Auditing a color-appearance equation before trusting it

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. Pin the corrected coefficient in the paper’s colorfulness equation and retain all six published fit statistics, including the unfavorable one.

The inputs are deterministic equation values. No camera, display, printer, or observer was measured in this study.

What the audit found

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.

The background experiment exposes the more important lesson. 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’s reported fits also resist a one-directional summary: brightness improves from R² = 0.86 to 0.95 and chroma from 0.87 to 0.96, while colorfulness declines from 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;
}

What this establishes—and what it does not

The audit establishes the numerical behavior of these declared equations and shows why a component should not be interpreted as the complete model. It also checks that the implementation carries the corrected coefficient 43 in the paper’s colorfulness relation.

The bounded audit itself is not a full forward transform, a standards- conformance test, or an observer experiment. A separate Python companion evaluates both six-correlate forward formulations for caller-supplied XYZ and viewing conditions; that makes the equations reusable but does not change what the audit establishes. The published correlations are retained source-paper values; the underlying observer datasets were not re-fitted here. A perceptual conclusion would require suitable observer data, not a larger equation sweep or another model output table.