Forgive the cryptic title. I couldn't get a decent explanation into the character limit and I thought this title would get some of the right people here.

I want to understand the following workflow. This is to help me get to grips with channels, and is the first of two hard questions

Say you have two photos of the same size. Import them into one document on separate layers. Make the top photo an arbitrary transparency, say 40%. You can now see a blend of the two photos. Export to .png or .bmp and discard the GIMP file. Delete one of the original photos. You now have your merged photo (in a lossless format) and only one of the originals, plus your knowledge about how the merge was made.

How can you use the remaining photo to recreate the other photo from the merged photo?

Can't be done. The merged photo contains only 60% of the original information about the bottom photo and if you have only the top photo available for reconstruction then the missing 40% is not recoverable. Likewise the merged photo contains only 40% of the info from the top photo and if the bottom photo is retained then there is no way to recover the missing 60%.

See http://en.wikipedia.org/wiki/Alpha_compositing

The channel values of a given pixel in the final photo is:

Cf: final result, Co: the picture over, Cu: the picture under, a: the opacity of the top picture.

So, if you want to recover Cu, knowing Cf and Co (typically, a watermarked picture and the watermark):

However,

- AFAIK Gimp isn't working in linear light but in some gamma-corrected color space so this is a bit theoretical,
- It's not that easy to implement with just layers and layers modes and avoiding round-off errors and value overflows.

... but I tried a long while ago, and it sort of works...

I stand corrected. (As the saying goes, a sure way to get something done is to claim that it is impossible.)

Thank you, this is a great start. I'm having difficulty getting my head round what I do with the number(s) I would get though. If I get a value for C_u for the Red channel, I think value for the Green and Blue channels will be the same, since the same thing was done to all channels.

So I'd import the remaining photo over the merged photo... get my C_u number ready... but then what? We're well into Maths territory. I've tried to understand the Wikipedia article, but it seems to be written for mathematicians rather than for me

I think that because we're just using 256 values per channel (8bit), the reversal procedure should be perfect, right?

Thank you again

In practice the final image is the assembly of the red/green/blue image, each obtained by stacking the red/green/blue versions the background and overlay images with the same opacity(*). So you can really do all the computation channel by channel.

Having only 256 values is a major problem, because it leads very quickly to round-off errors, and if you implement this using layer stacking you can't even have intermediate values outside of the range.

(*) which is somewhat a restriction of the Gimp/Photoshop data model, since actual materials do not have the same opacity at all wavelengths.

Would things be better working in 16bit and exporting to 8bit?

At least you reduce the roundoff errors.