Inversion method for a generic model of diffusely reflecting, opaque surfaces
4.13.1. Background and theory
The following method is proposed to derive generic surface reflectance models for painted or coated opaque surfaces. The approach can be considered an intermediate model downgrade and follows the principle of fallback approaches (see section 2.2). While simplified assumptions are made for the complex-valued refractive index, the diffuse reflection based on subsurface scattering still contains spectrally resolved information. Beyond that, even the assumption of a constant refractive index is still able to model scattering events in a spectrally resolved way.
Highly resolved spectral measurement data is often unavailable for practically relevant, coated or painted materials, e.g. lammelas of shading elements. Nonetheless, the manufacturer will usually still provide integrated reflectance values for specific ranges of the global radiation spectrum. The reflectance values usually represent either the total diffuse or the total hemispherical reflectance, i.e. they exclude or include specular reflection. The values are usually determined by using incidence light with a low, near-normal direction. The commonly applied standard ASTM E 903 (ASTM E 903, 1996) advises using incidence angles in the 6 to 12 degrees range. In order to determine the solar reflectance value, the reflectance values are weighted based on a global radiation standard (see section 4.4).
As discussed in section 4.3.3, the diffuse reflection of painted or coated material is commonly the result of subsurface scattering processes on pigments submerged in a binder. It is, therefore, a reasonable approximation to apply a generic model for these materials, assuming a constant or smooth refractive index for the binder with little or no absorption. The subsurface scattering on the pigments, providing the characteristic absorption and colour of the surface, is then only modelled by using the subsurface reflectance function ππ π , see section 4.11.
The actual shape of the spectral reflectance function ππ π is determined in an optimisation process. Again, simulated annealing (section 5.2) is utilized for this purpose. In the optimisation process, the reflectance measurement is performed virtually and in accordance with the relevant standard (ASTM E 903, 1996). The spectral distribution of the incidence light is modelled according to the standard the measurement data refers to. The intensity and spectral distribution of the reflected rays are tracked, and the function ππ π is altered until the quantities measured in the model match the provided empirical data. The optimisation is performed by moving the knots of the implemented cubic-spline subsurface reflectance function (see section 4.11). The use of the spline functions to approximate the reflectance spectra takes into account that solid-state absorption (unlike gas absorption) usually exhibits a continuous and smooth spectral dependence.
The empirical reflectance data can be provided for hemispherical reflectance or for diffuse reflectance. In the latter case, a filter is used to exclude the specular reflection (gloss) from the measurement. The same features are implemented in the method to ensure that the model measurement mimics the real-world measurement.
As the empirical data commonly only covers four different reflectance values (UV, visible, NIR and total) and the cubic spline has comparably many degrees of freedom, an additional optimisation objective is generated by setting a target for the colour of the surface. For this purpose, the spectrum of all diffusely reflected rays is converted into an RGB triple. This is performed by first converting the wavelengths into the XYZ colour space of CIE 1931 and then further into the RGB colour space Light, sun and optics – applied principles, models and methods RadiCal, D. RΓΌdisser 106 (Fairman et al., 1997; Service, 2016; Wikipedia, 2022). Since the brightness of the colour is already determined by the visible reflectance value, only the nomalised colour value is defined as the optimisation target. A third, less significant optimisation target measures the variability of the reflectance functions. It is simply defined as the average vertical distance between successive spline knots.
The three optimisation targets and an additional precision term are formulated as loss functions to assess the quality of the fitting.
They are defined as:
The weighted loss functions determine the entire cost function for the optimisation process:
π
_πππ_πππ π represents the deviation for the reflectance in the given spectral ranges, πππ_πππ is the deviation of the nomalised colour value and π£πππ_ππππππ‘π¦ provides a penalty for strongly fluctuating functions. Finally, ππππ_ππππππ‘π¦ adds a penalty related to the precision of the current solution. In order to increase the performance of the optimisation, the first optimisation cycles are performed with fewer samples. The potential statistical error of the results is considered by adding three times the standard error to the cost function. This very effectively prevents early solutions from being identified as optimal solutions. The coefficients ππ are used to balance the four components of the cost function. In the optimisations performed, the applied cost function weighting was: π1 = π2 = π4 β = 1 and π3 = 1 1000 β .
The method has proven to provide stable results matching the targeted reflectance quantities with good accuracy. However, given the limited restrictions for the optimisation process and the relatively high degree of freedom, the simulation results can certainly not be considered unique. In general, multiple different solutions will be able to meet the optimisation targets. Even if the generated subsurface reflection includes a certain degree of randomness and deviates from the actual spectral reflectance, it will still be a good approximation for the unknown function as it exhibits the same integrated reflectance values within the specific, provided spectral ranges and is based on a continuous reflectance function. In commonly applied alternative methods, a constant diffuse reflection coefficient ππ ππ, which is integrated over the entire global radiation range, is generally used.
4.13.2. Implementation
The implementation is contained in the module rc_TsubsurfaceOpt. The method OPT_ASTM_E903_Reflectance can be called to perform an optimisation of subsurface reflectance function represented by a cubic spline with N Knots.
As can be seen in the implementation, currently, an optimisation method performing virtual measurements according to the ASTM E 903 standard is implemented. In order to align the model with the empirical data, the optimisation method offers the same functionality as the βreal-world measurementβ method. The incidence direction, spectrum, size of the gloss filter and type of reflectance value can be defined freely. Beyond that, the colour target, the number of free knots for the spline optimisation and the polarisation state of the incidence beam have to be provided.
4.13.3. Testing and validation
The feasibility of the inversion method is demonstrated based on three different surface coatings for Venetian blinds. The materials reflect the materials of the actual testing specimens used in the full system validation (chapter 8).
Figure 67 shows the material data as provided by the manufacturer. The table provides the integrated reflectance values for the total, UV, visible and NIR ranges for three different coatings/colours. The manufacturer also specifies the testing method and spectrum used. In the product data sheet, the Light, sun and optics – applied principles, models and methods RadiCal, D. RΓΌdisser 108 material of the coating is referred to as PE (polyester). The specifications are inconsistent, as PE is usually used for polyethylene and not polyester. As most polymers exhibit refractive indices in the range of 1.4 to 1.6 in the relevant spectral range, a constant value of 1.5 is commonly assumed to model polymers generically. Alternatively, additional information on the gloss at specified angles could be used to approximate the material’s refractive index. A precise measurement of the gloss value can be used to narrow down the refractive index range for the applied spectral range, as the gloss value is a measure for the specular reflection (see sections 4.3.2 and 4.3.3).
The generic value of 1.5 is assumed for the refractive index for the inversions performed here. The absorption of the material representing the binder of the coating is considered to be negligible; therefore, the complex-valued refractive index is defined as πΜ = 1.5 β 0π.
In order to model the diffuse subsurface reflection caused by the pigments, the subsurface reflectance functions ππ π (ο¬) are determined in the optimisation process described above. The results are depicted in Figure 68. As shown in Table 9, the integrated reflectance values for the different spectral ranges of optimised subsurface functions match the targeted values with reasonable accuracy. The deviations are in the range of a few tenths of a per cent. The reason for the deviation is that the optimisation based on smooth spline functions additionally has to consider the colour target. Without this objective, the optimisation would be able to provide almost perfectly matching reflectance values. Without additional color measurements available, the nomalised colour target was set to πππ = πππππ = πππ’π = 1.0 for all three materials, representing a desaturated colour (white or grey).
The subsurface reflectance function for the white coating shows strong fluctuations in the UV range. This is required to fulfil the steep drop from 80.9% reflectance in the visible to only 12.2% in the UV range (stated in the manufacturer’s datasheet). In order to assess the quality and relevance of the optimised functions, it has to be considered that the visible and lower parts of the NIR spectral range are most relevant. The UV range contributes only 3%, and the range above 1460 nm accounts for only 10% of global radiation (see Table 2).
