**2. Methods**

Three-dimensional graphs are widely used in science communication to demonstrate the relationship between conflicting parameters. The proposed voxelization method goes a step further by discretizing the continuous data of each dimension to create unit voxels (analogous to pixels in 3-D shapes). The size of a voxel can be defined by identifying the acceptability or detectability of the minimum value for each dimension. The minimum identifiable value is often characterized as a just-noticeable difference (JND) in psychophysical studies. It is also possible to convert continuous data to discrete data by selecting arbitrary unit sizes when a JND cannot be identified. Once the dimensions of a unit voxel are identified, they can be plotted in a 3-D graph, as shown in Figure 1.

Key dimensions of the proposed demonstration method for museum studies are damage, light intensity, color quality, and energy efficiency. Damage to artwork can be quantified using the Berlin model [3] or the amount of light absorbed by the painting [19]. In the Berlin model, the damage caused by optical radiation is calculated as a function of effective radiant irradiance.

$$E\_{dm} = \int\_{\lambda} E\_{e,\lambda} \times s(\lambda)\_{dm,rel} \times d\lambda \tag{1}$$

where *Edm* (unit: W/m<sup>2</sup> ) is the effective irradiance that causes damage, *Ee,<sup>λ</sup>* is the spectral irradiance (unit: W/m<sup>2</sup> ), *s*(*λ*)*dm,rel* is the relative spectral responsivity of a material normalized at 300 nm, so that *s*(*λ*)*dm,rel* = 1.0 for *λ* = 300 nm, and *λ* is wavelength (unit: nm) [3]. The alternative damage calculation method is the ratio of the light absorbed by the surfaces under a test light source to the light absorbed by the surfaces under a reference illuminant

$$A = \frac{\int E\_{\varepsilon\lambda, \text{test}}(\lambda) \times (1 - R(\lambda)) \times d\lambda}{\int E\_{\varepsilon\lambda, \text{ref}}(\lambda) \times (1 - R(\lambda)) \times d\lambda} \tag{2}$$

where *A* is a unitless relative absorption value reported as a percentage, *Ee,λ,test*(*λ*) is the test light source irradiance, *Ee,λ,ref*(*λ*) is the reference source irradiance, and *R*(*λ*) is the reflectance factor of a pigment. The test light source *Ee,λ,test*(*λ*) should be rescaled so that the light reflected from the painting under the test and reference light sources are equal. Equalizing the reflected light from the painting under the test and reference light source ensures the luminance is the same in both conditions so that the comparison is not affected by luminance related color appearance phenomena, such as the Hunt Effect [21] and Bezold–Brücke hue shift [22].

Both the Berlin Model and relative absorption calculation method account for the Grotthuss–Draper law, which states that only light that is absorbed can cause photochemical activation. The difference between the two methods is that the relative absorption *A* offers an easy-to-interpret measure for damage, but it does not account for the Planck–Einstein relation (lower wavelength radiation has higher energy potential). On the other hand, the Berlin Model uses a damage curve (action spectra) normalized to 300 nm, which may undermine the Grotthuss–Draper law and can be hard to interpret.

In the proposed voxelization method, the light intensity can also be quantified by using appropriate metrics, such as illuminance (unit: lx) or irradiance (unit: W/m<sup>2</sup> ). Although illuminance is relevant for the human visual system, irradiance can be used to account for the difference between spectral sensitivity of the materials and the spectral luminous efficiency function (visual system's response to light). The color quality of the painting can be quantified using colorimetric tools, such as color rendition metrics, or more precise tools, such as color difference, chroma, and hue shift formulae. Color shift formulae can provide detailed and specific information about the magnitude and direction of color shifts between two lighting conditions. In the following voxelization example, the two lighting conditions will be a reference white illuminant (i.e., daylight and incandescent lamps are

considered ideal in museums for color quality purposes) and a test light source (SPDs generated by a mpLED). incandescent lamps are considered ideal in museums for color quality purposes) and a test light source (SPDs generated by a mpLED). Test SPDs were generated by the linear optimization of a seven-channel mpLED

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Test SPDs were generated by the linear optimization of a seven-channel mpLED lighting system. The spectrum of each channel, shown in Figure 2, were combined by iteratively mixing each channel at 20% dimming intervals, resulting in 279,936 (6<sup>7</sup> ) test SPD combinations. The color differences in the appearance of 24 Macbeth ColorChecker test samples [23] between each test SPD combination and reference incandescent halogen light source were calculated using CAM02-UCS [24]. The root mean square (RMS) of the 24 color difference values (∆*E*'RMS) were calculated to get an average score of the color shifts. lighting system. The spectrum of each channel, shown in Figure 2, were combined by iteratively mixing each channel at 20% dimming intervals, resulting in 279,936 (67) test SPD combinations. The color differences in the appearance of 24 Macbeth ColorChecker test samples [23] between each test SPD combination and reference incandescent halogen light source were calculated using CAM02-UCS [24]. The root mean square (RMS) of the 24 color difference values (Δ*E*'RMS) were calculated to get an average score of the color shifts.

**Figure 2.** The spectral power distribution of the seven-channel multi-primary LED system is used in the linear optimization to generate data that are analyzed for the proposed 3-D representation of metric trade-offs. **Figure 2.** The spectral power distribution of the seven-channel multi-primary LED system is used in the linear optimization to generate data that are analyzed for the proposed 3-D representation of metric trade-offs.

An incandescent halogen light source spectrum was used as a reference since they are still widely used in museums [25]. Macbeth ColorChecker samples include a range of saturated, desaturated, chromatic, and achromatic samples, which can be representative of a wide range of artwork, and it is widely used in color and museum lighting research [26,27]. The color quality of every nominal white light was quantified using an ANSI/IES TM-30 fidelity index *R*f, a gamut index *R*g, and a local chroma shift in hue bin 1 (*R*cs,h1) [28]. In addition, relative absorption *A* (light absorbed by a pigment under the test light source divided by the light absorbed by a pigment under the reference halogen light source), illuminance (*E*v), irradiance (*E*e), the luminous efficacy of radiation (LER), correlation color temperature (CCT), and the distance from the Planckian locus (Duv) [29] were calculated An incandescent halogen light source spectrum was used as a reference since they are still widely used in museums [25]. Macbeth ColorChecker samples include a range of saturated, desaturated, chromatic, and achromatic samples, which can be representative of a wide range of artwork, and it is widely used in color and museum lighting research [26,27]. The color quality of every nominal white light was quantified using an ANSI/IES TM-30 fidelity index *R*<sup>f</sup> , a gamut index *R*g, and a local chroma shift in hue bin 1 (*R*cs,h1) [28]. In addition, relative absorption *A* (light absorbed by a pigment under the test light source divided by the light absorbed by a pigment under the reference halogen light source), illuminance (*E*v), irradiance (*E*e), the luminous efficacy of radiation (LER), correlation color temperature (CCT), and the distance from the Planckian locus (Duv) [29] were calculated for each test SPD.

### for each test SPD. **3. Results**

ments.

**3. Results**  The data generated by the linear optimization method have been sorted and analyzed for metric correlation. Since most of the LED combinations (236,502 out of 279,936) were not nominally white, color quality metrics that require a test light source to be close to the Planckian locus (i.e., *R*f, *R*g, CCT, Duv) were not used in the analysis. However, it is pos-The data generated by the linear optimization method have been sorted and analyzed for metric correlation. Since most of the LED combinations (236,502 out of 279,936) were not nominally white, color quality metrics that require a test light source to be close to the Planckian locus (i.e., *R*<sup>f</sup> , *R*g, CCT, Duv) were not used in the analysis. However, it is possible to filter out the non-white SPD combinations to utilize color quality metrics that are developed for white lights, with a caveat of reduced damage reduction for individual pigments.

sible to filter out the non-white SPD combinations to utilize color quality metrics that are developed for white lights, with a caveat of reduced damage reduction for individual pig-The data generated by the optimization method were voxelated using the most important measures for museum lighting: damage to artwork, color appearance, and

efficiency. For example, a 3-D voxelated volume (*VV*1) was calculated using the relative

energy efficiency. For example, a 3-D voxelated volume (*VV*1) was calculated using the relative absorption for a test color sample (Macbeth ColorChecker sample #24), the RMS color difference of 24 Macbeth ColorChecker samples ∆*E*'RMS, and the LER, as shown in Figure 3. Measures in each dimension were discretized by rounding values to a unit size of 1 (e.g., LER of 200.3 lm/W and 200.7 lm/W were rounded to 200 lm/W and 201 lm/W, respectively, and they fell into two different voxels). All the test SPD combinations that fell into the same voxel were considered identical. Therefore, the number of unique voxels (*VV*1 = 45,813) represents the number of unique SPD combinations that can be generated within the seven-channel mpLEDs. It is important to note that the uniqueness of each voxel depends on the voxel size criteria. For example, if the LER was voxelated using 5 lm/W as the unit voxel size, the number of voxels would drastically decrease. Therefore, the absolute magnitude of the volume does not have an inherent meaning. ference of 24 Macbeth ColorChecker samples Δ*E*'RMS, and the LER, as shown in Figure 3. Measures in each dimension were discretized by rounding values to a unit size of 1 (e.g., LER of 200.3 lm/W and 200.7 lm/W were rounded to 200 lm/W and 201 lm/W, respectively, and they fell into two different voxels). All the test SPD combinations that fell into the same voxel were considered identical. Therefore, the number of unique voxels (*VV*1 = 45,813) represents the number of unique SPD combinations that can be generated within the seven-channel mpLEDs. It is important to note that the uniqueness of each voxel depends on the voxel size criteria. For example, if the LER was voxelated using 5 lm/W as the unit voxel size, the number of voxels would drastically decrease. Therefore, the absolute magnitude of the volume does not have an inherent meaning.

absorption for a test color sample (Macbeth ColorChecker sample #24), the RMS color dif-

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**Figure 3.** The 3-D voxelated volume *VV*1 shows the trade-offs between damage to artwork (light absorption ratio of test light source to reference), color quality (shifts in the appearance of 24 Macbeth ColorChecker samples between test SPDs and the reference light source), and energy efficiency of the light source (luminous efficacy of radiation; unit: lm/W). Voxels are shown as circles **Figure 3.** The 3-D voxelated volume *VV*1 shows the trade-offs between damage to artwork (light absorption ratio of test light source to reference), color quality (shifts in the appearance of 24 Macbeth ColorChecker samples between test SPDs and the reference light source), and energy efficiency of the light source (luminous efficacy of radiation; unit: lm/W). Voxels are shown as circles for representation purposes only.

The data departed from normality at the 0.05 significance level as tested by the Shapiro–Wilk test, and a non-parametric test (Spearman's rank correlation coefficient) was used to analyze the correlation between the dimensions of the voxelated *VV*1 volume. The data departed from normality at the 0.05 significance level as tested by the Shapiro– Wilk test, and a non-parametric test (Spearman's rank correlation coefficient) was used to analyze the correlation between the dimensions of the voxelated *VV*1 volume. While the correlation between absorption *A* and ∆*E*'RMS were low (*ρ* = 0.111), the LER was inversely correlated to absorption (*ρ* = −0.757) and ∆*E*'RMS (*ρ* = −0.427).

While the correlation between absorption *A* and Δ*E*'RMS were low (*ρ* = 0.111), the LER was inversely correlated to absorption (*ρ* = −0.757) and Δ*E*'RMS (*ρ* = −0.427). Figure 3 illustrates the relationship between color quality, damage, and energy efficiency, where the top far corner is the ideal condition (low absorption, small color shifts, and high efficacy). The visual illustration makes it clear that the ideal SPDs are increasingly scarce compared to other SPDs that perform worse in terms of either damage, color shifts, or energy efficiency. Since the relative absorption *A* > 100 denotes additional dam-Figure 3 illustrates the relationship between color quality, damage, and energy efficiency, where the top far corner is the ideal condition (low absorption, small color shifts, and high efficacy). The visual illustration makes it clear that the ideal SPDs are increasingly scarce compared to other SPDs that perform worse in terms of either damage, color shifts, or energy efficiency. Since the relative absorption *A* > 100 denotes additional damage, and the large color differences are not desired, it is possible to zoom into the graph by limiting the *x* and *y* axes (relative absorption (*A* < 100) and color difference (∆*E*'RMS < 20), respectively), as shown in Figure 4.

age, and the large color differences are not desired, it is possible to zoom into the graph

sample (Macbeth ColorChecker sample #24), the TM-30 fidelity index *R*f, and the irradiance *E*e, as shown in Figure 5. In the second volume, which is the graphical representation of the same data, there were *VV*2 = 2,265 unique voxels. While the absolute volume size does not have an inherent meaning, comparing two or more light sources—using identical

A second example (*VV*2) was calculated using relative absorption *A* for a test color

for representation purposes only.

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**Figure 4.** Limiting the *x* and *y* axes of the voxelated 3-D volume *VV*1 can highlight areas of interest, such as low relative absorption (*A* < 100) and color difference (Δ*E*'RMS < 20). Voxels are shown **Figure 4.** Limiting the *x* and *y* axes of the voxelated 3-D volume *VV*1 can highlight areas of interest, such as low relative absorption (*A* < 100) and color difference (∆*E*'RMS < 20). Voxels are shown as circles for representation purposes only.

voxel dimension metrics—can provide more information about the performance of the light sources for a specific set of pigments (or the overall color quality of a painting).

A second example (*VV*2) was calculated using relative absorption *A* for a test color sample (Macbeth ColorChecker sample #24), the TM-30 fidelity index *R*<sup>f</sup> , and the irradiance *E*e, as shown in Figure 5. In the second volume, which is the graphical representation of the same data, there were *VV*2 = 2,265 unique voxels. While the absolute volume size does not have an inherent meaning, comparing two or more light sources—using identical voxel dimension metrics—can provide more information about the performance of the light sources for a specific set of pigments (or the overall color quality of a painting). **Figure 4.** Limiting the *x* and *y* axes of the voxelated 3-D volume *VV*1 can highlight areas of interest, such as low relative absorption (*A* < 100) and color difference (Δ*E*'RMS < 20). Voxels are shown as circles for representation purposes only.

as circles for representation purposes only.

**Figure 5.** Another 3-D voxelated volume *VV*2 shows the trade-offs between damage to artwork (light absorption ratio of test light source to reference), color quality (ANSI/IES TM-30 fidelity index *R*f), and illumination intensity (irradiance *E*e). Voxels are shown as circles for representation purposes only. **Figure 5.** Another 3-D voxelated volume *VV*2 shows the trade-offs between damage to artwork (light absorption ratio of test light source to reference), color quality (ANSI/IES TM-30 fidelity index *R*<sup>f</sup> ), and illumination intensity (irradiance *E*e). Voxels are shown as circles for representation purposes only.

Voxelated 3-D volume *VV*2 also shows the relationship between competing target

0.010) or irradiance *E*e (*ρ* = 0.017). However, irradiance *E*e and relative absorption *A* were

Voxelated 3-D volume *VV*2 also shows the relationship between competing target parameters. The fidelity index *R*<sup>f</sup> was not correlated with either relative absorption *A* (*ρ* = 0.010) or irradiance *E*<sup>e</sup> (*ρ* = 0.017). However, irradiance *E*<sup>e</sup> and relative absorption *A* were highly correlated (*ρ* = 0.996), which is not surprising since absorption increases with light intensity.

It should be noted that the graphical distributions were applied to a single pigment using relative absorption and to multi-pigments (e.g., a painting with numerous colors) using color quality metrics to demonstrate the different use cases of the proposed method. The proposed model can provide a more analogous analysis if all the dimensions are chosen at the individual pigment scale (e.g., absorption for a single color and color difference in the pigment under reference and test light sources). On the other hand, the proposed model can also be used to gain a holistic understanding of a painting by using a high-level approach (e.g., average absorption ratio by the pigments used in a painting, the average color difference of pigments in the painting under reference and test light sources). While the provided examples do not include the exposure time—an important dimension of damage to artwork—it is possible to incorporate the total radiant exposure as a metric to the voxelization method. The total exposure can be calculated by multiplying exposure time *<sup>t</sup>* (unit: hr) with irradiance (*E*<sup>e</sup> <sup>×</sup> *<sup>t</sup>*, unit: W h/m<sup>2</sup> ) or illuminance (*E<sup>v</sup>* × *t*, unit: lx h). Alternatively, the radiant exposure can be quantified using the Berlin model (*H*dm, unit: W h/m<sup>2</sup> ) [3].
