CAM16 equation audit

Why audit an appearance-model equation?

Color appearance models connect colorimetry with perceptual attributes under specified viewing conditions. Because their equations are coupled, changing one term can have a consequence that is obscured when the term is quoted by itself. The source paper makes the same point about its own proposal: “changes made to one part of a color appearance model can have unexpected repercussions in other parts.” The practical question is therefore not only whether an equation was transcribed correctly, but also whether the interpretation survives when the surrounding terms are restored.

What the source paper argues

Hellwig and Fairchild revisit how CIECAM02 and CAM16 relate brightness to lightness, and trace the nonlinearity between them to a transcription rather than to a measurement: “the nonlinear relationship between lightness, J, and brightness is an artifact of how the Hunt model was transcribed to CIECAM97s.”

They state the consequence as a thought experiment. Asked to pick the gray card halfway between black and white by lightness, and then again by brightness, an observer picks the same card. CAM16 cannot: “the card that CAM16 predicts to have middle lightness will always be lighter than the card that CAM16 predicts to be middle brightness.”

Replacing that nonlinearity forces a reevaluation of the chroma, colorfulness, and saturation equations, which is where the background-dependence question below comes from. The paper also identifies a limit case in the current formulation: below a background luminance factor of 20, “the chroma of all colors increases, approaching infinity as Y_B approaches zero,” a behavior it attributes to the N_cb term.

That is the setting for this audit. It reproduces a deliberately small subset of those deterministic consequences, includes the proposal’s worse reported fit to the LUTCHI colorfulness data, and says where a calculation about equations stops and a statement about observers would begin.

Normalized brightness

Within one fixed viewing-condition contract, normalized CAM16 brightness is:

Q / Q_white = sqrt(J / 100)

The proposed relation is linear:

Q / Q_white = J / 100

Both map J = 0 to black and J = 100 to white. Inside that interval they differ: at J = 25, CAM16 gives 0.5 while the linear relation gives 0.25; at J = 50, they give approximately 0.707 and 0.5. The calculation shows the consequence of the two definitions. It does not decide which better predicts an observer.

Isolated and coupled background behavior

Holding every other term fixed, the N_cb^0.9 contribution relative to Y_background = 20 reduces to:

isolated factor = (20 / Y_background)^0.18

That is only one factor inside CAM16 chroma. Restoring the other background-dependent terms gives:

C(Y_background) / C(20) =
    (n_ref / n)^0.18
  × [(1.64 - 0.29^n) / (1.64 - 0.29^n_ref)]^0.73
  × (J_ref / 100)^[(z(n) - z(n_ref)) / (2 z(n_ref))]

n = Y_background / 100
z(n) = 1.48 + sqrt(n)

The implementation holds the adapted responses fixed and sweeps reference lightness from J = 10 through 90.

At Y_background = 5, the coupled result stays below the isolated factor. At 1 and 0.1, it crosses that factor as lightness changes. The direction and size of the difference therefore depend on both background and reference lightness. Quoting 2.595× alone would conceal that interaction.

Corrected coefficient and retained tradeoff

Equation 23 was corrected after first online publication. The paper prints the notice directly beneath the equation: “[Correction added on 22nd April 2022, after first online publication. Equation (23) correction has been updated.]” Two forms are therefore in circulation, and a reproduction agrees only when it uses the same one. This implementation pins the corrected form, whose leading coefficient is 43:

M = 43 N_c e_t sqrt(a² + b²)

For N_c = e_t = 1 and a 3-4-5 opponent vector, direct substitution gives 43 × 5 = 215. The test suite asserts that value, so a silent revert to an uncorrected coefficient fails the build rather than shifting every published colorfulness figure at once.

The paper reports the following squared correlations:

The last row matters: the proposal improves the reported brightness and chroma fits but worsens colorfulness on the listed dataset. The paper does not treat that as a defect to be hidden. It argues the proposed colorfulness must scale with the achromatic white signal A_W so that colorfulness and brightness stay in proportion as scene luminance changes—the condition for saturation to remain invariant to luminance level—and concludes on that basis that “the worse performance on the LUTCHI colorfulness data by the proposed colorfulness formulas is permissible.”

These are values reported by the paper, not correlations independently reproduced from observer records in this portfolio.

The straight brightness line is the proposed relation; the curved line is CAM16. In the center panel, the gap and crossing between the isolated term and the coupled range are the finding—not plotting uncertainty. The right panel preserves all three reported comparisons.

What this calculation cannot answer

The CSV is fully regenerable from the published equation module. The tests pin the curve sizes, representative points, coupled endpoints, corrected coefficient, and domain rejections.

The separate Python companion implements both six-correlate forward paths for declared XYZ and viewing conditions, making the formulation comparison usable outside this report’s fixed grid. That software extension does not add perceptual validation. Observer validation would still require suitable observer data and a study designed for that question; extending either numerical grid would not cross that boundary.

Neither this audit nor the companion maps results into CAM16-UCS. That follows the paper’s own limit on how far its proposal has been carried: “The uniform color space CAM16-UCS was not considered in this article and certainly needs to be revised and refit to experimental data given the changes proposed here.”

Source

Luke Hellwig and Mark D. Fairchild, “Brightness, Lightness, Colorfulness, and Chroma in CIECAM02 and CAM16,” Color Research & Application 47 (2022), 1083–1095, doi:10.1002/col.22792.

Equation 23 carries a correction added 22 April 2022, after first online publication. The corrected form is the one implemented here; quotations above are from the corrected article.