Why isn't bilateral filtering shift-invariant?

1. What is a shift invariant filter?
2. Is median filtering shift invariant?
3. What is the purpose of bilateral filter?
4. Is bilateral filter a Gaussian filter?

What is a shift invariant filter?

In plain terms, a time-invariant filter (or shift-invariant filter) is one which performs the same operation at all times. It is awkward to express this mathematically by restrictions on Eq.(4.2) because of the use of as the symbol for the filter input.

Is median filtering shift invariant?

Yes. For a 1D signal, shift invariance of a filter implies the following.

What is the purpose of bilateral filter?

A bilateral filter is a non-linear, edge-preserving, and noise-reducing smoothing filter for images. It replaces the intensity of each pixel with a weighted average of intensity values from nearby pixels.

Is bilateral filter a Gaussian filter?

The bilateral filter is almost like a Gaussian filter, except that the Gaussian is modulated by a function that computes the similarity between the central pixel (where the filter is applied) and a pixel in its neighborhood (that is used in blurring).