Mathematical model

Background

The Light Damage Calculator estimates the fading of colourants on heritage objects exposed to light by combining photochemical fading kinetics with optical colour theory. For each colourant type, the database contains initial CIE XYZ tristimulus values (X_o, Y_o, Z_o) and sensitivity coefficients (K_UV, K_noUV) that characterize fading rates with and without ultraviolet (UV) radiation. These coefficients were derived from published lightfastness ratings, where one just-noticeable difference (JND) corresponds to a colour difference (ΔE) of approximately 1.7 (ISO 105-A02:1993, grey scale step 4). The mathematical model, as described on this page, employs the Kubelka-Munk theory, which relates colourant concentration to optical properties to predict how the loss of colourant through photochemical degradation affects an object’s colour. This approach allows estimation of colour changes even at high light doses, where significant fading occurs. The model assumes first-order kinetics for the photochemical degradation of colourants, where the rate of fading is proportional to the fraction of colourant remaining (c_t). It also assumes that optical properties scale linearly with colourant concentration. While the model provides reasonable estimates across a wide range of exposures, it does not currently account for more complex object behaviours at high light doses, such as darkening or hue shifts. In its present form, the model treats the colour of all objects as fading monotonically toward white.

Calculations

Light dose

Light exposure is quantified as the total amount of light energy that reaches a surface over time. This cumulative exposure, known as the light dose, helps determine the extent of photochemical damage, given the object’s responsivity to light. While instantaneous light levels are measured in lux (lumens per square meter), the total light dose is expressed in lux hours (lx·h), which combines intensity and time. Light dose is calculated separately for exhibit lighting and for flash photography as

Where

Because the working values are typically large, both dose components (D_lighting and D_flash) are divided by 106 to convert them into megalux hours (Mlx·h). The two dose components are then summed to give the total future dose (D_f):

The prior dose for a colourant with existing light damage is approximated as

Where

D_p = prior dose
c_p = prior fade fraction
K = sensitivity coefficient

Values of D_p are calculated separately for K with UV exposure (K_UV) and without UV exposure (K_noUV) with the user input of prior fade fraction (c_p).

Fraction of initial colourant

The model assumes first-order kinetics for the photochemical degradation of colourants, meaning the instantaneous fading rate is proportional to the amount of colourant remaining. This leads to an exponential decay in the fraction of colourant (c_t), with increasing light dose as

The sensitivity coefficient (K) determines how quickly this decay occurs, with separate values K_UV and K_noUV to reflect the higher fading rate when UV radiation is present in the light source.

Reflectance of the faded colourant

The Kubelka-Munk theory provides a way to relate an object’s optical properties (meaning how it absorbs and scatters light) to its composition. In this model, it connects the concentration of remaining colourant to the object’s reflectance. While the full theory accounts for both absorption and scattering coefficients, this implementation assumes that these properties scale proportionally with colourant concentration, leading to a simplified relationship.

The transformation begins with the measured colour of the object, expressed as CIE XYZ tristimulus values, and converts it to reflectance values. The equations solve for how the reflectance changes as the colourant concentration decreases during fading. Separate calculations for X, Y and Z components maintain the relationships between these coordinates as the colour changes.

The following equations outline the transformation of X, Y and Z into the corresponding reflectance value (R). Then the Kubelka-Munk equation relating reflectance to colourant concentration is solved, using the solution for the positive root of the quadratic equation:

Helper variables are used for calculating B_X as

and similarly for B_Y

and B_Z

In the above equations, values of X_ref, Y_ref, Z_ref are tristimulus values of the reference white illuminant. The Light Damage Calculator uses values for CIE D65/2° viewing conditions, giving

Reflectance to tristimulus values (CIE XYZ)

The calculated reflectance values for the faded colourant are converted back to tristimulus values using the reference white illuminant as

These are calculated for the following conditions:

• Current (based on prior exposure selection)
• Future, with UV (using K_UV)
• Future, no UV (using K_noUV)

In the following sections, these colours are transformed to sRGB for rendering swatches and CIE L*a*b* values for calculating colour difference.

CIE XYZ to sRGB conversion

To display the calculated colours on computer screens, the CIE XYZ values must be converted to sRGB coordinates, the standard colour space for digital displays. The calculator renders colour swatches to visualize the initial colour and fading behaviour of colourants. One exception is the ISO Blue Wool standards, which are rendered in grey scale to align with the ISO 105-A02:1993 method for assessing colour difference, even though the actual samples are different shades of blue. This neutral presentation emphasizes the magnitude of the difference rather than the hue, helping users focus on the degree of fading rather than the specific appearance of the colour.

This involves two steps: first, a matrix transformation that maps the CIE XYZ values to linear sRGB values, then, a non-linear gamma correction that accounts for how human vision and computer displays respond to different light intensities. To start, the CIE XYZ values are normalized as

and these values are then transformed by the following matrix operation:

Gamma correction is then applied with separate equations for very dark colours (using a linear response) and brighter colours (using a power function):

The resulting sRGB values are converted to an 8-bit integer range (0 to 255) for digital output.

CIE XYZ to CIE L*a*b* conversion

The CIE L*a*b* colour space was designed to be more perceptually uniform, meaning that equal perceived distances in the space correspond to roughly equal perceived differences in colour. This makes it particularly suitable for calculating visible colour differences. The L* coordinate represents lightness (from black to white), while a* and b* represent colour dimensions: red-green and yellow-blue, respectively.

The conversion from CIE XYZ involves non-linear transformations that mimic the eye’s response to light intensity. The cube root functions approximate how we perceive brightness differences, while the constants ε and κ handle very dark colours, where human vision becomes less sensitive to small differences. Conversion to CIE L*a*b* is provided though the following series of functions:

These equations use helper functions (f_x, f_y, f_z) that handle the non-linear response of human vision to brightness:

The tristimulus values are scaled relative to the reference white illuminant to account for viewing conditions (CIE D65/2°) as follows:

Within these equations, the constants ε and κ provide a different calculation path for very dark colours, where human vision becomes less sensitive to brightness differences:

Colour difference

A simple colour difference formula (ΔE₇₆) calculates the geometric distance between two colours in the CIE L*a*b* space. While newer formulas, such as CIEDE2000 (ΔE₀₀), exist that better match human perception, the original ΔE₇₆ formula is currently used for its simplicity and reasonable accuracy. Colour difference is determined from a pair of CIE L*a*b* values as

Where

A value of approximately 1.7 corresponds to a JND under good viewing conditions, providing a practical benchmark for assessing the significance of colour differences.