The work is devoted to the investigation of the best and reasonable methods to reconstruct the scattering properties of inks and paints widely used in modern electronic devices containing various light guiding plates and fibers. Typically scattering properties for inks have a volume scattering nature however it can be complicated with surface scattering because of the microroughness of ink layers. Both the volume and surface scattering reconstruction are considered in the given paper. Typically scattering properties of inks are described with bidirectional scattering functions (BSDF) and the BSDF is required from the internal side of the material of light-guiding elements. It cannot be measured directly because neither detector of the measuring device nor the light source cannot be placed inside the material. The only way to solve the problem of BSDF extraction for such cases is to build computer models of volume and surface scattering for inks and then with optimization procedures reconstruct parameters for these models which can provide surface scattering close enough to scattering of real samples of inks and paints. Light propagation in ink media is very complex requiring a long calculation time. As a result, the selection of an effective optimization procedure is the key point of the task. Several more typical real samples with printed inks are selected for investigation. The achieved results of BSDF extraction show very good agreement with data obtaining by real measurement. The comparison is based on numerical comparison of light scattering of ink samples as well as its visual appearance.
The light-guiding plates with complex scattering properties are widely used in modern optical systems. Examples of such devices can be light guiding plates for LCD, luminaires light-guiding elements in cars dashboards, different types of measuring devices. The ink and paint layers are used in such devices to provide desired complex angular and spatial light distributions. The accurate simulation of such layers is complicated because of the difficulty to describe light transformation on them. A more direct way is to describe the optical scattering properties of ink or paint with bi-directional scattering functions (BSDF). The function can be obtained with measurements however most cases it is a complex, expensive, and sometimes impossible task. Figure 1 shows several standard schemes of ink usage which eliminates evident problems related to measurements of ink scattering properties. There are three options for printing ink on a transparent dielectric plate. In the first case, ink is printed to the top plate face, in the second case – to the bottom face, and in the last case, ink is printed for both plate sides. It is evident in all cases to provide correct light propagation inside of light guiding plate it is necessary to define light scattering function inside of material as for reflectance as well for transmittance. However EMAIL: (V.
Sokolov); ddzhdanov@mail.ru (D.
Zhdanov);
2021 Copyright for this paper by its authors. precise measurements inside of material are not possible because internal total reflections of light which will add parasitic light to measured scattering function apart from it is not possible to place detector of measuring device inside of the material.
The problems related to measurements mentioned above practically exclude direct BSDF
measurements to obtain scattering properties for ink materials. To solve the problem typically computer
model of scattering is used to describe volume and surface scattering for ink and paint layers. However,
building an analytical computer model of volume or/and surface scattering it is required detailed
information about media and surface properties and structure which as a rule are absent. In most cases,
there is a material sample, for example in a form of a plane-parallel plate, but the parameters of the
individual ink components such as information about ink pigment (particle) concentration, shape,
optical properties of ink binder, height distribution of ink layer surface and other elements of the ink
layer are absent. These parameters are extracted with help of optimization procedures. The purpose of
the optimization procedure, in this case, is to obtain the BSDF of the whole sample maximally close to
the measured BSDF. Note measurements of BSDF for the whole sample without separation of ink
properties from a plate to which ink is printed is a rather trivial and simple task. After the definition of
ink parameters, it is possible to calculate BSDF of the ink layer separately from the plate. In computer
modeling, it is not a problem to place as a source of illumination as well as light detectors inside of
material required for ink BSDF extraction. This process generally is described in [
There are a lot of methods for the reconstruction of BSDF for Volume Scattering media [
The figure presents in very general terms the scattering mechanism in a heterogeneous ink medium. Light scattering in a binder medium occurs when light propagating inside collides with scattering agents (particles) having optical properties different from the binder (refractive index, dispersion, etc.). The nature of this scattering is stochastic and most often the form of angular scattering is closer to diffuse and depends on many factors, such as the size of particles, their shape and size, particle concentration, and optical properties. The size of particles (ink pigments) is usually very small and comparable with the wavelength of light. An accurate determination of light scattering on such small particles can be modeled using wave optics only. However, purely wave solutions are extremely complex (they require colossal computational resources and memory) and, as a rule, are not used in general programs intended for optical modeling. Usually, mixed, hybrid approaches using both ray and wave optics are used for the calculation, see figure 3.
The propagation of light between the particles is determined using statistical ray tracing methods
based on the average path length of the beam with the bulk medium, and the light scattering is
determined using the phase scattering function (angular intensity). The form of this function is
determined using the wave theory of light propagation. There are two main and well-known methods
for calculating the phase scattering function. The first assumes that the shape of the scattering particles
is spherical. In this case, the Mie or Rayleigh models are used to calculate the phase function, depending
on the size (diameter) of the scattering particles. If the particle size is much less than the wavelength,
Rayleigh scattering is used and if the particle size is commensurate with the wavelength, then the Mie
theory is used. From the point of view of the problem within the framework of this work, we are
primarily interested in the input parameters with which the Mie / Rayleigh model operates. These
parameters are shown in the Figure 4.
where g is the anisotropy factor of the Henyay-Greenstein phase function; - defines the angle between
the incident and the scattered beam. To fully define this model, there are two more parameters
describing light scattering on particles:
Scattering - the fraction of the energy of a parallel light beam diffusely scattered over 1mm
path in a medium containing these particles with defined volumetric concentration.
Absorption - the fraction of the energy of a parallel beam of light absorbed (converted into heat)
per 1mm path in a medium containing these particles with a certain volumetric concentration (PVC)
It should be noted that the last two parameters are defined in the format of the Lumicept software
package used for research in this work [
In the given paper both Mie/Rayleigh and Henyay-Greenstein models are used for BSDF reconstruction, the results of the investigation and conclusions about optimal model choice are presented below. 2.2.
For some samples ink and paint layers have rough surfaces. In the case the application of the single
Volume Scattering model is not sufficient for accurate BSDF reconstruction. Thus, an addition of some
surface scattering model for rough surface is required. The problem is investigated in the work. There
are a lot of approaches as analytical as well numerical to reconstruct BSDF properties of rough surface
scattering properties, some overview and description of the different approaches is presented in [
The base idea of BSDF reconstruction is not new and, for example, it can be examined in [
1. It should be noted that volume scattering uses wave optics (at least partially), which means that spectral calculations are required to correctly simulate the propagation of light in media with volume scattering. In addition, the propagation of light in such environments is quite complex and often with long ray paths. In other words, light calculations are rather slow, many times slower than for objects with surface scattering (for example, in comparison with rough surfaces). 2. The number of parameters describing volume scattering is much larger, at least in comparison with the parameters describing a rough surface, which in some cases have to be added to the optimization process of scattering reconstruction if the sample with volume scattering has working rough surfaces. 3. Often scattering media have color, which significantly complicates the process of BSDF reconstruction, since one has to use the spectral representation of the parameters used in the reconstruction. 4. Most of the parameters have different ranges of acceptable values, which is an unfavorable factor for use in most optimizers. 5. Often measured data (BSDF of the whole sample) which is used as objective function for optimization process contains mirror components (for example, brightness peaks when measuring reflective ink from the side of a substrate with a smooth surface) that can negatively affect the optimization process.
These specificities were considered in optimization scheme used for BSDF reconstruction. The scheme is shown in Figure 6.
Optimization scheme presented in Figure 6 is quite typical for BSDF reconstruction:
1. The only input measured data for BSDF reconstruction process on the volume scattering layer
is BSDF measured with a GCMS-4 spectroradiophotometer [
Considering factors mentioned just above, a simplified approach was used in the optimization
process to calculate the objective function: The measured BSDF was assigned to an infinitely thin
(“single-layer”) surface. This surface was illuminated with parallel light. And then the angular light
intensity was calculated in Lumicept [
At the third step of the reconstruction process, a computer model of the sample with a scattering medium is constructed. The choice of initial parameters depends on the chosen volume scattering model (Mie/Rayleigh and Henyay-Greenstein).
For the constructed model of the ink sample, BSDF is calculated under the same conditions as for the measured BSDF.
After BSDF calculation it is compared with BSDF calculated on the base of the measured one and the root mean square deviation (RMSD) is calculated between these BSDF. Note that RMSD can be specified in both absolute and relative forms, in absolute shape: Here is the measured intensity and is calculated. Index i means the value of the intensity for a defined direction of observation, illumination, and wavelength in case the spectral values are used for evaluation. If the color is not important (samples are white) then the intensity is calculated in the photometric coordinate system. In the developed methodology, all directions of observation (in the range of ±90) are included in the calculation of RMSD and optionally, any direction of illumination. To speed up the optimization process, one or two directions of illumination close to the normal are usually used. In the case of diffuse scattering and in the absence of specular peaks, it is convenient to use the absolute RMSD formula; in the presence of brightness peaks, the relative RMSD is preferable. In the latter case, it is better to convert intensity values to a logarithmic scale (to reduce the parasitic influence of the brightness peaks.
The calculated optimization criterion (RMSD) is an input parameter for the SCIPY optimizer, which, based on this criterion, decides to either end the optimization process or continue it. In the latter case, the optimizer calculates new optimization parameters, and the process repeats from the third step. It is important to note here that the parameters of volume scattering have different dimensions - the range of variation, for example, if the refractive index changes from 1.4 - 2.5, then its imaginary part, which describes the absorption, changes in the range 1e-3 - 1e-7, this is an unfavorable factor for optimizers. To eliminate this problem, you can enter weighting coefficients of the influence on each parameter, but this will lead to additional manual work, which we would like to minimize. In the implemented method, it was proposed to bring all optimization parameters to a single range from 0 to 1. In other words, each parameter involved in the optimization process is converted to a unit interval when entered into the optimizer and then returns back to its natural value when constructing an optical model of a medium with volume scattering. As shown by numerous (2) (3) experiments, this approach gives good convergence. The question of choosing the optimal optimization method will be considered separately. 7. After completion of the optimization, up to reaching the goal or other criteria for stopping the optimization process (the maximum number of iterations, for example), a model of the sample with volume scattering is prepared and it is based on the parameters of the best optimization step and, using a special plug-in, BSDF is calculated for the layer with volume scattering, as shown in Figure 7.
To compare scattering light obtained with measured BSDF versus computer model of ink the following scheme is used: The volume scattering layer is alternately illuminated with parallel light at different angles of incidence from both sides of the scattering medium (from the side of the air and the filler material). For each lighting condition, the angular distribution of light is recorded with the help of special detectors (receivers) that register the angular distribution of light, which is converted into the distribution of the luminance factor and collected in BSDF - four components in the most general case describing the scattering of reflected and refracted light from both sides of the scattering layer, which can be replaced by a single-layer (infinitely thin) surface with BSDF, see Figure 8.
At the final step, after ink BSDF simulation, a model of the sample with a “one-sheet” surface is built to which the reconstructed BSDF is assigned. Then the angular transmission and/or reflection of the entire plate is calculated. Then a similar calculation is started for BSDF measured with GCMS-4 spectroradiophotometer. These results are presented in the shape of a single graphs in the next chapters. 4.2.
To visualize the reconstructed ink BSDF, a scheme like to real models aimed for color investigation
is used. The model of special measuring box like JUDGE-II by X-Rite has been prepared. It has surfaces
close to diffuse and several luminescent tube lamp emulating daylight. The several simplified objects
(plate, sphere and torus) are placed into the measured box. On the surfaces of these test objects either
the reconstructed BSDF or its analog measured with GCSM-4 is assigned. Then with the help of a
special simulator (“Path Tracing” rendering available in Lumicept [
Three samples have been selected for investigation. They are marked as “sample #1”, “sample #2” and “sample #3” Their simplified models, input data are presented in the figure 10. 1. Description of sample #1: It is a white ink/paint with reflective properties close to diffuse. The working surface of the ink is on the side of the substrate (inside the dielectric material). In other words, a direct measurement of such a BRDF is impossible. All surfaces of the sample (scattering layer and substrate) are smooth (polished). From the known data, there is the refractive index of the substrate and BSDF measured with GCMS-4. It is measured for both reflected (BRDF) and transmitted light (BTDF) under illumination from the side of the substrate, ink layer thickness equal 0.063mm. Features: The presence of specular reflection from the substrate, which is parasitic for the BSDF reconstruction. 2. Description of sample #2: This sample is similar to the first one in terms of the type of measurements (BRDF, BTDF) under illumination from the side of the substrate. The goal is also the same to restore the ink BRDF from the side of the substrate. In addition to the measured BSDF, a substrate material (acryl) with a refractive index close to 1.5 is known. Features: The presence of specular reflection from the substrate, microroughness on the upper surface of the ink. 3. Description of sample #3: This sample is the most difficult case to restore scattering properties, ink of sample is colored and it is necessary not only to reconstruct the shape of the reflected light scattering, but also the desired color. The situation is simplified by the fact that the ink in a real device is illuminated in a rather narrow spectral range from 570 to 630 nanometers, thus the process of BSDF reconstruction can be limited to a narrow spectral range in comparison with white samples. The working surface of the paint is on the side of the substrate (inside the dielectric material). All surfaces of the sample (scattering layer and substrate) are smooth (polished). From the known data, there is the refractive index of the substrate and BRDF measured with GCMS-4 spectroradiophotometer under illumination from the side of the substrate.
It should be noted that this sample was the simplest of the investigated set, where the scattering properties of the ink are very close to the ideal (Lambertian) diffuser. Since the parameters of the internal structure of the paint are not known, a wide range of parameters were chosen for reconstruction. First, this is the volumetric concentration of particles (a common parameter for all methods). The rest of the parameters were assigned in accordance with the chosen model: Mie/Rayleigh or HenyayGreenstein. In the case of the Mie/Rayleigh model, these were: particle radius and their complex refractive index (both are real and imaginary parts). For Henyay-Greenstein, the following were chosen anisotropy factor, scattering and absorption. As the objective function, measured BTDF was used, see Figure 1. Note that even though the purpose of reconstruction is ink BRDF inside the material, we used the transmission component only to speed up the calculation and also due to the fact that the reflective component contains specular peaks of the luminance factor due to Fresnel reflection of light from the substrate. The latter effect is negative for the determination of the objective function and can worsen the convergence of the optimization process and it is better to avoid this if possible.
“Simplex” method [
The numerical comparison of angular intensity for measured and reconstructed BSDF for Mie/Rayleigh model are presented on plots in the Figure 11.
Figure 11 contains plots comparing angular reflection and transmittance (angular distribution of luminous intensity) for sample # 1 for measured and reconstructed BSDF. In the case of measurements, the sample was modeled as a single-layer surface with the assignment of the measured BSDF. In the case of the reconstructed BSDF, the model for the calculation was real with two-layers (solid) with the assignment of BSDF to one of the surfaces (upper) when the lower one was illuminated with parallel light at different angles (0, 15, 30, 45, 60). Simulated data are showed with dashed lines and measured with solid ones. “BRDF” shows angular intensity of reflected light and “BRTD” – angular transmitted with the sample.
Figure 12 is similar to the previous one but presents Henyay-Greenstein model of Volume Scattering.
The images of figure 13 presents visualization of measured and simulated ink on test samples. Note the explanation about models used for images generation was done in 4.2 chapter.
There is a roughness on the boundary “ink-air” in the sample #2. Note that this should not strongly
affect the reconstructed BSDF, since it must be determined from the side of the substratum, but the
roughness has to be considered in the reconstruction process, because it affects the measured BSDF of
the entire sample. This significantly complicates the reconstruction process due to the increase in the
set of parameters for optimization. In the case of the Mie/Rayleigh model, the following parameters
were chosen for reconstruction: the volume concentration of particles, absorption (the imaginary part
of the refractive index of the particles), and the real part of the refraction of particles. The rest of the
ink parameters were selected based on trial reconstruction attempts. To optimize the roughness, the
numerical method “Normals” (as the simplest) with the “Cauchy” distribution function and “sigma”
angular parameter for “Cauchy” function was chosen [
The numerical comparison of angular intensity for measured and reconstructed BSDF for Mie/Rayleigh model are presented on plots in Figure 14 and for Henyay-Greenstein method in Figure 15. The all plots are similar to Figures 11 and 12 for sample #1
BSDF reconstruction for sample # 3 is similar to # 1, # 2 with one small but noticeable difference the ink is colored, and color reproduction is required. This is a serious complication, since many ink parameters affect the reproduction of light color and this characteristic is very sensitive. Ideally, each parameter of volume scattering should be modeled as a spectrogram, however there is no necessary measured data for this, and it is practically impossible to add color (spectrograms) to the optimization process of BSDF reconstructing due to the catastrophic increase in the parameters for optimization. To somehow solve this problem, the following approach was used: 1. In the measured BRDF the spectral distribution of the integral coefficient of diffuse reflection (averaged over all directions of illumination) was extracted. It is marked as Kd(). 2. The further process depends on the volume scattering model. In the case of the Mie/Rayleigh model, this data can be used as a spectrogram for particle absorption (imaginary refractive index). In accordance with the theory, the imaginary refractive index connects the energy of the input () and transformed (reflected, transmitted) light () by (4) formula:
() = () −4 () (4) where i is the imaginary part of the refractive index and d is a distance (some constant value can be set in our case). Using the above characteristics, it is possible to link it with the integral reflectance of the input and reflected light:
() = (()) (5)
Substituting (5) into (4), one can easily find the spectrogram for the imaginary part of the refractive index: () = − ( ()) (6)
4
The calculated absorption spectrogram (imaginary refractive index) is used in the optimization process of reconstruction by modifying its average (over all wavelengths) value. In other words, the shape of the spectrogram does not change during BSDF reconstruction process. For Henyay-Greenstein volume scattering model the similar approach is used with one difference: the integral reflectance spectrogram Kd() is plugged directly into the Scattering parameter and the mean value of the spectrogram is optimized.
BSDF for sample # 3 is measured from the substratum side and contains luminance peaks due to
Fresnel reflection. In order to reduce their parasitic influence on the optimization process, the reflection
functions calculated for the measured and simulated BSDFs are converted to a logarithmic scale and
then the relative RMSD (defined with (3) formula) between them is calculated and it is an objective
optimization function. The calculation was carried out on a limited spectral range of 570-630nm, which
is justified since, according to the input characteristics, the lighting system of the device, where this ink
is used, operates in this limited range. The “Simplex” method [
1 Particle radius 250nm 2 Particle Volume Concentration (PVC) 14.0% 3 Particle refraction index (real part) 1.8 4 Absorption (imaginary part of refraction index) 0.0001* Achieved RMSD = 13.7% (RMSD between measured and reconstructed angular intensity reflected on sample #3). It was calculated according to (3) formula and for all wave lengths used in optimization process.
Table 6 Sample #3. Value of Volume Scattering parameters after reconstruction for Henyay-Greenstein model
1 Anisotropy factor -0.57 2 Particle Volume Concentration (PVC) 31.87% 3 Scattering 64.95* 4 Absorption 0.0052 Achieved RMSD = 12.1%
Note the parameters of the substrate material (thickness, refractive index) were set constant in accordance with the input sample specification presented in chapter 5 and did not participate in the BSDF reconstruction process. The thickness of the paint layer was set = 100μm. “*” marks the values represented by the average values of the spectrograms
The numerical comparison of angular intensity for measured and reconstructed BSDF for both Mie/Rayleigh and models is presented on plots in Figure 17. Only one BSDF component of reflected light (BRDF) is presented.
Plots in Figure 17 presents angular intensity in spectral format (for 3 wave lengths – 600nm, 650nm, 730nm).
Visualization of sample #3 is presented in Figure 18.
Note x, y in Figure 18 are chromaticity values taken for the center of sphere test object.
Reconstruction of all three real samples covered by ink shows good agreement with measurements results. Firstly, it is seen on a quantitative level comparing plots with reconstructed BSDF versus BSDF obtained with direct measurements. Secondly, it is proved with the visual evaluation of generated images for test objects with ink where we can see close similarity in shape of highlight zones, absence of any artifacts and color reproduction of colored sample. Both methods of the phase function of scattering agents presentation (Henyay-Greenstein and Mie/Rayleigh) are investigated. In the case of sample #1, both approached give practically the same good output. In the case of sample #2 with microroughness, Mie/Rayleigh method showed better agreement with measurements while in the case of sample #3 direct specification of spectral scattering available in the Henyay-Greenstein approach gives better color reproduction of colored ink. So, both approaches are worth to be considered for application in the reconstruction of ink scattering properties.
The work was supported by RFBR, Grant № 20-01-00547.