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I'm reading this paper and one thing sticks out at me.

Let's say I use the approximate transducer:

R = 54.09288 ·G^0.41850  

Then I scale it:

R̂ = R * l

Then I go back to Ĝ :

Ĝ  =  7.2232 · 10^−5 · R̂  ^2.3895

Combining these 3 calculations, we get:

Ĝ  =  7.2232 · 10^−5 · R̂  ^2.3895
   =  7.2232 · 10^−5 · (R*l)  ^2.3895 
   =  7.2232 · 10^−5 · R  ^2.3895 * l  ^2.3895 
   =  7.2232 · 10^−5 · (54.09288 ·G^0.41850)  ^2.3895 * l  ^2.3895 
   = G *  l  ^2.3895

Here, l ^2.3895 is a constant. So the entire contrast mapping procedure appears to amount to a linear scaling of the physical contrast.

OTOH, in the paper, it says:

Since the contrast response R is perceptually linearized, scaling effectively enhances low physical contrast W, for which we are the most sensitive, and compresses large contrast magnitudes, for which the sensitivity is much lower.

This seems to contradict what I wrote above.

Am I missing something?

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