Window operator

A bilateral filter is non-linear, edge-preserving and noise-reducing smoothing filter for images. The intensity value at each pixel in an image is replaced by a weighted average of intensity values from nearby pixels. This weight can be based on a Gaussian distribution. Crucially, the weights depend not only on Euclidean distance of pixels, but also on the radiometric differences. For example, the range difference such as color intensity, depth distance, etc. This preserves sharp edges by systematically looping through each pixel and adjusting weights to the adjacent pixels accordingly.

The bilateral filter is defined as

$I^{\text{filtered}}(x)=\frac{1}{W_p}\sum _{x_i\in \mathrm{\Omega }}I(x_i)f_r(\Vert I(x_i)-I(x)\Vert )g_s(\Vert x_i-x\Vert ),$

where the normalization term

$W_p=\sum _{x_i\in \mathrm{\Omega }}{f}_r(\Vert I(x_i)-I(x)\Vert )g_s(\Vert x_i-x\Vert )$

ensures that the filter preserves image energy and

• $I^{\text{filtered}}$ is the filtered image;
• $I$ is the original input image to be filtered;
• $x$ are the coordinates of the current pixel to be filtered;
• $\mathrm{\Omega }$ is the window centered in $x$;
• $f_r$ is the range kernel for smoothing differences in intensities. This function can be a Gaussian function;
• $g_s$ is the spatial kernel for smoothing differences in coordinates. This function can be a Gaussian function;

Adobe Photoshop implements a bilateral filter in its surface blur tool. GIMP implements a bilateral filter in its Filters-->Blur tools; and it is called Selective Gaussian Blur'.

See also

• Gaussian filter
• Gaussian function
• Gaussian blur
• Convolution

External links

• Kunal N. Chaudhury Constant-time filtering
• Kunal N. Chaudhury, Daniel Sage, and Michael Unser Java plugin, Fast bilateral filtering
• Haarith Devarajan, Harold Nyikal, Bilateral Filters, in: Image Scaling and Bilateral Filtering 2006 course
• Sylvain Paris, Pierre Kornprobst, Jack Tumblin, Frédo Durand, Bilateral Filtering: Theory and Applications, preprint
• Sylvain Paris, Pierre Kornprobst, Jack Tumblin, Frédo Durand, A Gentle Introduction to Bilateral Filtering and its Applications, SIGGRAPH 2008 class
• Ben Weiss, Fast Median and Bilateral Filtering, SIGGRAPH 2006 preprint
• Carlo Tomasi, Roberto Manduchi, Bilateral Filtering for Gray and Color Images (shorter HTML version), proceedings of the ICCV 1998