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I understand the mathematical formalism behind how neural networks work as a classifier or perform regression analysis. But I face difficulty to realize how they are such a great denoising instrument. Even a simple fully connected single hidden layer network can denoise appreciable amount of noise from a relatively simple enough signal (with proper training of course). Do neural networks mimic one or more mathematical operations of filter (such as convolution) and that's why they are so effective in denoising application?

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in general CNN can generate images, so we can feed them noised images and images without noise at output and it can learn how to figure out pixel color based on his neighbours, actually you can't reconstruct perfectly because noise cause removal of some information from image and all you can do it to find "as best as possible" values to restore original image

for more info this clip is very informative https://www.youtube.com/watch?v=z4vl3Z6NFW8

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