Bilateralfilter

The bilateral filter, often written as bilateral filter, is a non-linear, edge-preserving smoothing operator for still images and videos. It was introduced by Carlo Tomasi and Radu Manduchi in 1998 and has since become a standard tool in image processing. Unlike linear blur, the bilateral filter reduces noise while maintaining sharp edges.

The method combines a spatial (domain) kernel with a range kernel that depends on image intensities. For each pixel i, the output is a weighted average of nearby pixels j, where the weight is the product of a spatial Gaussian exp(-|i-j|^2/(2 sigma_d^2)) and a range Gaussian exp(-|I(i) - I(j)|^2/(2 sigma_r^2)). As a result, pixels with similar color to i and close in space contribute more, while dissimilar or distant pixels contribute less, preserving edges.

Because it is non-linear, the bilateral filter cannot be expressed as a simple convolution, and computation scales with the neighborhood size. In practice, it tends to be more expensive than linear filters, although several fast approximations exist, including separable variants and transform-based methods.

Applications include denoising while preserving edges in photography, pre-processing for computer vision tasks, and medical imaging. It is also used as a building block in more advanced filters and in real-time video processing with optimized implementations.

Extensions include color bilateral filtering, joint or cross bilateral filtering, and higher-dimensional versions. The filter has inspired many related algorithms aimed at edge-preserving smoothing with improved speed or specialized noise models.