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，
权重系数w(i,j,k,l)取决于定义域核
和值域核
的乘积
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)
CopyRight：
% Douglas R. Lanman, Brown University, September 2006.% dlanman@brown.edu, 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 deviations
I1=bfilter2(I,w,sigma);
subplot(1,2,1);imshow(I);subplot(1,2,2);imshow(I1)
实验结果：
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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