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Gaussian filtering and bilateral filtering principle, matlab implementation and result comparison

2022-07-08 02:15:00 【Strawberry sauce toast】

This paper introduces the principles of Gaussian filtering and bilateral filtering in detail and gives MATLAB Realization , Finally, compare the effect of Gaussian filtering and bilateral filtering .

Catalog

One 、 Filtering principle

1.1 One dimensional Gaussian distribution

1.2 Two dimensional Gaussian distribution

1.3 Gaussian filtering summary

Two 、 Bilateral filtering

1. The principle of bilateral filtering

2. How to realize bilateral filtering “ Edge preserving denoising ” Of ？

3. MATLAB Realize bilateral filtering

4. About sigma Value selection

4.1 Space domain sigma selection

4.2 range sigma selection

3、 ... and 、 The results of bilateral filtering and Gaussian filtering are compared

3.1 Simulation image comparison

3.2 Real image comparison

3.3 summary

One 、 Filtering principle

Reference resources ：https://blog.csdn.net/nima1994/article/details/79776802（ The principle of Gaussian filtering ）

Focus on ：

The essence of spatial filtering is in the window “ Take the average ”, The larger the radius of the template , The stronger the blur effect .

Gaussian filtering , Gaussian function is used to calculate the template weight , It is based on ： In continuous images , The closer the point is, the closer the relationship is , The farther away the point, the more distant the relationship , therefore , The weighted average is more reasonable , That is, the closer to the center point, the greater the weight of the point , The farther the distance, the smaller the weight of the point .

1.1 One dimensional Gaussian distribution

One dimensional Gaussian distribution （ Normal distribution ） Just meet this condition , Pictured 1.

chart 1 Normal distribution “ A bell curve ”

One dimensional Gaussian distribution formula is shown in formula （1）, The mean is $\mu$ Determines the center point of the curve , $\sigma$ Determines the width of the curve , $\sigma$ The larger the curve, the wider .

$f= \frac{ 1}{ \sigma\sqrt{2\pi} }e^{ -(x-\mu)^{ 2}/2\sigma^{2}}$ …………（1）

MATLAB Realize one-dimensional Gaussian function ：

function g = Gaussian(X,avg,sigma)
%%  Functional specifications ： One dimensional Gaussian distribution 
%%  Parameter description ：avg --  mean value , Determine the location of the center point 
%          sigma --  Standard deviation , Determine the curve width 
   temp_1 = 1 / (sigma * sqrt(2*pi));
   temp_2 = -(X - avg).^2./(2*sigma^2);
   g = temp_1 * exp(temp_2);
   g = g ./ sum(g(:));
end

Use the above functions to draw different $\sigma$ Value corresponds to the curve , The code is as follows ：

X = -15:0.1:15;
avg = 0;  %  The center point of the image is 0
sigmas = [1;2;3;4];
for i = 1:length(sigmas)
    g = Gaussian(X,avg,sigmas(i));
    plot(X,g)
    hold on
end
legend('\sigma = 1','\sigma = 2','\sigma = 3','\sigma = 4');

The image is like this 2, obviously , $\sigma$ Determines the width of the curve , $\sigma$ The larger the curve, the wider , And the image is about x = avg（ Here for 0） symmetry .

chart 2 Different $\sigma$ Value corresponding to one-dimensional Gaussian distribution

Additional explanation ： I did before min-max Standardization , Will be different sigma The value range of the curve corresponding to the value is mapped in [0,1] Within the scope of , But now I find that this practice ignores sigma Influence of value on curve height .

（ Properties of Gaussian distribution ： Below the curve and x The sum of the areas above the shaft is 1）sigma The value of not only determines the width of the curve , It also determines the height of the curve ,sigma The bigger it is , The wider the curve, the shorter ;sigma The smaller it is , The narrower the curve, the higher .

1.2 Two dimensional Gaussian distribution

take x = x,y Substitute into the formula （1） after , Two dimensional Gaussian distribution formula can be obtained by multiplying the two （2）：

$f = \frac{ 1}{ 2\pi\sigma^{ 2}} e^{ -((x-\mu_{1})^{2}+(y-\mu_{2})^{2})/2\sigma^{2}}$ …………（2）

Here is the order $\mu_{1} = 0, \mu_{2} = 0$ A simplified two-dimensional Gaussian distribution function formula can be obtained （3）：

$f = \frac{ 1}{ 2\pi\sigma^{ 2}} e^{ -(x^{2}+y^{2})/2\sigma^{2}}$ …………（3）

According to the formula （3） You can calculate the weight of each point in the Gaussian filter template .

Two dimensional Gaussian function , The code is as follows ：

function [G] = Gaussian(R,sigma)
%%  Functional specifications ： Two dimensional Gaussian function , It can be used to calculate the Gaussian filter weight matrix 
%           Gauss filtering ： Only airspace information is considered , Gray similarity is not considered 
%%  Input parameters ：R --  Formwork radius 
%          sigma --  Standard deviation 
    [X,Y] = meshgrid(-R:0.1:R);
    temp1 = 1/(2*pi*sigma^2);
    temp2 = -(X.^2 + Y.^2)/(2*sigma^2);
    G = temp1 * exp(temp2);
    
    G = G./sum(G(:));  %  The sum of the weights of the template must be equal to 1.

    %  Draw a template diagram 
    figure
    surf(X,Y,G);
    shading interp;
end

The size is 15*15, The standard deviation is 2 The two-dimensional Gaussian distribution template of is shown in the figure 3 Shown

chart 3 The size is 11*11 Two dimensional Gaussian distribution

Use a size of 5*5, The standard deviation is 0.3 Gaussian blur is applied to the image containing weak targets and clouds , Pictured 4 Shown .

chart 4 The size is 5*5, The standard deviation is 0.3 Template Gaussian fuzzy filtering results

1.3 Gaussian filtering summary

Gaussian filtering only considers the spatial similarity between image pixels , That is, in continuous images , The closer the point is, the closer the relationship is , The farther away the point, the more distant the relationship .

Two 、 Bilateral filtering

1. The principle of bilateral filtering

Reference resources ：https://blog.csdn.net/Chaolei3/article/details/88579377

Bilateral filtering is a nonlinear method , Compared with Gaussian filtering , Bilateral filtering not only considers the spatial similarity of images , The gray similarity is also considered , You can achieve “ Edge preserving denoising ” Purpose .

Bilateral filtering consists of two parts ： Spatial matrix and range matrix , Spatial matrix can be analogous to Gaussian filtering , Used for blur denoising ; The range matrix is obtained according to the gray similarity , Used to protect edges .

Calculation formula of spatial matrix （4）, Range matrix calculation formula （5）

$d(i,j,k,l)=e^\frac{ -(i-k)^{2}+(j-l)^{2}}{ 2\sigma_{d}^{2}}$ …………（4）

$r(i,j,k,l)=e^\frac{ -||f(k,l)-f(i,j)||^{2}}{ 2\sigma_r^{2}}$ …………（5）

Formula description ： Definition (i, j) Is the coordinate of the center point ,(k, l) For the point (i, j) Is any point in the neighborhood of the center .

The formula （4）（5） Multiplication is the calculation formula of bilateral filter weight matrix （6）

$w(i,j,k,l)=e^{\frac{ -(i-k)^{2}+(j-l)^2}{ 2\sigma_{d}^{2}}+\frac{ -||f(k,l)-f(i,j)||^{2}}{2\sigma_r^{2}}}$ …………（6）

Last , Calculate the weighted average value as the filtered value of the central point coordinate

$g(i,j)=\frac{ \sum_{(k,l)\in{S}}f(i,j)w(i,j,k,l)}{ \sum_{(k,l)\in{S}}w(i,j,k,l)}$ ………… （7）

2. How to realize bilateral filtering “ Edge preserving denoising ” Of ？

Reference resources ：https://blog.csdn.net/Jfuck/article/details/8932978?utm_medium=distribute.pc_relevant.none-task-blog-BlogCommendFromBaidu-2.control&depth_1-utm_source=distribute.pc_relevant.none-task-blog-BlogCommendFromBaidu-2.control

By formula （4）, The weight distribution of the spatial domain weight matrix is still “ Bell shape ”, That is, the closer to the center , The greater the weight ;

By formula （5）（6）, It is known that $y = e^{-x}$ Yes （0,1） The subtraction function of points , That is, the greater the gray difference , The less weight , The worse the smoothing effect , conversely , The smaller the gray difference , The greater the weight , The better the smoothing effect . therefore , When the center point is at the edge , Large gray difference , The spatial domain smoothing effect is weakened , The edges are preserved ; When the neighborhood gray value is close to the center point , The gray difference is close to 0, The weight of the range is close to 1, At this time, the filtering effect depends on the spatial domain weight matrix , The effect is consistent with Gaussian filtering .

The formula （7） The meaning of ： The value of the center point after filtering = Weight matrices .* original image （ And ensure that the sum of weights is 1, normalization ）（ Refer to the filtering principle ）

chart 5 Bilateral filtering “ Edge preserving denoising ” Schematic diagram

3. MATLAB Realize bilateral filtering

function b = BF_Filter(img,r,sigma_d,sigma_r)
%%  Functional specifications ： Bilateral filtering , Used to calculate the image after bilateral filtering 
%           Bilateral filtering is a nonlinear method , At the same time, the spatial information and gray similarity of the image are considered .
%           A new weight matrix is formed by spatial matrix and range matrix , among , Spatial matrix is used for fuzzy denoising , The range matrix is used to protect the edge .
%%  Input parameters ：img --  Image to be filtered 
%          r --  Formwork radius ,e.g. 3*3 The radius of the template is 1
%          sigma_d --  Standard deviation of spatial matrix 
%          sigma_r --  Range matrix standard deviation 
    
    %  Determine whether it is a grayscale image 
    if(size(img,3)>1)
        img = rgb2gray(img);
    end
    [x,y] = meshgrid(-r:r);
    
    %  Airspace weight matrix  size = (2r+1)*(2r+1)
    w_spacial = exp(-(x.^2 + y.^2)/(2*sigma_d.^2));

    [m,n] = size(img);
    img = double(img);
    
    %  Expand the image ,size = (m+2r)*(n+2r)
    f_temp = padarray(img,[r r],'symmetric');
    
    %  Slide the window and filter 
    b = zeros(m,n); %  The filtered image 
    for i = r+1:m+r
        for j = r+1:n+r
            temp = f_temp(i-r:i+r,j-r:j+r);
            w_value = exp(-(temp - img(i-r,j-r)).^2/(2*sigma_r^2));  % size = (2r+1)*(2r+1)
            w = w_spacial .* w_value;
            s = temp.*w;
            b(i-r,j-r) = sum(s(:))/sum(w(:));  
        end
    end
end

4. About sigma Value selection

4.1 Space domain sigma $\sigma_d$ selection

The size of the nucleus is usually $(6\multi\sigma_d+1)$ , namely $r=6\sigma_d$

Because it is far from the center $(3\multi\sigma_d+1)$ The coefficient outside the size is smaller than the coefficient at the midpoint , It can be considered that the other points have little connection with the central point , The weight coefficient can be regarded as 0.

4.2 range sigma $\sigma_r$ selection

chart 6 $y=e^{-x}$ Images

When $\sigma_r=255$ , The range of gray difference is [0,255] when ,x Value range of $x\leqslant 1$ , The range of values is ：[0.3679,1];

When $\sigma_r=122.5$ , The range of gray difference is [0,255] when ,x Value range of $x\leqslant 4$ , The range of values is ：[0.01832,1];

therefore ,sigma The bigger it is , The smaller the range of weight value , At this time, even if the difference between the neighborhood gray value and the center point gray value is large , Its corresponding weight will also be large , This is related to bilateral filtering “ Edge preserving ” Against the original intention of ;sigma The smaller it is , The larger the range of weight value , At this time, when the difference between the gray values of the center point of the neighborhood gray value range is large , The smaller the corresponding weight , Can weaken “ smooth ”, achieve “ Edge preserving ” effect .

Sum up ,sigma The bigger it is , The more blurred the edge ,sigma-->∞ when ,x-->0, The weight -->1, Multiplied by Gaussian template, it can be considered as equivalent to Gaussian filtering ;

sigma The smaller it is , The clearer the edge ,sigma-->0 when ,x-->∞, The weight -->0, When multiplied by Gaussian template, the coefficients are all equal , Equivalent to the source image .

3、 ... and 、 The results of bilateral filtering and Gaussian filtering are compared

3.1 Simulation image comparison

Generate an image with obvious edges , Pictured 7 Shown .

chart 7 Simulation image

img_B = zeros(250,250);
img_B(:,40:50) = 1;
img_B(:,180:200) = 1;
img_B(100:105,:) = 1;
imshow(img_B)

The results of Gaussian filtering and bilateral filtering are shown in the figure 8 Shown , obviously , Bilateral filtering can better preserve the edge information of the image .

chart 8 Comparison results of simulation images of bilateral filtering and Gaussian filtering

%%  Gauss filtering 
r = 7;
sigma_d = 1;
G = Gaussian(r, sigma_d);
g = imfilter(img_B, G, 'symmetric');
%%  Bilateral filtering 
sigma_r = 3;
b = BF_Filter(img_B,r,sigma_d,sigma_r);
figure
subplot(1,2,1)
imshow(b);title([num2str(2*r+1),'*',num2str(2*r+1),' Bilateral filtering results ','\sigma_d=1 ,',' \sigma_r=3']);
subplot(1,2,2)
imshow(g);title([num2str(2*r+1),'*',num2str(2*r+1),' Gaussian filtering result ','\sigma_d=1']);

3.2 Real image comparison

From the results , For images with structural backgrounds such as clouds and weak targets , Compared with Gaussian filtering , Bilateral filtering can well preserve the edge of the image （ structure ） Information .

chart 9 Original real image

chart 10 Comparison results of real images with bilateral filtering and Gaussian filtering

img = imread('cloud.bmp');
if(size(img,3)>1)
    img = rgb2gray(img);
end
r = 7;
sigma_d = 0.5;
G = Gaussian(r, sigma_d);
img_g = imfilter(img, G, 'symmetric');

sigma_r = 3;
img_b = BF_Filter(img,r,sigma_d,sigma_r);
subplot(1,3,1);
imshow(img);title(' Original picture ');
subplot(1,3,2)
imshow(img_g,[]);title(' Gaussian filtering result ');
subplot(1,3,3);
imshow(img_b,[]);title(' Bilateral filtering results ');

3.3 summary

Combined with the subject of weak and small target detection , Bilateral filtering is suitable for predicting rich edges （ structure ） Background prediction of information , Choose the right one through experiments sigma value , Predict the background , Original picture - Forecast background = Goal map （STI）, And then from STI Detect the target in the figure , The next step needs to be explored .

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