Editing Digital image processing (section)

===Improvement of image quality method===

Image quality can be influenced by camera vibration, over-exposure, gray level distribution too centralized, and noise, etc. For example, noise problem can be solved by [[smoothing]] method while gray level distribution problem can be improved by [[histogram equalization]].

'''[[Smoothing]] method'''

In drawing, if there is some dissatisfied color, taking some color around dissatisfied color and averaging them. This is an easy way to think of Smoothing method.

Smoothing method can be implemented with mask and [[convolution]]. Take the small image and mask for instance as below.

image is
<math>
\begin{bmatrix}
2 & 5 & 6 & 5\\
3 & 1 & 4 & 6 \\
1 & 28 & 30 & 2 \\
7 & 3 & 2 & 2
\end{bmatrix}
</math>

mask is <math>
\begin{bmatrix}
1/9 & 1/9 & 1/9 \\
1/9 & 1/9 & 1/9 \\
1/9 & 1/9 & 1/9
\end{bmatrix}
</math>

After convolution and smoothing, image is
<math>
\begin{bmatrix}
2 & 5 & 6 & 5\\
3 & 9 & 10 & 6 \\
1 & 9 & 9 & 2 \\
7 & 3 & 2 & 2
\end{bmatrix}
</math>

Observing image[1, 1], image[1, 2], image[2, 1], and image[2, 2].

The original image pixel is 1, 4, 28, 30. After smoothing mask, the pixel becomes 9, 10, 9, 9 respectively.

new image[1, 1] = <math>\tfrac{1}{9}</math> * (image[0,0]+image[0,1]+image[0,2]+image[1,0]+image[1,1]+image[1,2]+image[2,0]+image[2,1]+image[2,2])

new image[1, 1] = floor(<math>\tfrac{1}{9}</math> * (2+5+6+3+1+4+1+28+30)) = 9

new image[1, 2] = floor({<math>\tfrac{1}{9}</math> * (5+6+5+1+4+6+28+30+2)) = 10

new image[2, 1] = floor(<math>\tfrac{1}{9}</math> * (3+1+4+1+28+30+7+3+2)) = 9

new image[2, 2] = floor(<math>\tfrac{1}{9}</math> * (1+4+6+28+30+2+3+2+2)) = 9

'''Gray Level Histogram method'''

Generally, given a gray level histogram from an image as below. Changing the histogram to uniform distribution from an image is usually what we called [[histogram equalization]].

[[File:Gray level histogram.jpg|thumb|Figure 1]]

[[File:Uniform distribution.jpg|thumb|Figure 2]]

In discrete time, the area of gray level histogram is <math>\sum_{i=0}^{k}H(p_i)</math>(see figure 1) while the area of uniform distribution is <math>\sum_{i=0}^{k}G(q_i)</math>(see figure 2). It is clear that the area will not change, so <math>\sum_{i=0}^{k}H(p_i) = \sum_{i=0}^{k}G(q_i)</math>.

From the uniform distribution, the probability of <math>q_i</math> is <math>\tfrac{N^2}{q_k - q_0}</math> while the <math> 0 < i < k </math>

In continuous time, the equation is <math>\displaystyle \int_{q_0}^{q} \tfrac{N^2}{q_k - q_0}ds = \displaystyle \int_{p_0}^{p}H(s)ds</math>.

Moreover, based on the definition of a function, the Gray level histogram method is like finding a function <math>f</math> that satisfies f(p)=q.

{| class="wikitable"
|-
! Improvement method
! Issue
! Before improvement
! Process
! After improvement
|-
|-
| Smoothing method
| noise
with Matlab, salt & pepper with 0.01 parameter is added<br /> to the original image in order to create a noisy image.
| [[File:Helmet with noise.jpg|thumb|]]
| 
# read image and convert image into grayscale
# convolution the graysale image with the mask <math>
\begin{bmatrix}
1/9 & 1/9 & 1/9 \\
1/9 & 1/9 & 1/9 \\
1/9 & 1/9 & 1/9
\end{bmatrix}
</math>
# denoisy image will be the result of step 2.

| [[File:Helmet without noise.jpg|thumb|]]
|-
|-
| Histogram Equalization
| Gray level distribution too centralized
| [[File:Cave scene before improvement.jpg|thumb|]]
| Refer to the [[Histogram equalization]]
| [[File:Cave scene after improvement.jpg|thumb|]]
|-
|}