Computing the CAM16 brightness and background comparison

Scope

This module evaluates normalized brightness, the isolated N_cb^0.9 term, one bounded coupled background expression, and the corrected colorfulness coefficient. Its inputs are already-computed correlates and relative background values; it does not perform chromatic adaptation or return a full CAM16 appearance specification.

The standalone Python comparator supplies six-correlate forward J, Q, C, M, s, h paths for standard CAM16 and the Hellwig–Fairchild 2022 proposal when a full XYZ-to-appearance calculation is needed.

Calculation flow

input J and relative background values
              │
              ├── normalized brightness: sqrt(J/100) and J/100
              │
              ├── isolated background factor: (Yb_ref/Yb)^0.18
              │
              └── coupled background expression
                    ├── n-dependent base
                    ├── z(n) = 1.48 + sqrt(n)
                    └── reference-lightness exponent

The calculation evaluates 21 brightness points (J = 0…100 in steps of five), eight backgrounds, and nine positive reference-lightness values for each background. This produces 72 coupled points.

Operating conditions

The isolated factor is defined only for finite relative backgrounds in (0,100]. The coupled expression additionally requires reference lightness in (0,100]; zero is rejected because it appears as the base of a generally non-zero exponent. Normalized brightness accepts finite J in [0,100].

The coupled sweep holds adapted responses fixed. Its result applies to this controlled isolation of the background-dependent terms, not to arbitrary XYZ stimuli or viewing conditions.

Corrected colorfulness relation

The corrected relation is:

M = 43 N_c e_t hypot(a, b)

N_c and e_t must be finite and non-negative; the opponent coordinates must be finite. hypot avoids the unnecessary intermediate overflow risk of forming a² + b² directly. A non-finite result is rejected.

Numerical checks

The checks cover:

• the different brightness midpoints at J = 25 and 50;
• exact isolated factors at four background values;
• the two endpoints of the Y_background = 0.1, J = 10…90 coupled range;
• the fact that 2.595× lies inside, rather than bounds, that range;
• the coefficient 43 using a 3-4-5 opponent vector;
• 21 brightness, 8 isolated, and 72 coupled report points;
• all six source-paper performance values; and
• rejection of invalid domains and non-finite inputs.

These checks cover the numerical implementation. Perceptual validation requires observer data and is outside this calculation.