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The influence of color interreflections on lighting simulations.

1 INTRODUCTION

Most lighting simulation programs represent color as combinations of red, green, and blue. Users specify colors as RGB triplets, the programs perform separate radiative flux transfer calculations for each color band, and the results are displayed on color monitors with RGB primaries.

In mathematical terms, these programs very coarsely sample the spectral power distributions of the light sources and the spectral reflectance and transmittance distributions of the surface materials. It could be argued that this approach is justified on the grounds that the human visual system is broadly responsive to red, green, and blue light (as shown by the CIE color matching functions [bar.r], [bar.g], and [bar.b]), and that photometric quantities such as illuminance and luminance are based on these responses.

We assume then that these lighting calculations, and in particular interreflections between colored surfaces, can be accurately calculated using three separate color bands. It has not however been previously demonstrated that this assumption is correct.

2 SUMPNER'S PRINCIPLE

Imagine an infinitely large room with an infinite array of direct lighting. (We can think of this as a luminous ceiling.) The light emitted by the luminaires will directly illuminate the floor. Of this light, 20 percent will be reflected upwards to the ceiling, while the rest will be absorbed. The ceiling will reflect 20 percent of the indirect light towards the floor, which will reflect 20 percent back towards the ceiling and so on, until all of the light is absorbed.

[FIGURE 1 OMITTED]

Calculating the total illuminance ET of the floor is simple. Given a surface reflectance p and direct illuminance ED due to the luminaires, it is:

[E.sub.T] = (1+[rho]+[[rho].sup.2]+[[rho].sup.3]+...)[E.sub.D] = [E.sub.D]/(1-[rho]) (1)

In mathematical terms, this is a MacLaurin series expansion for 0[less than or equal to][rho][less than or equal to]1. It is also the basis of Sumpner's Principle (Cuttle 1991), which estimates mean room surface illuminance. (This equation does however accurately describe the sur-face illuminance of an integrating sphere.)

For our example of [rho] = 0.20, this becomes [E.sub.T] = 1.25 [E.sub.D]. In other words, 25 percent of the total illuminance of the floor is due to interreflected light between the floor and the ceiling. As the reflectance r is increased, more and more of the total illuminance ET is due to light interreflected from the room surfaces.

This begs the question, however. We refer to both the gray and red surfaces as having reflectances of 20 percent, but what does this really mean?

3 REFLECTANCE AND COLOR

ANSI/IES RP-16 (IES 2010) defines the reflectance of a surface as "the ratio of the reflected flux to the incident flux," where flux may be either radiant flux (measured in watts) or luminous flux (measured in lumens). This seems simple enough, but we also have the definition of radiant flux:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

and luminous flux: as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

in terms of spectral radiant flux [[PHI].sub.e,[lambda]] (measured in watts per nanometer), where [K.sub.m] = 683 lumens per watt for photopic vision.

What this implicitly says is important:

The reflectance of a colored surface depends on the spectral power distribution of the illuminant.

As an example, suppose we measure the reflectance of a red surface with the spectral reflectance distribution shown in Fig. 1 as being 20 percent when illuminated by [D.sub.65] daylight. If we illuminate the same surface with quasimonochromatic blue light from a 465 nm LED however, the measured reflectance is less than one percent.

[FIGURE 2 OMITTED]

Equation 1 assumes that the reflectance r is constant for each reflection. This however is true only for gray surfaces. Each reflection from a colored surface modifies the SPD of the incident light. In colloquial terms, the light reflected from a red surface becomes redder with each subsequent reflection (Fig.2).

Equation 1 therefore needs to be generalized to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

for colored surfaces, where [E.sub.D,[lambda]] is the spectral direct irradiance.

It is thus evident that the horizontal illuminance of the red room will be less than that of the gray room with a [D.sub.65] illuminant. We now have to address the issue of calculating these values.

4 CALCULATING COLOR

Professional lighting designers think strictly in terms of "white light" when performing lighting calculations, with no thought given to the SPD [[PHI].sub.1] of the natural or electric light sources. This is unfortunately all but mandated by their lighting simulation programs, which model color as RGB triplets.

ITU Recommendation BT.709 (ITU 2002) specifies the CIE 1931 xy chromaticities of the primary colors and [D.sub.65] white point of high-definition television monitors and computer displays as:
Color                  x        y

Red                   0.640    0.330

Green                 0.300    0.600

Blue                  0.150    0.060

White([D.sub.65])    0.3127   0.3290


where the white point is defined as the chromaticity generated by additively mixing the three primary colors at maximum intensity. Lighting simulation programs therefore enable users to specify material and light source colors as BT.709 red-green-blue (RGB) triplets or equivalent CIE 1931 XYZ color space values according to the transformation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where k is a scaling constant.

By default, lighting simulation programs assume that all "white" light sources have a correlated color temperature (CCT) of 6500 Kelvin. They further perform their calculations for direct and interreflected light using three color bands: red, green, and blue.

[FIGURE 3 OMITTED]

Referring to 4, this becomes:

[E.sub.T] = [K.sub.m](([E.sub.D.R]/(1-[[rho].sub.R]))+([E.sub.D.G]/(1-[[rho].sub.G]))+([E.sub.D.B]/(1-[[rho].sub.B])) (6)

where the subscripts R, G, and B indicate the respective values for the three color bands.

This represents a very coarse discretization of the SPD and SRD. The question is whether this discretization results in significant errors in calculating photometric quantities for architectural environments.

We can estimate these errors by increasing the resolution of the SPD discretization and using:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

for [DELTA][lambda]= 10 nm over the visible spectrum. We also however need to know or estimate the SRDs of common architectural materials so that we can discretize them as well.

5 SPECTRAL RECONSTRUCTION

There is a large body of multispectral imaging literature devoted to the reconstruction of SRDs from sparsely sampled spectra. While the techniques vary, the basic idea is to measure the SRDs of many different materials with a high spectral resolution and store a representation of them in a database. Given the sparsely sampled SRD of an unknown material, its high resolution SRD can be interpolated from the SRDs in the database whose sparsely sampled values most closely match those of the unknown material.

Fairman and Brill (2004) presented a technique based on principal components analysis (PCA) of 3,534 measured SRDs for color samples from the Munsell Book of Color, the Swedish Natural Color System and the OSA-UCS color atlas. Given any set of ITU-R BT.709 RGB triplets whose chromaticity is within the intersection of the color gamuts of the BT.709 primaries and these color samples, a reasonably accurate SRD can be reconstructed for any AX using only the mean and first three principal components of the dataset for an assumed illuminant (such as CIE D65).

The mean and first three principal components of Fairman and Brill's dataset (Fig. 3 and Fig. 4) may appear remarkably smooth in comparison to the illuminant SPDs, but they are consistent with numerous other studies. Maloney (1986) noted that there are strong molecular constraints on the variability of SRDs for natural materials (including wood, paper, minerals, and biological pigments), while Westland and others, (2000) reported a band limit for natural and man-made surfaces of approximately 0.015 to 0.020 cycles per nanometer. They further reported (again consistent with other studies) that 96 percent of SRD variance for natural materials and 98 percent for man-made materials can be accounted for by a linear model with only 3 parameters.

[FIGURE 4 OMITTED]

We do not however need to reconstruct SRDs as accurately as possible. Instead, we only need reasonable approximations for common architectural materials when given their CIE xyY or ITU-R BT.709 color space values. We further need a spectral reconstruction algorithm that is based on a large measured sample of architectural materials, and which generates continuously variable SRDs over the color gamut of ITU-R BT.709. Fairman and Brill's technique meets these requirements. (An implementation of this technique is shown in Fig. 5.; the program FullSpectrum is freely available from http://www.helios32.com.)

[FIGURE 5 OMITTED]

6 ERROR ANALYSIS

Given that most lighting simulation programs enable the user to choose material colors in the ITU-R BT.709 color space with red, green, and blue values ranging from 0.0 to 1.0, it makes sense to divide the [RGB.sub.709] color cube into ten equally-spaced steps in each dimension and perform the following:

1. Convert [RGB.sub.709] values to CIE XYZ tristimulus values

2. Calculate interreflected [RGB.sub.709] values using (6)

3. Calculate [RGB.sub.709] luminance [L.sub.709]

4. Calculate reflectance spectrum according to Fairman and Brill (2004)

5. Calculate interreflected reflectance spectrum values using (7)

6. Calculate reflectance spectrum luminance LRs

7. Calculate error [epsilon] = [L.sub.RS]/[L.sub.709]

The [RGB.sub.709] luminance value is given by:

[L.sub.709] = 0.2026 R + 0.7152 G + 0.722 B (8)

where the parameters are the middle row of the inverse of the XYZ-to-RGB transformation matrix (5). Similarly, the reflectance spectrum luminance [L.sub.RS] is calculated by multiplying the reflectance spectrum by the photopic luminous efficiency function V([lambda]).

Figure 6 shows the range of [RGB.sub.709] values (scaled to the range of 0.0 to 1.0) over which the error a is between 0.80 and 1.25. Various studies have shown that the accuracy of lighting calculations is approximately [+ or -] 10 percent for electric lighting and [+ or -] 20 percent for daylighting, so these are reasonable error limits.

What these charts do not show are the worst case values. For example:
Color      [R.sub.709]   [G.sub.709]   [B.sub.709]   Error

Red               0.90          0.00          0.00    0.22

Green             0.00          0.90          0.00    0.66

Blue              0.00          0.00          0.90    0.66

Cyan              0.00          0.90          0.90    3.63

Yellow            0.90          0.90          0.00    4.41

Magenta           0.90          0.00          0.90    0.39


It would be unusual to have an architectural environment where everything is painted in a single primary or complementary color, but it is still possible.

What the charts do show however is more important: the ITU-R BT.709 color space used by lighting simulations for predicting photometric quantities is acceptably accurate for most architectural environments. Fully 60 percent of the [RGB.sub.709] color gamut yields results that are within the defined error limits.

7. CONCLUSIONS

Most lighting simulation programs represent color as RGB triplets in the ITU-R BT.709 color space. Implicit in this representation is the assumption that interreflections between colored surfaces can be accurately calculated using three separate color bands. We have used Sumpner's Principle and an infinite room to examine the errors due to this approximation when compared to realistic spectral reflectance distributions.

It has been shown that, apart from contrived architectural environments where everything is painted in a single primary or complementary color, the use of three color bands (red, green, and blue) in lighting calculations should yield acceptably accurate photometric quantities.

REFERENCES

[CIE] Commission Internationale de L'Eclairage. 2004. CIE 15:2004 Colorimetry. Third edition. Vienna, Austria: Commission Internationale de l'Eclairage. 79 p.

Cuttle C. 1991. Sumpner's principle: A discussion. Light Res Tech. 23(2):99-106.

Fairman HS and Brill MH. 2004. The principal components of reflectances. Color Res App. 29(2):104-110.

[IESNA] Illuminating Engineering Society of North America. 2010. ANSI/IES RP-16, Nomenclature and Definitions for Illuminating Engineering. New York, NY: Illuminating Engineering Society of North America. 117 p.

[ITU] International Telecommunications Union. 2002. ITU Recommendation BT. 709-5 Parameter Values for the HDTV Standards for Production and International Programme Exchange. International Telecommunication Union. 30 p. Available at: http://www.itu.int/dmspubrec/itu-r/rec/bt/R-REC-BT.709-5-200204-I!!PDF-E.pdf

Maloney LT. 1986. Evaluation of linear models of surface spectral reflectance with small number of parameters. J Opt Soc Am A. 3(10):1673-1683.

Westland S, Shaw J, Owens H. 2000. Colour statistics of natural and man-made surfaces. Sensor Rev. 20(1):50-55.

Ian Ashdown, PEng, FIES (1)

(1) by Heart Consultants Limited

doi: 10.1582/LEUKOS.2011.07.03002
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Author:Ashdown, Ian
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Geographic Code:1USA
Date:Jan 1, 2011
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