How do I "reverse" a gradient transparency?

I'd like to know how to restore the area of an image that has been put into a gradually fading transparency.

I have seen many tutorials around on how to fade out the borders of an image, but what if I start with a picture with a fading border and want to reverse the process back into a fully opaque picture?

Examples:

With the exception of the very edges of the picture which I believe are unrecoverable, there's still information there that can be used to reverse most of the transparency no?

Any tips are appreciated.

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