: 9/10