Spectral-fidelity calculations

This companion explains the numerical core used by the spectral-sensitivity study. The published code starts after archive ingest: wavelength axes have already been aligned, sensitivity curves selected, chart reflectances paired, and measured RGB prepared. It does not read RAW files, discover sessions, or decide which historical records belong together.

Study · Report · Aggregate data · Validation controls

Data flow

aligned SSF + illuminant + reflectance + paired RGB
    └─ spectral_closure → white-card gate → one global scale → RMS/correlation

aligned SSF + CIE color-matching functions
    └─ spectral_quality → normalized subspace fit → Luther residual → QI

aligned SSF + illuminant + reflectance + CIE functions
    └─ spectral_smi → synthetic RGB/XYZ → fitted 3×3 → ΔE76/CIEDE2000 → SMI

These branches share spectral inputs but do not measure the same property.

Physical closure

Implementation: spectral_closure.hpp and spectral_closure.cpp.

Inputs

The module requires finite vectors on one strictly increasing, uniformly spaced wavelength grid. Every sensitivity channel and illuminant has one value per wavelength; every patch has one reflectance vector and one measured RGB triplet. At least one patch and two wavelength samples are required.

Uniform spacing is an operating condition, not a cosmetic preference. The implementation uses equal sample weights. The common step width would multiply every prediction equally and is absorbed by the fitted exposure scale, but a nonuniform grid would require explicit integration weights and is rejected.

White-card pairing gate

For a flat perfect diffuser, the predicted channel response is

Wc = Σλ Sc(λ) E(λ)

Measured and predicted R/G and B/G ratios are compared. Their largest relative disagreement must not exceed the declared gate. Closure stops after a failure because fitting the chart under an unconfirmed illuminant pairing would produce a precise residual for the wrong experiment.

One global exposure scale

For each patch and channel,

Pp,c = Σλ Sc(λ) E(λ) Rp(λ)

The exposure scale minimizes squared error over the complete patch-by-channel set:

k = Σp,c Mp,c Pp,c / Σp,c Pp,c²

The implementation rejects a non-representable or zero-energy denominator. It then reports, per channel,

relative RMSc = sqrt(meanp((Mp,c − kPp,c)²)) / |meanp(Mp,c)|

and Pearson correlation across patches. The mean measured response must be nonzero because the relative RMS would otherwise be undefined.

A per-channel scale is calculated for diagnosis only. The reported predictions always use the one global k; three fitted scales would remove channel balance from the residual.

Luther-condition quality

Implementation: spectral_quality.hpp and spectral_quality.cpp.

Let A contain the sampled red, green, and blue sensitivities as columns, and let q be one sampled CIE color-matching function. Each input column and target is first divided by its L2 norm. The least-squares system is

G = AᵀA
β = G⁻¹Aᵀq
r(q) = ||q − Aβ||₂ / ||q||₂

The code solves the 3×3 Gram system with Cramer’s rule and rejects a basis whose determinant is too small relative to the product of its diagonal terms. At least four wavelengths are required so the three-parameter fit is overdetermined.

The combined residual and quality index are

rcombined = sqrt((rx² + ry² + rz²) / 3)
QI = 1 − rcombined

Because the sensitivity columns are normalized, rcombined is invariant to nonzero positive scaling of one or all channels. The invariant object is the normalized subspace metric—not the camera’s physical amplitude response, SMI, or closure residual.

ISO 17321-style SMI

Implementation: spectral_smi.hpp and spectral_smi.cpp. It reuses the published colorimetry implementation for the 3×3 fit, CIELAB conversion, and color differences.

Spectral synthesis

Unlike closure, this branch accepts a nonuniform grid and uses trapezoidal weights. For weight wλ, illuminant E, reflectance R, sensitivity S, and CIE color-matching function C,

camera channel = Σλ wλ E(λ) R(λ) S(λ)
reference XYZ  = Σλ wλ E(λ) R(λ) C(λ)

The perfect-diffuser reference is scaled to Y = 100 and supplies the CIELAB white. The camera and target patch sets then enter an unconstrained 3×3 least-squares fit. A second fit constrains camera white to map exactly to the reference white, exposing how much that condition changes the result.

Scores

The primary score is

SMI = 100 − slope × mean ΔE76

with declared slope 5.5. CIEDE2000 statistics are evaluated on the same fitted patches but remain separate diagnostics. They cannot be substituted into the SMI formula.

This is an ISO 17321-style implementation, not a bit-exact Annex-B equivalence. Chart selection, illuminant, wavelength grid, matrix constraint, and optimization convention all belong with the result.

Invalid inputs

The public modules refuse conditions under which their result would be undefined or misleading:

The closure white-card mismatch is different: it returns an unresolved result with no patch predictions because the failed pairing is a scientific outcome, not malformed input.

What the public tests establish

• test_spectral_closure.cpp recovers an exact global scale, proves the white gate precedes the fit, forces a visible channel imbalance, and exercises the invalid-input cases.
• test_spectral_quality.cpp uses one observer target outside the sensitivity span, pins the exact combined residual, checks scaling invariance to 1e-12, and rejects invalid bases and vectors.
• test_spectral_smi.cpp compares an exactly colorimetric synthetic camera with a wavelength-shifted counterexample, pins the score definition, checks white preservation, and exercises the declared rejections.

These are library-level tests on synthetic arrays. They establish what the published numerical functions compute. They do not validate command-line orchestration, private archive selection, monochromator behavior, or the physical accuracy of a retained sensitivity curve.