{"id":1022,"date":"2023-10-24T09:12:28","date_gmt":"2023-10-24T09:12:28","guid":{"rendered":"https:\/\/uneedtalk.com\/config\/tredword\/?page_id=1022"},"modified":"2023-12-01T13:12:41","modified_gmt":"2023-12-01T13:12:41","slug":"inversion-method-for-a-generic-model-of-diffusely-reflecting-opaque-surfaces","status":"publish","type":"page","link":"https:\/\/radi-cal.org\/method\/inversion-method-for-a-generic-model-of-diffusely-reflecting-opaque-surfaces\/","title":{"rendered":"Inversion method for a generic model of diffusely reflecting, opaque surfaces"},"content":{"rendered":"
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Inversion method for a generic model of diffusely reflecting, opaque surfaces<\/h1>\n

4.13.1. Background and theory<\/h2>\n

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 \ud835\udf0c\ud835\udc60\ud835\udc60, see section 4.11.
The actual shape of the spectral reflectance function \ud835\udf0c\ud835\udc60\ud835\udc60 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 \ud835\udf0c\ud835\udc60\ud835\udc60 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. R\u00fcdisser 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:<\/p>\n<\/div><\/div>\n

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