This research paper contains computer codes for different problems and explains how each of the problems can be solved using basic mathematical explanations as well as it covers the histogram equalization and how it affects the original image, the implementation of high-boost filtering using the equation of the high-boost and the represented mask for it, the implementation of Laplacian for image sharpening using a different mask and how to enhance the image using the median filtering technique. The tool used in this paper is MATLAB desktop version “MATLAB R2022b”.
Explanation of each case with mathematical calculation step by step with an example of each algorithm.
Take the input image and find the histogram by using the probability of each pixel in the image: Original image
| 0 | 2 | 4 | 7 |
|---|---|---|---|
| 5 | 3 | 6 | 4 |
| 1 | 7 | 5 | 7 |
| 7 | 2 | 5 | 4 |
Probability
| p(0) | 0.0625 |
|---|---|
| p(1) | 0.0625 |
| p(2) | 0.125 |
| p(3) | 0.0625 |
| p(4) | 0.1875 |
| p(5) | 0.1875 |
| p(6) | 0.0625 |
| p(7) | 0.25 |
Draw the histogram of the image by taking the number of the pixel as Y-axis and the probability as X- the axis:
Swap the pixels based on the cumulative probability mass function:
| s0 | 0.4375 | 0 |
|---|---|---|
| s1 | 0.875 | 1 |
| s2 | 1.75 | 2 |
| s3 | 2.1875 | 3 |
| s4 | 3.5 | 4 |
| s5 | 4.8125 | 5 |
| s6 | 5.25 | 6 |
| s7 | 7 | 7 |
Apply the New values of pixels to the new image:
Equalized image
| 0 | 2 | 4 | 7 |
|---|---|---|---|
| 5 | 2 | 5 | 4 |
| 1 | 7 | 5 | 7 |
| 7 | 2 | 5 | 4 |
Get the probability of the Equalized image to draw the histogram-equalized:
Probability
| p(0) | 0.0625 |
|---|---|
| p(1) | 0.0625 |
| p(2) | 0.125 |
| p(3) | 0 |
| p(4) | 0.1875 |
| p(5) | 0.25 |
| p(6) | 0 |
| p(7) | 0.25 |
g(x,y)=f(x,y)+k*[f(x,y)-f ̅(x,y)]
Apply the blur using the mask: Input Image f(x,y)
| 0 | 2 | 4 | 7 | 1 |
|---|---|---|---|---|
| 5 | 3 | 6 | 4 | 3 |
| 1 | 7 | 5 | 7 | 5 |
| 7 | 2 | 5 | 4 | 6 |
| 2 | 1 | 0 | 6 | 9 |
Mask
| 0.11 | 0.11 | 0.11 |
|---|---|---|
| 0.11 | 0.11 | 0.11 |
| 0.11 | 0.11 | 0.11 |
Blurred Image f ̅(x,y)
| 1 | 2 | 3 | 3 | 1 |
|---|---|---|---|---|
| 2 | 4 | 4 | 4 | 3 |
| 3 | 4 | 4 | 5 | 3 |
| 2 | 3 | 4 | 4 | 4 |
| 1 | 1 | 2 | 3 | 2 |
Subtract The Input Image from the blurred Image:
Result Image [f(x,y)-f ̅(x,y)]
| 0 | 0 | 1 | 4 | 0 |
|---|---|---|---|---|
| 3 | 0 | 2 | 0 | 0 |
| 0 | 3 | 1 | 2 | 2 |
| 5 | 0 | 1 | 0 | 2 |
| 1 | 0 | 0 | 3 | 7 |
Multiple the Result Image with K constant; where K = 2:
Result Image kx[f(x,y)-f ̅(x,y)]
| 0 | 0 | 2 | 8 | 0 |
|---|---|---|---|---|
| 6 | 0 | 4 | 0 | 0 |
| 0 | 6 | 2 | 4 | 4 |
| 10 | 0 | 2 | 0 | 4 |
| 2 | 0 | 0 | 6 | 14 |
Add the Result Image to the Input Image:
Output Image g(x,y)
| 0 | 2 | 6 | 17 | 1 |
|---|---|---|---|---|
| 11 | 3 | 10 | 4 | 3 |
| 1 | 13 | 7 | 11 | 9 |
| 17 | 2 | 7 | 4 | 10 |
| 4 | 1 | 0 | 12 | 23 |
g(x,y)=f(x,y)-[∇^2 f(x,y)]
Take the Input Image and make convolution with mask:
| 0 | 2 | 4 | 7 | 1 |
|---|---|---|---|---|
| 5 | 3 | 6 | 4 | 3 |
| 1 | 7 | 5 | 7 | 5 |
| 7 | 2 | 5 | 4 | 6 |
| 2 | 1 | 0 | 6 | 9 |
Mask
| 1 | 1 | 1 |
|---|---|---|
| 1 | -8 | 1 |
| 1 | 1 | 1 |
The result from [∇^2 f(x,y)]:
[∇^2 f(x,y)]
| 10 | 2 | -10 | -38 | 6 |
|---|---|---|---|---|
| -27 | 6 | -9 | 6 | 0 |
| 16 | -22 | -2 | -18 | -16 |
| -43 | 12 | -8 | 11 | -17 |
| -6 | 8 | 18 | -24 | -56 |
Take the resulting image and subtract it from the input image to get a sharper image: g(x,y)
| -10 | 0 | 14 | 45 | -5 |
|---|---|---|---|---|
| 32 | -3 | 15 | -2 | 3 |
| -15 | 29 | 7 | 25 | 21 |
| 50 | -10 | 13 | -7 | 23 |
| 8 | -7 | -18 | 30 | 65 |
Take the Input Image and make convolution with mask:
| 0 | 2 | 4 | 7 | 1 |
|---|---|---|---|---|
| 5 | 3 | 6 | 4 | 3 |
| 1 | 7 | 5 | 7 | 5 |
| 7 | 2 | 5 | 4 | 6 |
| 2 | 1 | 0 | 6 | 9 |
Mask
| * | ||
By using a median filter, it reduces the noise for the image: Output Image
| 2 | 3 | 4 | 4 | 3 |
|---|---|---|---|---|
| 2 | 4 | 5 | 5 | 4 |
| 5 | 5 | 5 | 5 | 5 |
| 2 | 2 | 5 | 5 | 6 |
| 2 | 2 | 2 | 6 | 6 |
