Source, Material, and Observer

A useful spectral model keeps the illumination, sample, and observer separate. This allows one measured material spectrum to be evaluated under different sources without confusing a change in lighting with a change in the material. The separation is valid only while the measurement geometry and physical operating conditions remain applicable.

Define the three factors

Let $S(\lambda)$ describe the incident spectral distribution and $H(\lambda)$ the sample’s spectral power throughput. In a simple transmission model, $H=T$; in a compatible reflectance measurement, $H$ represents the measured reflection quantity. For a linear, nonfluorescent response:

Here $H$ acts on spectral power, not the complex electric-field amplitude used in coherent-wave transfer functions. A passive transmitted or collected-power fraction lies between zero and one. Reflectance factors defined relative to a reference geometry need their own interpretation and are not universally bounded by one.

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The observer is represented by a weighting function $\Phi$ or by three color-matching functions. A scalar channel has response:

This integral is linear in the stimulus. It is not a complete model of perceived appearance: adaptation, spatial context, and subsequent nonlinear processing are absent.

Normalize a luminous ratio

Compare the filtered and unobstructed source using the same wavelength limits and photopic weighting:

The ratio expresses the retained luminous contribution in the defined geometry; multiply by 100 for percent. A neutral throughput $H=0.40$ gives 40% for any source with a positive denominator. A selective filter generally gives different results under different sources, even when its measured throughput curve is unchanged.

Preserve one common XYZ scale

For relative object colorimetry, define the reference as $H=1$ and normalize its $Y$ to 100:

The same $k$ multiplies all three integrals. Independently normalizing X, Y, and Z would change the coordinate system. For the CIE 1931 observer, $\bar y=V$ and $Y/100$ equals the luminous ratio over matching limits.

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D65 supplies a defined daylight spectral distribution. The standard 2° and 10° observers represent different color-matching field conditions; they are not a general switch between central and peripheral vision. Keep the illuminant, observer, wavelength coverage, and reference white attached to every result.

Identify where the cascade stops applying

Angle, polarization, temperature, sample condition, and collection geometry can alter $H$. Interchanging sources is useful when those conditions stay fixed and the sample response remains linear. Fluorescence requires coupling excitation wavelengths to emission wavelengths; a single pointwise multiplier cannot describe that conversion.

Measured spectra must also resolve important source and filter structure. Align the wavelength grids, use compatible units, and integrate with the actual interval widths. Missing spectral tails are missing evidence, not demonstrated zeros. The calculator uses an explicitly synthetic passive throughput curve to show the framework; an application decision needs representative measured data and its uncertainty.