How to remove the background without losing information if it is available as a separate image?

Question

Design

There are:
1) A picture with an object on the background.
2) Background separately.
Everything is colorful, different colors, both the object and the background. The object is very blurry in places. You need to cut the object so that it is on a transparent background, and remove the existing background.
How to do it? And is it possible in principle, at least theoretically?

For example, I made a small picture with some kind of complex object:

And a separate background for her:

If you make a difference, then the traces still remain

Да и что дальше делать не ясно.

Dividing doesn't work either.

Фон, конечно, в белый превращается, да, но следы фона на объекте остаются. И его цвета меняются.
Да и что потом с белым делать? Ведь сам объект-то не полупрозрачный внутри, просто белый перевести в прозрачность как-то грубо.

Answer 1

The maximum you can do is to overlay one image on another in difference mode, and then apply the result as a mask to the first image. But it won't be what you want.

Answer 2

Here it looks like the background and the object are blended with some kind of alpha channel.
Those. final pixel color C = C1*Alpha+C2*(1-Alpha)
Here C1 is for example an object, and C2 is a background.
According to the condition of the problem, C and C2 are present, if there is also Alpha, then C1 can be calculated as follows:
C1 = (C - C2*(1-Alpha))/Alpha
Those . You can't just subtract one image from another.