Opposite of linear light

[wonderer]

Hey mates

I'm not properly a newbie in Photoshop, but I've never been in the need of looking for such a solution.
I'll try to be the most detailed possible.

Layer 1: it's a monotonic color filled layer (no matter if one or more colors)
Layer 2: it's a 50% grey layer in linear light blending mode

I paint on layer 2 with black and white (brush opacity decreased to some decent value to "dodge and burn").
Then I create layer 3 as result of previous ones (shift+alt+ctrl+e).

I guess it's quite clear and easy until this point.

What I need now is:
given layer 3 and layer 2 is there any way to retrieve layer 1?

My attempt was to create layer 4 as an inverted of layer 2 and blend it over layer 3 in linear light.
It looks that it's working for grey tones between 110 and 180 (0-255 scale - 50% grey is 128) while it fails for lighter and darker tones. (this is just an empiric result, it could probably vary according to your image) :banghead:

Is anyone out there aware of what i should do to get the result I'm looking for? :question:
Thanks in advance

 

[raikkonen]

Tell you what, it's impossible generally. Linear light is a combination of linear burn and linear dodge.
"Dodge applies to values lighter greater than middle gray, and burn to darker values" - says Wiki, my love.

Say, we need to blend 255,0,0 (pure red) with 76,76,76 (dark grey).

255,0,0 + 76,76,76 = (Linear Burn) = (255+76-255),(76-255),(76-255) = 76,0,0

Lets blend the result with the inverted second layer. 76,0,0 + 179,179,179 = 255,179,179.

That's not 255,0,0.

 

[wonderer]

First of all thanks a lot for your answer
Your math explanation sounds convincing, but still I need to find a solution to this problem.
Is there any resource which illustrates blending modes math? It would be really useful in order to achieve the result I'm trying to obtain.

 

[raikkonen]

http://en.wikipedia.org/wiki/Blend_modes

UPD: that link is better. 1 in math there is 255.
http://dunnbypaul.net/blends/