An isolated CAM16 background term does not bound the coupled expression

The question

What do the proposed brightness relation and the background-dependent chroma terms actually predict, and which conclusions remain valid when an isolated factor is put back into the coupled expression that contains it?

What was calculated

The experiment has three parts:

1. Compare normalized CAM16 brightness, sqrt(J/100), with the proposed linear relation, J/100, over J = 0…100.
2. Evaluate the isolated N_cb^0.9 background factor relative to Y_background = 20, then evaluate the complete background-dependent chroma expression over J = 10…90.
3. Apply the corrected coefficient in the paper’s colorfulness equation and compare the three fit results reported for both formulations.

This is a numerical equation study using deterministic inputs rather than new measurements.

Results

The two brightness relations agree at black and white but assign different midpoints. CAM16 reaches half normalized brightness at J = 25; the proposed linear relation reaches it at J = 50.

At Y_background = 0.1, the isolated factor is 2.595×, but the coupled expression spans 2.120–2.687× as reference lightness varies from J = 90 to J = 10. The isolated term sits inside that range. It is neither a lower nor an upper bound on the complete expression under this sweep.

The paper reports higher coefficients of determination for brightness (R² = 0.86 to 0.95) and chroma (0.87 to 0.96), and a lower one for colorfulness (0.81 to 0.71).

Left: the square-root and linear brightness relations. Center: the isolated background factor and the coupled range across J = 10…90. Right: the source paper’s published fits, including the colorfulness regression.

double cam16_relative_chroma_fixed_adapted_response(
    double y_background, double reference_j,
    double reference_y_background) {
  require_relative_background(y_background);
  require_relative_background(reference_y_background);
  if (!std::isfinite(reference_j) || reference_j <= 0.0 ||
      reference_j > 100.0) {
    throw std::runtime_error(
        "CAM16 equation audit: reference J must be within (0,100]");
  }

  const double n = y_background / 100.0;
  const double reference_n = reference_y_background / 100.0;
  const double z = 1.48 + std::sqrt(n);
  const double reference_z = 1.48 + std::sqrt(reference_n);
  const double isolated =
      std::pow(reference_n / n, 0.18);
  const double chroma_scale = std::pow(
      (1.64 - std::pow(0.29, n)) /
          (1.64 - std::pow(0.29, reference_n)),
      0.73);
  const double lightness_scale = std::pow(
      reference_j / 100.0,
      (z - reference_z) / (2.0 * reference_z));
  const double result = isolated * chroma_scale * lightness_scale;
  if (!std::isfinite(result)) {
    throw std::runtime_error(
        "CAM16 equation audit: coupled chroma expression overflow");
  }
  return result;
}

Scope of the result

The calculation shows the numerical behavior of the selected equations and why the isolated background term does not describe the complete coupled expression. It uses the corrected coefficient 43 in the paper’s colorfulness relation.

This bounded sweep is not a full forward transform, a standards-conformance test, or an observer experiment. The reported coefficients of determination come from the source paper; the observer datasets were not re-fitted here. Determining which formulation better predicts appearance requires suitable observer data.