# Bilateral filter

A bilateral filter is a 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 (e.g. range differences, 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

$I^\text{filtered}(x) = \frac{1}{W_p} \sum_{x_i \in \Omega} I(x_i)f_r(\|I(x_i)-I(x)\|)g_s(\|x_i-x\|),$

where the normalization term

$W_p = \sum_{x_i \in \Omega}{f_r(\|I(x_i)-I(x)\|)g_s(\|x_i-x\|)}$

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;
• $\Omega$ 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;

As mentioned above, the weight $W_p$ is assigned using the spatial closeness and the intensity difference.[1] Consider a pixel located at $(i, j)$ which needs to be denoised in image using its neighbouring pixels and one of its neighbouring pixels is located at $(k, l)$. Then, the weight assigned for pixel $(k, l)$ to denoise the pixel $(i, j)$ is given by: $w(i, j, k, l)= e^{(-\frac{(i-k)^2+ (j-l)^2}{2 \sigma_d^2}- \frac{\|I(i, j)- I(k, l)\|^2}{2 \sigma_r^2})}$

where σd and σr are smoothing parameters and I(i, j) and I(k, l) are the intensity of pixels $(i, j)$ and $(k, l)$ respectively. After calculating the weights, normalize them. $I_D(i, j)=\frac{\sum_{k, l}{I(k, l) * w(i, j, k, l)} } { \sum_{k, l}{w(i, j, k, l)} }$

where $I_D$ is the denoised intensity of pixel $(i, j)$.

## Parameters

• As range parameter σr increases, the bilateral filter gradually approaches Gaussian convolution more closely because the range Gaussian widens and flattens, which means that it becomes nearly constant over the intensity interval of the image.
• On increasing the spatial parameter σd, the larger features get smoothened.

## Limitations

The bilateral filter in its direct form can introduce several types of image artifacts:

• Staircase effect - intensity plateaus that lead to images appearing like cartoons [1]
• Gradient reversal - introduction of false edges in the image [2]

There exist several extensions to the filter that deal with these artifacts. Alternative filters, like the guided filter [3], have also been proposed as an efficient alternative without these limitations.

## Implementations

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'.

## Related models

The Bilateral filter was shown to be an application of the short time kernel of the Beltrami flow [2] see also [3]