[Digital image processing] Bilateral Filters

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basic introduction

It is a nonlinear bilateral filtering may be able to retain a blurred image and edge information technology.

It is possible to decompose the image into different scales without halos (haloes) after modification, which makes it ubiquitous computing (Ubiquitous), for example color, style converted photographic applications retune light, to noise (tone mapping, style transfer, relighting, and denoising).

Successful bilateral filtering mainly in some of the following:

Its formula is simple: each pixel is replaced by a weighted average of its neighbors. This aspect is important because it is easy to gain an intuitive understanding of their behavior to suit the specific application requirements and achieve it.
It only depends on two parameters, these parameters indicates the size and contrast characteristics to be retained.
It can be used in a non-iterative. This makes it very easy to set parameters because their impact is not accumulated through many iterations.
Since the effective value of the policy, and, even in the case where the graphics hardware is available, it may be calculated on a large image at interactive speed.

Specific expression

Bilateral filtering and general Gaussian filter (convolution) are closely linked.

Gaussian Convolution

\ (I_P \) represents the value of the image on the pixel position p of, for grayscale, then the individual values
\ (F [I] \) represents an image \ (the I \) through the filter \ (F. \) Processed output
\ (S \) represents the set of all possible positions of the images, the spatial domain is named
\ (R & lt \) represents the set of all possible pixel values, which is named range domain
\ (\ sum_ {p \ in S} \) represents the sum for all the image pixels using the index p
\ (| \ Cdot | \) represents the absolute value
\ (|| \ || CDOT \) represents the L2 norm used herein \ (PQ || || \) represents the Euclidean distance between the position corresponding to two pixels p and q
\ (\ Sigma \) represents the neighborhood range

Bilateral Filtering

Bilateral filtering in a method similar to a Gaussian convolution is defined as a weighted average of neighboring pixels. The difference is that bilateral filtering takes into account the differences in the value of neighbors, smooth when you can keep edge information.

The key idea bilateral filtering that affect other for a pixel, it should only appear in a similar position, should have similar values .

An important feature of the bilateral filtering is multiplied by weights: If any of the weights is close to zero, no smoothing occurs. In a large space algorithm Gaussian distribution Gaussian distribution with a small range of the combination of, for example, despite the large spatial extent, but it can only achieve a limited smoothing effect. Range of weights will be forced to retain the outline.

\ (\ sigma_s \) and \ (\ sigma_r \) represents the image \ (the I \) amount of filtering (i.e. filter range)
\ (G _ {\ sigma_s} \) represents the Gaussian spatial weighting to reduce the influence of distant pixels
\ (G _ {\ sigma_r} \) represents the Gaussian weighting range, to reduce the influence of the pixel q and the larger the luminance value of the pixel p when the difference, caused by