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Which approach is better or there is mathematical justification for using Bilater filter and Fourier Transform to decompose a image into High Frequency and Low Frequency Component.

Both Bilateral Filter and Fourier Transform can be used for getting High and Low Frequency component of an image.

Is one approach is better than the other or are both same in getting high and low frequency component of a image, or are they used in some particular context of image.

I know that Bilateral Filter uses range weights and Spatial weights into consideration for getting high and low frequency components.

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The bilateral filter is just that: a filter. It does not decompose the image into anything. It just attenuates "medium" frequency content in a clever manner. In other words, it smooths things that are mostly smooth further, and leaves sharp edges sharp. What you are left with has more low and high frequency content (on a relative basis) than what you started with.

The Fourier transform, on the other hand, actually does reveal the entire spectrum for an image. Unlike a filter, it does not change the information content of an image. It just rearranges it into the frequency, rather than spatial, domain. The bilateral filter does not reveal the spectral content of an image, so really only the Fourier transform (or one of its related transforms, like cosine and wavelet) can be used to get the high and low frequency components of an image.

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