MATLAB image histogram equalization, namely spatial filtering

1. Histogram equalization

1.1 The theoretical analysis

Define symbols in discussion
$$\begin{array}{rl}& P\in R^{m\times n},primarychartimageMomentfront\\ & S\in R^{m\times n},straightFangchartallequilibriumturnAndafterOfchartimage\\ & [0,L-1],imageplainspotOftakevalueFanaround\\ & p(r),primarychartimageGeneralrateThe\; secretdegreeLetterCount\\ & T(r),ashdegreeturninLetterCount\\ & round(x),FourHouse5、\; ...\; andEnter\; intoLetterCount\end{array}$$
First, calculate the probability density function of the original image （PDF）
$$p(r)=\frac{1}{mn}\sum I(P_{ij}=r),r=0,\cdots ,L-1$$
Next, calculate the grayscale conversion function
$$T(r)=round((L-1)\sum _{i=0}^{r}p(i))$$
Then, according to the gray conversion function, the image after histogram equalization is obtained
$$S_{ij}=T(P_{ij})$$

1.2 Result analysis

The original image is as follows 3-2.1 Shown , In the original image, there are a large number of pixels whose pixel values are almost 0（ The histogram is shown in 3-2.2）. after image process toolbox The processed image is as follows 3-2.3 Shown , The image itself has a large number of pixel values 0, Therefore, the result of histogram equalization will make the original result 0 The pixels of move to 50-100 Within the range of （ See histogram 3-2.4）, The picture is too bright .

Specifically matlab The algorithm of histogram equalization with its own function is different from the textbook , therefore 3-2.5（ The histogram equalization result obtained by using the algorithm in the textbook ） and 3-2.5（ Use it directly matlab The histogram equalization result obtained by the self-contained function ） There must be a difference . But the results of both methods are not ideal , It means that the image is not suitable for histogram equalization .

1.3 result

3-2.1 Original image

3-2.2 Original image histogram

3-2.3 Use matlab Own function for histogram equalization

3-2.4 Use matlab The histogram of the image is obtained by histogram equalization with its own function

3-2.5 Own histogram equalization processing image

3-2.6 Own histogram equalization of the image histogram after processing

2. Laplace filtering

2.1 The theoretical analysis

Use the following formula to find the Laplace transform of the image
$$\begin{array}{rl}{\nabla }^{2}f(x,y)=& 8f(x,y)-f(x+1,y+1)-f(x+1,y)-\\ & f(x+1,y-1)-f(x,y+1)-f(x,y-1)-\\ & f(x-1,y+1)-f(x-1,y)-f(x-1,y-1)\end{array}$$
The mask is converted into a mask to obtain the following mask
$$\left[\begin{array}{ccc}-1& -1& -1\\ -1& 8& -1\\ -1& -1& -1\end{array}\right]$$
Set the original image to $f(x,y)$, The filtered function is $f_{mask}(x,y)$, The final image is $g(x,y)$
$$g(x,y)=f(x,y)+f_{mask}(x,y)$$

2.2 Result analysis

First, the image 3-5.1 Sharpen , obtain 3-5.2. You can see that the edges of the sharpened image have been significantly enhanced , Noise also increases to a certain extent .

Then the image 3-5.3 Sharpen , obtain 3-5.4. Images 3-5.3 Image for text , The high frequency component of the image only appears at the edge of the text , Therefore, the sharpening process will only enhance the edges of the text , No more noise like normal images .

Comparing the results of two image sharpening, we can think that image sharpening is very suitable for processing text images .

2.3 result

3-5.1 The original image of the moon

3-5.2 The result of lunar image enhancement

3-5.3 Text original image

3-5.4 Text image enhancement results

3. Unsharp mask

3.1 The theoretical analysis

Define symbols in discussion ：
$$\begin{array}{rl}& f(x,y),primarychartimage\\ & \bar{f}(x,y),thetooflatslidefilterwaveOfchartimage\\ & g_{mask}(x,y),increasestrongsenduseOfMaskmembrane\\ & g(x,y),increasestrongafterOfchartimage\end{array}$$
First, use the following smoothing filter to get $\bar{f}(x,y)$
$$\frac{1}{9}\times \left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$$
Then according to $\bar{f}(x,y)$ and $f(x,y)$ obtain $g_{mask}(x,y)$
$$g_{mask}(x,y)=f(x,y)-\bar{f}(x,y)$$
The final enhanced image is obtained from the mask （ Coefficient in the experiment k = 1）
$$g(x,y)=f(x,y)+k{g}_{mask}(x,y)$$

3.2 Result analysis

The original image is 3-6.1, The original image is smoothed and filtered to obtain an image 3-6.2. because $g_{mask}(x,y)$ The displayed results are difficult to observe with the naked eye, so they are not displayed . The final enhanced image is 3-6.3. In fact, if $g_{mask}(x,y)$ The coefficient before is 1, The present situation , Image enhancement is difficult to show , However, if the coefficient is increased, a lot of noise will appear .

3.3 result

3-6.1 Original image

3-6.2 Image after smooth filtering

3-6.3 Enhanced image

appendix 1： Histogram equalization

%  Read images 
pic = imread('./DIP3E_Original_Images_CH03/Fig0308(a)(fractured_spine).tif');
% pic = int64(pic);

%  Original image and histogram 
count = count_histogram(pic);
x = 0:255;
subplot(3, 2, 1);
imshow(pic);
subplot(3, 2, 2);
bar(x, count);

%  Use matlab Its own histogram equalization 
pic_hiseq = histeq(pic);
subplot(3, 2, 3);
imshow(pic_hiseq);
subplot(3, 2, 4);
imhist(pic_hiseq);

%  Use their own histogram equalization 
pic_new = histogram_equalization(pic);
subplot(3, 2, 5);
imshow(pic_new, []);
count = count_histogram(pic_new);
x = 0:255;
subplot(3, 2, 6);
bar(x, count);

%  Function for calculating histogram 
function count = count_histogram(pic)
    [m, n] = size(pic);
    count = zeros(1, 256);
    for i = 1:m
        for j = 1:n
            count(pic(i, j) + 1) = count(pic(i, j) + 1) + 1;
        end
    end
end

%  Histogram equalization of the image 
function new_pic = histogram_equalization(pic)
    [m, n] = size(pic);
    count = count_histogram(pic);
    new_pic = zeros(m, n);
    
    s = zeros(1, 256);
    for i = 1:256
        for j = 1:i
            s(i) = s(i) + 255 * count(j) / (m * n);
        end
        s(i) = round(s(i));
    end
    
    for i = 1:m
        for j = 1:n
            new_pic(i, j) = s(pic(i, j) + 1);
        end
    end
end

appendix 2： Sharpening image code

pic = imread("./DIP3E_Original_Images_CH03/Fig0338(a)(blurry_moon).tif");

mask = [-1, -1, -1; -1, 8, -1; -1, -1, -1];
pic_new = imfilter(pic, mask);
pic_new = pic_new + pic;


%  Processing images of the moon 
subplot(2, 2, 1);
imshow(pic);
subplot(2, 2, 2);
imshow(pic_new);

%  Working with text images 
pic_word = imread("./DIP3E_Original_Images_CH03/Fig0340(a)(dipxe_text).tif");

pic_word_new = pic_word + imfilter(pic_word, mask);

subplot(2, 2, 3);
imshow(pic_word);
subplot(2, 2, 4);
imshow(pic_word_new);

appendix 3： Unsharp mask enhanced image

pic = imread("./DIP3E_Original_Images_CH03/Fig0343(a)(skeleton_orig).tif");

smooth_mask = [1, 1, 1; 1, 1, 1; 1, 1, 1] ./ 9;

%  Smooth filtering 
smooth_pic = imfilter(pic, smooth_mask);
%  Get mask 
g_mask = pic - smooth_pic;
%  Image enhancement 
new_pic = pic + g_mask ;

subplot(1, 3, 1);
imshow(pic);
subplot(1, 3, 2);
imshow(smooth_pic);
subplot(1, 3, 3);
imshow(new_pic);