Bilateral Filters（双边滤波算法）原理及实现（一）

姓名：张右润

学号：19021210648

转载自：https://blog.csdn.net/piaoxuezhong/article/details/78302920

【嵌牛导读】双边滤波是一种非线性滤波器，它可以达到保持边缘、降噪平滑的效果。和其他滤波原理一样，双边滤波也是采用加权平均的方法，用周边像素亮度值的加权平均代表某个像素的强度，所用的加权平均基于高斯分布[1]。最重要的是，双边滤波的权重不仅考虑了像素的欧氏距离（如普通的高斯低通滤波，只考虑了位置对中心像素的影响），还考虑了像素范围域中的辐射差异（例如卷积核中像素与中心像素之间相似程度、颜色强度，深度距离等），在计算中心像素的时候同时考虑这两个权重。

【嵌牛鼻子】数字图像处理 图像滤波

【嵌牛提问】双边滤波和传统滤波相比有哪些好处？

【嵌牛正文】

双边滤波算法原理

双边滤波是一种非线性滤波器，它可以达到保持边缘、降噪平滑的效果。和其他滤波原理一样，双边滤波也是采用加权平均的方法，用周边像素亮度值的加权平均代表某个像素的强度，所用的加权平均基于高斯分布[1]。最重要的是，双边滤波的权重不仅考虑了像素的欧氏距离（如普通的高斯低通滤波，只考虑了位置对中心像素的影响），还考虑了像素范围域中的辐射差异（例如卷积核中像素与中心像素之间相似程度、颜色强度，深度距离等），在计算中心像素的时候同时考虑这两个权重。 公式1a,1b给出了双边滤过的操作，Iq为输入图像，Ipbf为滤波后图像：

mark下双边滤波里的两个权重域的概念：空间域（spatial domain S）和像素范围域（range domain R）,这个是它跟高斯滤波等方法的最大不同点。下面是我找到的对比说明，更好地理解双边滤波，首先是高斯滤波的情况：

然后对比再看一下双边滤波的过程：

双边滤波的核函数是空间域核与像素范围域核的综合结果：在图像的平坦区域，像素值变化很小，对应的像素范围域权重接近于1，此时空间域权重起主要作用，相当于进行高斯模糊；在图像的边缘区域，像素值变化很大，像素范围域权重变大，从而保持了边缘的信息。

为了更加形象的说明两个权重的影响，作者还给出了二维图像的直观说明:

双边滤波算法实现

在原理部分，从双边滤波的公式就可以得到该算法的实现途径。由于直接的编码实现上述过程，其时间复杂度为O(σs2) ，比较耗时，所以后来出现了一些改进算法，比较经典的有：论文《Fast O(1) bilateral filtering using trigonometric range kernels》，提出了用Raised cosines函数来逼近高斯值域函数，并利用一些特性把值域函数分解为一些列函数的叠加，从而实现函数的加速[5，8]。

这里只对原始方法进行实现，从而有助于更加清楚的了解算法的原理。

matlab实现方法，这里也附一下核心代码

function output = bilateralFilter( data, edge, sigmaSpatial, sigmaRange, ...

    samplingSpatial, samplingRange )

if ~exist( 'edge', 'var' ),

    edge = data;

end

inputHeight = size( data, 1 );

inputWidth = size( data, 2 );

if ~exist( 'sigmaSpatial', 'var' ),

    sigmaSpatial = min( inputWidth, inputHeight ) / 16;

end

edgeMin = min( edge( : ) );

edgeMax = max( edge( : ) );

edgeDelta = edgeMax - edgeMin;

if ~exist( 'sigmaRange', 'var' ),

    sigmaRange = 0.1 * edgeDelta;

end

if ~exist( 'samplingSpatial', 'var' ),

    samplingSpatial = sigmaSpatial;

end

if ~exist( 'samplingRange', 'var' ),

    samplingRange = sigmaRange;

end

if size( data ) ~= size( edge ),

    error( 'data and edge must be of the same size' );

end

% parameters

derivedSigmaSpatial = sigmaSpatial / samplingSpatial;

derivedSigmaRange = sigmaRange / samplingRange;

paddingXY = floor( 2 * derivedSigmaSpatial ) + 1;

paddingZ = floor( 2 * derivedSigmaRange ) + 1;

% allocate 3D grid

downsampledWidth = floor( ( inputWidth - 1 ) / samplingSpatial ) + 1 + 2 * paddingXY;

downsampledHeight = floor( ( inputHeight - 1 ) / samplingSpatial ) + 1 + 2 * paddingXY;

downsampledDepth = floor( edgeDelta / samplingRange ) + 1 + 2 * paddingZ;

gridData = zeros( downsampledHeight, downsampledWidth, downsampledDepth );

gridWeights = zeros( downsampledHeight, downsampledWidth, downsampledDepth );

% compute downsampled indices

[ jj, ii ] = meshgrid( 0 : inputWidth - 1, 0 : inputHeight - 1 );

di = round( ii / samplingSpatial ) + paddingXY + 1;

dj = round( jj / samplingSpatial ) + paddingXY + 1;

dz = round( ( edge - edgeMin ) / samplingRange ) + paddingZ + 1;

% perform scatter (there's probably a faster way than this)

% normally would do downsampledWeights( di, dj, dk ) = 1, but we have to

% perform a summation to do box downsampling

for k = 1 : numel( dz ),

    dataZ = data( k ); % traverses the image column wise, same as di( k )

    if ~isnan( dataZ  ),

        dik = di( k );

        djk = dj( k );

        dzk = dz( k );

        gridData( dik, djk, dzk ) = gridData( dik, djk, dzk ) + dataZ;

        gridWeights( dik, djk, dzk ) = gridWeights( dik, djk, dzk ) + 1;

    end

end

% make gaussian kernel

kernelWidth = 2 * derivedSigmaSpatial + 1;

kernelHeight = kernelWidth;

kernelDepth = 2 * derivedSigmaRange + 1;

halfKernelWidth = floor( kernelWidth / 2 );

halfKernelHeight = floor( kernelHeight / 2 );

halfKernelDepth = floor( kernelDepth / 2 );

[gridX, gridY, gridZ] = meshgrid( 0 : kernelWidth - 1, 0 : kernelHeight - 1, 0 : kernelDepth - 1 );

gridX = gridX - halfKernelWidth;

gridY = gridY - halfKernelHeight;

gridZ = gridZ - halfKernelDepth;

gridRSquared = ( gridX .* gridX + gridY .* gridY ) / ( derivedSigmaSpatial * derivedSigmaSpatial ) + ( gridZ .* gridZ ) / ( derivedSigmaRange * derivedSigmaRange );

kernel = exp( -0.5 * gridRSquared );

% convolve

blurredGridData = convn( gridData, kernel, 'same' );

blurredGridWeights = convn( gridWeights, kernel, 'same' );

% divide

blurredGridWeights( blurredGridWeights == 0 ) = -2; % avoid divide by 0, won't read there anyway

normalizedBlurredGrid = blurredGridData ./ blurredGridWeights;

normalizedBlurredGrid( blurredGridWeights < -1 ) = 0; % put 0s where it's undefined

blurredGridWeights( blurredGridWeights < -1 ) = 0; % put zeros back

% upsample

[ jj, ii ] = meshgrid( 0 : inputWidth - 1, 0 : inputHeight - 1 ); % meshgrid does x, then y, so output arguments need to be reversed

% no rounding

di = ( ii / samplingSpatial ) + paddingXY + 1;

dj = ( jj / samplingSpatial ) + paddingXY + 1;

dz = ( edge - edgeMin ) / samplingRange + paddingZ + 1;

% interpn takes rows, then cols, etc

% i.e. size(v,1), then size(v,2), ...

output = interpn( normalizedBlurredGrid, di, dj, dz );

参考：

https://en.wikipedia.org/wiki/Bilateral_filtering

http://people.csail.mit.edu/sparis/bf_course/

http://people.csail.mit.edu/sparis/bf/

http://blog.csdn.net/fightingforcv/article/details/52723376

http://blog.csdn.net/mumusan2016/article/details/54578038

http://blog.csdn.net/majinlei121/article/details/50463514

http://kaiminghe.com/eccv10/index.html  （Guided Image Filtering）

http://blog.csdn.net/dangchangying/article/details/14451963

原文：https://blog.csdn.net/piaoxuezhong/article/details/78302920