Bilateral filter

A bilateral filter is a non-linear, edge-preserving, and noise-reducing smoothing filter for images.

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.

$\begin{aligned} b f(I)_{\mathbf{p}} &=\frac{1}{W_{\mathbf{p}}} \sum_{\mathbf{q} \in N(\mathbf{p})} G_{\sigma_{\mathrm{s}}}(\|\mathbf{p}-\mathbf{q}\|) G_{\sigma_{\mathrm{r}}}\left(\left|I_{\mathbf{p}}-I_{\mathbf{q}}\right|\right) I_{\mathbf{q}} \\ W_{\mathbf{p}} &=\sum_{\mathbf{q} \in N(\mathbf{p})} G_{\sigma_{\mathrm{s}}}(\|\mathbf{p}-\mathbf{q}\|) G_{\sigma_{\mathrm{r}}}\left(\left|I_{\mathbf{p}}-I_{\mathbf{q}}\right|\right) \end{aligned}$

空间距离：当前点与中心点的欧式距离

$\mathrm{e}^{-\frac{\left(x_{i}-x_{C}\right)^{2}+\left(y_{i}-y_{C}\right)^{2}}{2 \sigma^{2}}}$
灰度距离：指的是当前点的灰度与中心点灰度的差的绝对值

$\mathrm{e}^{-\frac{\left(\operatorname{gray}\left(x_{i}, y_{i}\right)-\operatorname{gray}\left(x_{c}, y_{c}\right)\right)^{2}}{2 \sigma^{2}}}$

2. Bilateral Grid

2.1 edge-aware brush

2.2 Defination

Data Structure

3d array:

the first 2 dimensions (x, y) correspond to 2D position in the image plane and form the spatial domain
the third dimension z corresponds to a reference range, typically is image intensity

Samping

$S_s$ : the sampling rate of spatial axes
$S_r$ : the sampling rate of the range axis.

The number of grid cells is inversely proportional to the sampling rate: a smaller sampling rate yields a larger number of grid cells

2.3 Basic Usage of a Bilateral Grid

Grid Creation

Slicing

Slicing is the critical bilateral grid operation that yields a piecewise-smooth output.

Given a bilateral grid $\Gamma$ and a reference image E, we extract a 2D value map M by accessing the grid at $(x/S_s, y/S_s, E(x,y)/s_r)$