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Beer-Lambert law

Chromogen intensity isn't linear in concentration; its logarithm is

Definition
Beer-Lambert says how much light gets through a stained section: I = I₀ · exp(−ε · c · l). I₀ is the light going in and I the light coming out; c is concentration, l path length, and ε the molar extinction coefficient of the absorbing molecule at that wavelength. The negative logarithm, Optical Density (OD), equals ε · c · l, which is linear in concentration.
Intensity is exponential in concentration
Doubling the stain doesn't halve the brightness
Measure in OD
Linear in concentration, additive across stains
I₀ is the empty slide
What the camera records with no absorber
Absorption only, not fluorescence
Different physics, different math

What the conversion to OD changes in practice

Take two neighboring regions on a brightfield IHC slide, one at chromogen concentration c and the other at 2c. On an 8-bit scale the lighter region reads 100 and the darker one reads 25. The naïve average, (100 + 25) / 2 = 62.5, inverts through Beer-Lambert to a concentration that is not 1.5c. It's higher than that, because the exponential weights the darker pixel more heavily.

Convert to OD first. The lighter region has OD = −log(100/255) ≈ 0.41 and the darker one OD = −log(25/255) ≈ 1.01. Their average, (0.41 + 1.01) / 2 = 0.71, is the OD of concentration 1.5c, up to the path-length and extinction-coefficient constants.

The effect isn't small. Averaging RGB intensities instead of OD can be off by 20% or more when concentration varies appreciably across a region, and quantitative IHC scoring routinely relies on per-region averages. The RGB-to-OD conversion engineLoading... is the standard preparation: run it before any region-averaging engine, and the measurements correspond to the biology.

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