BILATERAL FILTER

This filter aims to reduce the Gaussian noise without blurring the image, the filter is inversely proportional to distance and proportional with the pixel intensity

O (p) = q \in S \sum H (p, q) I_{q} w h ere

H (p, q) = \frac{1}{W ( p )} G_{σ_{S}} (d_{S} (p, q)) G_{σ_{r}} (d_{r} (I_{p}, I_{q}))

$d_{S} (p, q) = ∥ p - q ∥_{2} = (u_{p} - u_{q})^{2} + (v_{p} + v_{q})^{2}$ → Spatial distance
$d_{r} (p, q) = ∣ p - q ∣$ → Range (intensity) distance
$W (p) = \sum_{q \in S} G_{σ_{S}} (d_{S} (p, q)) G_{σ_{r}} (d_{r} (I_{p}, I_{q}))$ → normalization factor

This filter takes into account the bi-dimensional distance and the difference in pixel intensity of the points in the $S$ neighborhood

As shown in the image this filter is good in the edges cause it does not introduce blur in the image cause it takes into account range distance

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BILATERAL FILTER

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