双边滤波器的原理及实现

双边滤波器是什么？

双边滤波（Bilateral filter）是一种可以保边去噪的滤波器。之所以可以达到此去噪效果，是因为滤波器是由两个函数构成。一个函数是由几何空间距离决定滤波器系数。另一个由像素差值决定滤波器系数。可以与其相比较的两个filter：高斯低通滤波器(http://en.wikipedia.org/wiki/Gaussian_filter)和α-截尾均值滤波器（去掉百分率为α的最小值和最大之后剩下像素的均值作为滤波器），后文中将结合公式做详细介绍。

双边滤波器中，输出像素的值依赖于邻域像素的值的加权组合，

双边滤波器的原理及实现

权重系数w(i,j,k,l)取决于定义域核

双边滤波器的原理及实现

和值域核

双边滤波器的原理及实现

的乘积

双边滤波器的原理及实现

同时考虑了空间域与值域的差别，而Gaussian Filter和α均值滤波分别只考虑了空间域和值域差别。

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双边滤波器的实现（MATLAB）：function B = bfilter2(A,w,sigma)

% Douglas R. Lanman, Brown University, September 2006.
% [email protected], http://mesh.brown.edu/dlanman

具体请见function B = bfltGray(A,w,sigma_d,sigma_r)函数说明。

%简单地说:%A为给定图像，归一化到[0,1]的矩阵%W为双边滤波器（核）的边长/2%定义域方差σd记为SIGMA(1),值域方差σr记为SIGMA(2)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Pre-process input and select appropriate filter.function B = bfilter2(A,w,sigma)% Verify that the input image exists and is valid.if ~exist('A','var') || isempty(A)   error('Input image A is undefined or invalid.');endif ~isfloat(A) || ~sum([1,3] == size(A,3)) || ...      min(A(:)) < 0 || max(A(:)) > 1   error(['Input image A must be a double precision ',...          'matrix of size NxMx1 or NxMx3 on the closed ',...          'interval [0,1].']);      end% Verify bilateral filter window size.if ~exist('w','var') || isempty(w) || ...      numel(w) ~= 1 || w < 1   w = 5;endw = ceil(w);% Verify bilateral filter standard deviations.if ~exist('sigma','var') || isempty(sigma) || ...      numel(sigma) ~= 2 || sigma(1) <= 0 || sigma(2) <= 0   sigma = [3 0.1];end% Apply either grayscale or color bilateral filtering.if size(A,3) == 1   B = bfltGray(A,w,sigma(1),sigma(2));else   B = bfltColor(A,w,sigma(1),sigma(2));end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Implements bilateral filtering for grayscale images.function B = bfltGray(A,w,sigma_d,sigma_r)% Pre-compute Gaussian distance weights.[X,Y] = meshgrid(-w:w,-w:w);%创建核距离矩阵，e.g.%  [x,y]=meshgrid(-1:1,-1:1)% % x =% %     -1     0     1%     -1     0     1%     -1     0     1% % % y =% %     -1    -1    -1%      0     0     0%      1     1     1%计算定义域核G = exp(-(X.^2+Y.^2)/(2*sigma_d^2));% Create waitbar.h = waitbar(0,'Applying bilateral filter...');set(h,'Name','Bilateral Filter Progress');% Apply bilateral filter.%计算值域核H 并与定义域核G 乘积得到双边权重函数Fdim = size(A);B = zeros(dim);for i = 1:dim(1)   for j = 1:dim(2)               % Extract local region.         iMin = max(i-w,1);         iMax = min(i+w,dim(1));         jMin = max(j-w,1);         jMax = min(j+w,dim(2));         %定义当前核所作用的区域为(iMin:iMax,jMin:jMax)         I = A(iMin:iMax,jMin:jMax);%提取该区域的源图像值赋给I               % Compute Gaussian intensity weights.         H = exp(-(I-A(i,j)).^2/(2*sigma_r^2));               % Calculate bilateral filter response.         F = H.*G((iMin:iMax)-i+w+1,(jMin:jMax)-j+w+1);         B(i,j) = sum(F(:).*I(:))/sum(F(:));                  end   waitbar(i/dim(1));end% Close waitbar.close(h);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Implements bilateral filter for color images.function B = bfltColor(A,w,sigma_d,sigma_r)% Convert input sRGB image to CIELab color space.if exist('applycform','file')   A = applycform(A,makecform('srgb2lab'));else   A = colorspace('Lab<-RGB',A);end% Pre-compute Gaussian domain weights.[X,Y] = meshgrid(-w:w,-w:w);G = exp(-(X.^2+Y.^2)/(2*sigma_d^2));% Rescale range variance (using maximum luminance).sigma_r = 100*sigma_r;% Create waitbar.h = waitbar(0,'Applying bilateral filter...');set(h,'Name','Bilateral Filter Progress');% Apply bilateral filter.dim = size(A);B = zeros(dim);for i = 1:dim(1)   for j = 1:dim(2)               % Extract local region.         iMin = max(i-w,1);         iMax = min(i+w,dim(1));         jMin = max(j-w,1);         jMax = min(j+w,dim(2));         I = A(iMin:iMax,jMin:jMax,:);               % Compute Gaussian range weights.         dL = I(:,:,1)-A(i,j,1);         da = I(:,:,2)-A(i,j,2);         db = I(:,:,3)-A(i,j,3);         H = exp(-(dL.^2+da.^2+db.^2)/(2*sigma_r^2));               % Calculate bilateral filter response.         F = H.*G((iMin:iMax)-i+w+1,(jMin:jMax)-j+w+1);         norm_F = sum(F(:));         B(i,j,1) = sum(sum(F.*I(:,:,1)))/norm_F;         B(i,j,2) = sum(sum(F.*I(:,:,2)))/norm_F;         B(i,j,3) = sum(sum(F.*I(:,:,3)))/norm_F;                   end   waitbar(i/dim(1));end% Convert filtered image back to sRGB color space.if exist('applycform','file')   B = applycform(B,makecform('lab2srgb'));else     B = colorspace('RGB<-Lab',B);end% Close waitbar.close(h);

调用方法：

I=imread('einstein.jpg');I=double(I)/255;w     = 5;       % bilateral filter half-widthsigma = [3 0.1]; % bilateral filter standard deviationsI1=bfilter2(I,w,sigma);subplot(1,2,1);imshow(I);subplot(1,2,2);imshow(I1)

实验结果：

双边滤波器的原理及实现

参考资料：

1.《Computer Vision Algorithms and Applications》

2. http://de.wikipedia.org/wiki/Bilaterale_Filterung

3.http://www.cs.duke.edu/~tomasi/papers/tomasi/tomasiIccv98.pdf

4. http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/MANDUCHI1/Bilateral_Filtering.html

5. http://mesh.brown.edu/dlanman

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