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Training_Ad_2086 t1_j9gi7c1 wrote

Reply to comment by Bax_Cadarn in Why can’t you “un-blur” a blurred image? by so-gold

Well what you described isn't really a blur function (it'd be a brightness shift). But if we want to call it that then yes it is reversible there.

There are several other mathematical operations you can do that are just reversible like that. However none of them are anywhere close to actual blur functions.

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The_Hunster t1_j9i7uwt wrote

Given 1 dimensional images again. Is blurry more like taking the image "2,3,4" and turning them all to the average "3,3,3"? Which could of course be "1,3,5" or "4,3,2". Meaning you lose the original information. Would that be a good example of a blur function?

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MagiMas t1_j9j8gtn wrote

Yes, but it's usually done with a moving average.

So if you have the pixel values 1,3,2,4,3,1,5,2 you could always average in groups of three

1,3,2 => 2
3,2,4 => 3
2,4,3 => 3
4,3,1 => 2.66
3,1,5 => 3
1,5,2 => 2.66

so the blurred image is

2,3,3,2.66,3,2.66

An actual blur filter is usually a bit more complex, a gaussian blur for example weights the original pixels in different amounts (according to a gaussian curve). So instead of just taking the average of 1,3,2 you'd calculate

0.25 * 1 + 0.5 * 3 + 0.25 * 2 = 2.25

And you can choose how broad you want to make the window of values that you consider in the operation etc.

Crucially, if we write the simple blurring operation from the top as a set of equations with the original pixel values unknown and named as x_i:

(x_1 + x_2 + x_3) / 3 = 2
(x_2 + x_3 + x_4) / 3 = 3
(x_3 + x_4 + x_5) / 3 = 3
(x_4 + x_5 + x_6) / 3 = 2.66
(x_5 + x_6 + x_7) / 3 = 3
(x_6 + x_7 + x_8) / 3 = 2.66

you can see that we have 8 unknown values but only 6 equations. If you remember the maths of systems of equations from school we need 8 equations to fully determine 8 unknowns. So this problem is under-determined even in a case of such a simple blurring operation where we know exactly which kind of operation was done to the original image. In a more realistic scenario, where we don't know the exact type of blurring operation done to an image, it gets even less feasible to reverse the operation without somehow using prior knowledge of how unblurred content usually looks like (which is what neural networks are doing when they are used to scale up and unblur small images).

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