双边滤波——原理及matlab实现

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1、双边滤波简介：

双边滤波（Bilateral filter）是一种非线性滤波方法（空间权值+相似权值）——空间权值：模糊去噪；相似权值：保护边缘。

2、双边滤波原理

双边滤波具有两个权重：空间权重与相似权重

1）空间权重：与像素位置有关，为像素之间的距离（欧式距离，空间度量），故可定义为全局变量放在循环外，通常定义为

$c(\xi ,x)=e^{-\tfrac{1}{2}(\tfrac{d(\xi ,x)}{\sigma_{d}})^{2}}$

$d(\xi,x)=d(\xi-x)=||\xi -x||$

其中 $d(\xi,x)$ 表示两个像素间的距离（欧式距离）。该过程滤波如下：

$h(x)=k_{d}^{-1}(x)\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }f(\xi )c(\xi ,x)d\xi$

权值为：

$k_{d}(x)=\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }c(\xi ,x)d\xi$

2）相似权重：与像素值大小有关，为像素值之间的距离（辐射距离，相似性度量），根据像素值不同而不同，需要放在循环内，通常定义为

$s(f(\xi) ,f(x))=e^{-\frac{1}{2}(\frac{\sigma (f(\xi ),f(x))}{\sigma _{\tau }})^{2}}$

$\sigma(\xi ,x)=\sigma(\xi-x)=||\xi -x||$

其中 $\sigma (f(\xi ),f(x))$ 表示两个像素值之间的距离。该过程滤波如下：

$h(x)=k_{\tau }^{-1}(x)\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }f(\xi )s(f(\xi) ,f(x))d\xi$

权值为：

$k_{\tau }(x)=\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }s(f(\xi) ,f(x))d\xi$

3）两者结合，得到基于空间距离、相似程度整体考虑的双边滤波如下：

$h(x)=k^{-1}(x)\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }f(\xi )c(\xi ,x)s(f(\xi) ,f(x))d\xi$

权值为：

$k(x)=\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }c(\xi,x)s(f(\xi) ,f(x))d\xi$

3、双边滤波实现：

实际应用时，根据需要对积分采用离散形式表示，滤波半径根据需要自行设置。

在进行滤波前，需将数据转换为浮点型等。

4、matlab源代码

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%主函数%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 读取
f = imread(filename);

% 设置参数
r = 5;% 滤波半径
a = 3;% 全局方差
b = 0.1;% 局部方差

% 判断二维图还是三维图
if ismatrix(f)
    g = bfilt_gray(f,r,a,b);
else
    g = bfilt_rgb(f,r,a,b);
end

% 显示
subplot(121)
imshow(f)
subplot(122)
imshow(g)



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%灰度图双边滤波%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function g = bfilt_gray(f,r,a,b)
% f灰度图；r滤波半径；a全局方差；b局部方差
[x,y] = meshgrid(-r:r);
w1 = exp(-(x.^2+y.^2)/(2*a^2));
f = tofloat(f);%f = im2double(f);

h = waitbar(0,'Applying bilateral filter...');
set(h,'Name','Bilateral Filter Progress');

[m,n] = size(f);
f_temp = padarray(f,[r r],'symmetric');
g = zeros(m,n);
for i = r+1:m+r
    for j = r+1:n+r
        temp = f_temp(i-r:i+r,j-r:j+r);
        w2 = exp(-(temp-f(i-r,j-r)).^2/(2*b^2));
        w = w1.*w2;
        s = temp.*w;
        g(i-r,j-r) = sum(s(:))/sum(w(:));
    end
    waitbar((i-r)/m);
end
% g = revertclass(g);



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%彩色图双边滤波%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function g = bfilt_rgb(f,r,a,b)
% f灰度图；r滤波半径；a全局方差；b局部方差
[x,y] = meshgrid(-r:r);
w1 = exp(-(x.^2+y.^2)/(2*a^2));
f = tofloat(f);%f = im2double(f);

h = waitbar(0,'Applying bilateral filter...');
set(h,'Name','Bilateral Filter Progress');

fr = f(:,:,1);
fg = f(:,:,2);
fb = f(:,:,3);
[m,n] = size(fr);
fr_temp = padarray(fr,[r r],'symmetric');
fg_temp = padarray(fg,[r r],'symmetric');
fb_temp = padarray(fb,[r r],'symmetric');
[gr,gg,gb] = deal(zeros(size(fr)));


for i = r+1:m+r
    for j = r+1:n+r
        temp1 = fr_temp(i-r:i+r,j-r:j+r);
        temp2 = fg_temp(i-r:i+r,j-r:j+r);
        temp3 = fb_temp(i-r:i+r,j-r:j+r);
        dr = temp1 - fr_temp(i,j);
        dg = temp2 - fg_temp(i,j);
        db = temp3 - fb_temp(i,j);
        w2 = exp(-(dr.^2+dg.^2+db.^2)/(2*b^2));
        w = w1.*w2;
        gr(i-r,j-r) = sum(sum(temp1.*w))/sum(w(:));
        gg(i-r,j-r) = sum(sum(temp2.*w))/sum(w(:));
        gb(i-r,j-r) = sum(sum(temp3.*w))/sum(w(:));
    end
    waitbar((i-r)/n);
end
g = cat(3,gr,gg,gb);
% g = revertclass(g);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%可以用到的函数%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [out,revertclass]=tofloat(in)
identity=@(x) x;
tosingle=@im2single;

table={'uint8',tosingle,@im2uint8
'uint16',tosingle,@im2uint16
'int16',tosingle,@im2int16
'logical',tosingle,@logical
'double',identity,identity
'single',identity,identity};

classIndex=find(strcmp(class(in),table(:,1)));

if isempty(classIndex)
    error('unsupported input immage class.');
end

out=table{classIndex,2}(in);
revertclass=table{classIndex,3};

5、实验结果

双边滤波——原理及matlab实现