双边滤波器的原理及实现

http://blog.csdn.net/abcjennifer/article/details/7616663

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。

[cpp] view
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%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.');

end

if ~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;

end

w = 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F

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));

%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：

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I=imread('einstein.jpg');

I=double(I)/255;

w = 5; % bilateral filter half-width

sigma = [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