=Paper=
{{Paper
|id=Vol-3271/Paper4_CVCS2022
|storemode=property
|title=Predictions of the Reflectance Factor of Translucen Layered Dental Resin Composites Using Two-Flux Models: assessing the importance of the interface reflectance parameter
|pdfUrl=https://ceur-ws.org/Vol-3271/Paper4_CVCS2022.pdf
|volume=Vol-3271
|authors=Vincent Duveiller,Emmanuel Kim,Marie Locquet,Arthur Gautheron,Raphaëël Clerc,Jean-Pierre Salomon
|dblpUrl=https://dblp.org/rec/conf/cvcs/DuveillerKLGCS22
}}
==Predictions of the Reflectance Factor of Translucen Layered Dental Resin Composites Using Two-Flux Models: assessing the importance of the interface reflectance parameter==
https://ceur-ws.org/Vol-3271/Paper4_CVCS2022.pdf
Predictions of the Reflectance Factor of Translucent Layered Dental Resin
Composites Using Two-Flux Models: assessing the importance of the
interface reflectance parameter
Vincent Duveiller 1, Emmanuel Kim 2, Marie Locquet 2, Arthur Gautheron 3, Raphaël Clerc 1,
Jean-Pierre Salomon 4,5,6,7 and Mathieu Hébert 1
1
Université de Lyon, UJM-Saint-Etienne, CNRS, Institut d’Optique Graduate School, Laboratoire Hubert Curien
UMR 5516, F-42023, Saint-Etienne, France
2
Institut d’Optique Graduate School, Saint-Etienne, France
3
Université de Lyon, INSA-Lyon, Université Claude Bernard Lyon 1, UJM-Saint-Etienne, CNRS, Inserm,
CREATIS UMR 5220, U1294, F-69621, Lyon France
4
Factulté d’Odontologie de Nancy, Département des Dispositifs Médicaux et Biomatériaux Dentaires,
Université de Lorraine, France
5
Institut de Science des Matériaux de Mulhouse UMR 7361 CNRS, Université de Haute Alsace, France
6
Université de Strasbourg, France
7
Department of Restorative Dentistry, Division of Biomaterials and Biomechanics. Oregon Health and Science
University, Portland, Oregon, USA
Abstract
Two-flux models are practical tools for predicting the appearance of strongly scattering
materials, but may fail to predict the spectral reflectance factor of translucent (i.e. weakly
scattering) materials because of the simplifying assumptions upon which they are based. In a
previous study, we showed how the internal reflectance at the interface of translucent layers, a
parameter usually calculated assuming a Lambertian angular light distribution, can be adjusted
in order to consider a directional light distribution, thus improving the predictive performance
of the two-flux model applied to translucent dental resin composites. In this paper, we explore
the dependence of this parameter on the layer’s thickness and show how different values should
be considered for accurately predicting the spectral reflectance factor of either thin or thick
layers of translucent direct resin composites according to their thickness. We also extend the
model to the layering procedure of resin with different levels of opacity which usually belong
to the category of thick layers and show that the two-flux model can reach good prediction
accuracy for their spectral reflectance factor.
Keywords 1
translucent, Kubelka-Munk, four-flux, spectrophotometer, dental resin composite
1. Introduction
The appearance management of dental restorations opens up new prospects for the practice of
dentists and manufacturers of dental materials, such as the development of materials which more
faithfully mimic the optical characteristics of human enamel and dentin, the improvement of aesthetic
quality of dental restorations, or in the long term, the use of 3D printers to make dental restorations,
overall improving patient care.
However, dental materials, just as skin or many other biological tissues, are translucent, i.e., weakly
scattering on an optical point of view [1,2]. This undermines simple flux transfer models, e.g., two-flux
The 11th Colour and Visual Computing Symposium, September 8–9, 2022, Gjøvik, Norway
EMAIL: vincent.duveiller@institutoptique.fr (A. 1); emmanuel.kim@institutoptique.fr (A. 2); marie.locquet@institutoptique.fr (A. 3);
gautheron@creatis.insa-lyon.fr (A. 4); raphael.clerc@institutoptique.fr (A. 5); jpsalomon61@gmail.com (A. 6);
mathieu.hebert@institutoptique.fr (A. 7)
© 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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[3,4,5] or four-flux [6,7] models, in their ability to predict the spectral reflectance factor, and
consequently the color of dental materials under a certain lighting. While these models are well suited
to the study of highly scattering materials such as paints [8,9] or textiles [10], they often fail in
predicting accurately the reflectance factor of slices of translucent materials in which light can keep
some directionality when crossing the material. Solving methods of the Radiative Transfer Equation
[11-17] can overcome this issue, but they are ruled out because of the complex optical measurements
required for their calibration which makes them unfit to use in an optical device intended for dentists.
Hence, two-flux and four-flux models remain interesting for their ease of practical implementation, and
their predictive performance should be sought to be improved with translucent materials.
The two-flux model is very easy to use and only requires two measurements for its calibration.
According to this model, the optical characteristics of a given material are described by its spectral
absorption coefficient K(λ), its spectral scattering coefficient S(λ) and its refractive index n, often
assumed to be 1.5 as in this study [18]. However, it only considers the propagation of a Lambertian
diffuse flux inside the material, which is not consistent with reality. Indeed, measuring instruments
based on a directional-hemispherical (or hemispherical-directional geometry), capture the exiting
radiance at a specific angle (just as our eyes when observing a scene under diffuse light). Even if the
object receives a Lambertian illumination, the flux reaching the sensor does not necessarily come from
the whole hemisphere and is therefore not representative of the Lambertian flux inside the object. This
is especially the case for non-scattering materials (e.g., a glass plate) for which the received flux only
comes from one specific angle, or for translucent materials. For a highly scattering material, for which
it can be assumed that the radiance captured by a detector in any direction is proportional to the exitance
of the object, the two-flux model’s assumptions apply. In a previous study [19], we showed that the
internal reflectance at the material-air interfaces is a key parameter for the prediction accuracy of the
reflectance and transmittance factors. For a thin slice of translucent dental resin composite, a better
prediction accuracy is achieved when considering internal reflectances corresponding to directional
light, as if the slice was a non-scattering plate. The accuracy should be improved with a four-flux model
describing the propagation of both directional and diffuse light into the material. Proposed by Maheu
et al. [6], the four-flux model describes the propagation of a directional flux, its forward and backward
scattering, and the propagation of diffuse light, inside a layer. A material is optically characterized by
four spectral parameters: its spectral absorption coefficient k(λ), its spectral scattering coefficient s(λ),
the spectral forward scattering ratio ζ(λ) and the spectral average path length parameter ε(λ). Rozé et
al. [20,21] updated the four-flux model with look-up tables enabling to use the asymmetry parameter
of the Henyey-Greenstein scattering phase function [22] denoted g, instead of ζ and ε.
Nevertheless, the four-flux model does not solve the issue of the angular distribution of diffuse light
at the interfaces, from which the internal reflectance of the interface is computed. Indeed, two-flux and
four-flux optical models assume that the angular distribution of diffuse light is Lambertian at both
interfaces of the layer, whereas in a recent study, Gautheron et al. [23] showed that it is far from being
Lambertian in thin or weakly scattering layers and is not necessarily equal at both interfaces.
Because of the diversity of clinical situations encountered by dentists, optical models predicting the
color of dental materials must be accurate in various cases: for a single layer of material or layering of
two different materials for a wide range of thicknesses.
Layering procedure of resin composites aims to use a first deep layer (with a so-called dentin or
opaque composite which aims to mimic optical properties of dentin, namely high levels of opacity and
chromaticity) with a second superficial layer (with a so-called enamel or translucent composite which
aims to reproduce optical properties of enamel, namely high level of translucency, different levels of
lightness and opalescence).
Mikhail et al. assessed the accuracy of the two-flux model for predicting the reflectance factor of
dental resin composite samples in optical contact on drawdown cards for thicknesses ranging from 0.3
mm to 1.2 mm [18], and for layering of samples of a total thickness of approximately 4 mm [24].
Kristiansen et al. and Wang et al. assessed the color prediction capability of the two-flux model to
predict the color of 1.0 mm thick dental ceramic samples on 5.0 mm samples [25,26].
In this paper, we study the two-flux and four-flux models’ ability to predict the reflectance factor of
samples of dental resin composites for different thicknesses, the ability of the two-flux model’s matrix
formalism to predict the reflectance factor of layered resin composites, and highlight the influence of
the internal reflectance at the interface parameter in the predictive performance of these models for thin
samples and thick layered dental resin composites.
2. Materials and methods
This experiment was carried out with samples made of the Estelite Universal Flow Medium in A2
and OA2 shades (Tokuyama Company). In the Vita Classical colorimetric terminology of dental resin
composites, the A2 shade is a light color with low chroma level, which aims to replace enamel. The
OA2 shade corresponds to the same color, but is opaquer than the A2 shade. It aims to replace the dentin
within the tooth, whose optical properties it imitates. Estelite Universal Flow is a direct light cured
supra-nano filled resin composite. It contains spherical silica-zirconia filler (mean particle size: 200
nm), bis-GMA, Bis-MPEPP, TEGDMA, UMDA, Mequinol, Dibutyl hydroxyl toluene, and UV
absorber. Batch numbers of A2 and OA2 shades are respectively 0503 and 7095. The Estelite Universal
Flow Medium A2 material will be referred to as A2 and the Estelite Universal Flow Medium OA2
material will be referred to as OA2.
2.1. Samples preparation
For both shades A2 and OA2, eight flat cylindrical samples with different thicknesses were
fabricated. The flowable dental resin was injected between two glass-slides, whose spacing is controlled
with high precision wedges and defines the nominal thickness of the sample. Samples with thickness
0.4 mm, 0.5 mm, 0.8 mm, 1.0 mm, 1.2 mm, 1.5 mm, 1.6 mm, and 2.0 mm were made. The samples
were light cured with a L.E.D. light curing unit (Radii Xpert, SDI company at 1500 mW/cm2) according
to the curing scheme I.S.O. 4049:2009 (each sample was irradiated five times, 40 seconds each
irradiation, at 12-3-6-9 o’clock positions and ending in the center of the sample). The sample diameter,
determined by the volume of material deposited, is ranging from 20 mm to 22 mm. The thickness of
each sample was measured with a precision micrometer as resin composites shrink during the curing
process. Although the measured thicknesses were considered in the experiments, samples will be
referred to by their nominal thickness for clarity. The fabricated samples are presented on Figure 1 and
Figure 2.
Figure 1: Samples of the Estelite Universal Flow Medium material shade A2 on a drawdown card (no
optical contact). The picture is taken in daylight with an uncalibrated camera. Samples are sorted from
the thickest on the left to the thinnest on the right: 2.0 mm, 1.6 mm, 1.5 mm, 1.2 mm, 1.0 mm, 0.8
mm, 0.5 mm and 0.4 mm.
Figure 2: Samples of the Estelite Universal Flow Medium material shade OA2 on a drawdown card (no
optical contact). The picture is taken in daylight with an uncalibrated camera. Samples are sorted from
the thickest on the left to the thinnest on the right: 2.0 mm, 1.6 mm, 1.5 mm, 1.2 mm, 1.0 mm, 0.8
mm, 0.5 mm and 0.4 mm.
2.2. Optical measurements.
The Color i7 spectrophotometer from X-Rite (USA) was used to measure the spectral reflectance
factor and transmittance factor of each sample. This device is based on the d:8° measuring geometry in
reflectance mode and on the d:0° geometry in transmittance mode [27]. The highest possible
illumination aperture of 17 mm was selected, while the smallest measuring aperture of 6 mm was
selected in order to limit the edge-loss phenomenon, which is known to alter the measurement of
translucent materials [28,29]. Measurements were performed from 400 nm to 750 nm by steps of 10
nm with a UV filter blocking wavelengths lower than 400 nm to prevent UV to visible fluorescence,
which is not accounted for by the evaluated two-flux and four-flux optical models. Each measurement
was repeated seven times and the average is calculated. To measure the reflectance factor of layered
resin composites, each sample of the A2 shade was superimposed to several samples of the OA2 shade
successively in order to create A2/OA2 couples of total thickness ranging from 1.0 mm to 4.0 mm. The
optical contact was performed between samples using a clear immersion oil (Immersion Oil Type B
from Cargille), which has a refractive index of 1.5180 at 546.1 nm. The A2/OA2 pairs evaluated are
presented in table 1. The reflectance factor of an A2/OA2 couple being necessarily different from the
reflectance factor of an OA2/A2 couple, the A2 samples (enamel-like shade) were always placed in
front of the spectrophotometer, just as enamel is superposed to dentin in teeth. The spectral reflectance
factor of each sample was also measured in optical contact on a black background (Byko-chart with L*
= 8.20, a* = -0.07, b* = 0.28) and on a white background (Byko-chart with L*=91.5, a* = -0.46, b* =
4.65).
Table 1.
Pairs of A2/OA2 samples evaluated for the prediction of their reflectance factor. The nominal
thickness of each sample is given in mm.
A2 sample OA2 sample
0.5 0.5
0.5 1.0
1.0 0.5
1.5 1.0
0.5 0.5
0.4 1.5
1.5 2.0
1.0 1.0
0.5 1.5
0.8 2.0
1.5 2.0
1.0 1.5
1.2 2.0
1.5 2.0
1.6 2.0
2.0 2.0
2.3. Experimental protocol.
To evaluate the capability of two-flux and four-flux optical models to predict the spectral reflectance
and transmittance factors of translucent dental restorative materials, and the spectral reflectance factor
of layered enamel/dentin resin layers of different thicknesses, the optical parameters of both A2 and
OA2 materials according to the two-flux and four-flux models must be determined.
The inverse two-flux model provides close-form analytical formulae giving the material’s spectral
absorption coefficient K(λ) and spectral scattering coefficient S(λ) as functions of the measured spectral
reflectance and transmittance factors, by assuming that the refractive index of the material is known
(we assume n = 1.5), and the thickness h of the sample under study is known as well. K and S coefficients
can be extracted for each wavelength either from the measured reflectance and transmittance factors of
one sample, or from the reflectance factor measurements of the sample in optical contact against a black
and a white background (with different sets of analytical formulae for each method). These two methods
were implemented, and the method requiring reflectance and transmittance factor measurements for
calibration is denoted the “2F-RT” model while the method requiring reflectance factor measurements
on a black and on a white background for calibration is denoted the “2F-Rbw” model. In these two
formalisms, the internal reflectance of the interface, denoted ri, is involved. This parameter is calculated
using eq. 1.
$/" )/"
𝑟! = ∫&'( 𝑅"# (𝜃)𝐿(𝜃) sin(2𝜃)𝑑𝜃.∫&'( 𝐿(𝜃) sin(2𝜃)𝑑𝜃, (1)
where R21 is the material-air Fresnel reflectance and L(θ) the outgoing radiance at the interface within
a cone. Usually, one can assume that the radiance incident at the interfaces and reaching the sensor is
Lambertian. In this case, L(θ) is a constant and ri is calculated by integrating the Fresnel reflectance
over the hemisphere [30] as in eq. 2. For n = 1.5, ri = 0.5963.
$/"
𝑟! = ∫&'( 𝑅"# (𝜃) sin(2𝜃)𝑑𝜃, (2)
This value is used in the 2F-RT and 2F-Rbw models. In the latter, the white background being highly
scattering, justifies the assumption of a uniform radiance falling on the interface.
We also implemented the two-flux reflectance/transmittance model, but this time assuming that the
light falling on the interfaces and reaching the sensor is mainly directional (as in a non-scattering glass
plate). In this case, the Fresnel reflectance is not integrated over the hemisphere for the calculation of
ri, which gives ri = 0.04 at normal incidence for n = 1.5. This model is denoted the “dir2F-RT” model.
In contrast with two-flux models, four-flux models cannot be easily inversed. Therefore, a fitting
algorithm based on the fmincon Matlab® solver was implemented in order to minimize the root mean
square difference between the predicted and measured reflectance/transmittance factors with respect to
the model’s parameters. The four-flux model was implemented according to the formalism proposed
by Rozé et al. [20,21] (denoted “Rozé4F-RT”), and also according to the formalism proposed by
Eymard et al. [31] (denoted “Eymard4F-RT”). Compared to the four-flux as generalized by Maheu et
al. [6], Rozé et al. calculated look-up tables enabling to use the asymmetry parameter of the Henyey-
Greenstein phase function [22] denoted g as a fitting parameter instead of the forward scattering ratio ζ
and the average path length parameter ε, reducing the number of unknown from 4 to 3. Therefore, the
refractive index can be used as a fourth fitting parameter instead of considering n = 1.5. Eymard’s four-
flux model uses the Rozé4F-RT formalism, but considers the g parameter to calculate the internal
reflectance at the interfaces of the layer (see eq. 3).
&/" &/"
𝑟! = ∫()* 𝑅"# (𝜃)P$% (𝑔, 𝜃) sin(2𝜃)𝑑𝜃0∫()* P$% (𝑔, 𝜃) sin(2𝜃)𝑑𝜃 (3)
where PHG is the Henyey-Greenstein phase function. This enables to use ri as an indirect wavelength
dependent fitting parameter for each material accounting for scattering anisotropy at the interfaces
instead of considering a Lambertian angular light distribution, as it is done in the Rozé4F-RT model.
The parameters of the five aforementioned models are extracted using measurements of one sample
with thickness h called calibration sample, and are assumed constant regardless of the thickness of the
sample. Once calibrated, each model is used to predict the spectral reflectance and transmittance factors
of other samples with different thicknesses. The spectral reflectance factor of stacks of samples is also
predicted using the matrix formalism of the two-flux model. The prediction error is evaluated by
calculating the color distance between measured and predicted spectra according to the perceptually
stable CIEDE2000 color distance metric. Measured and predicted spectra are converted into CIE 1931
XYZ tristimulus values considering the color matching functions of the 2° standard observer and a D65
power spectral distribution as illuminant [32]. XYZ tristimulus values are then converted into the CIE
1976 L*a*b* color space by considering a perfect white diffuser under illuminant D65 as white
reference for the chromatic adaptation. To assess the validity of predictions, the notions of perceptibility
threshold and acceptability threshold, developed in [33-36] for dental resin composites and ceramics,
are considered. The perceptibility threshold corresponds to the color distance below which most human
observers cannot distinguish two colors. It is evaluated between 0.8 and 1.3 CIEDE2000 units
depending on the study. The acceptability threshold corresponds to the color distance under which two
colors can be distinguished, but the color difference is deemed acceptable by most observers. It is
evaluated between 1.8 and 2.25 CIEDE2000 units depending on the study.
3. Predictive performance of two-flux and four-flux models
The 2F-RT, dir2F-RT, 2F-Rbw, Rozé4F-RT and Eymard4F-RT models were applied to both A2 and
OA2 samples and their predictive performance was assessed.
3.1. Prediction of single samples.
Figure 3 (resp. Figure 4) shows the prediction accuracy for the reflectance and transmittance factors
of A2 (resp. OA2) shade predicted by the 2F-RT, dir2F-RT, 2F-Rbw, Rozé4F-RT and Eymard4F-RT
optical models when the calibration sample is the sample with 1.2 mm thickness (Figure 3a, resp. Figure
4a) and when it is the sample with 2.0 mm thickness (Figure 3b, resp. Figure 4b). In each graphic, the
lower grey rectangle represents the perceptibility threshold and the upper grey rectangle represents the
acceptability threshold.
Reflectance Transmittance
10 10
2F-RT
8 dir2F-RT 8
2F-Rbw
Rozé4F
CIEDE2000
CIEDE2000
6 Eymard4F 6
4 4
2 2
0 0
0.4 0.8 1.2 1.6 2 0.4 0.8 1.2 1.6 2
Thickness (mm) Thickness (mm)
Reflectance Transmittance
10 10
2F-RT
8 dir2F-RT 8
2F-Rbw
Rozé4F
CIEDE2000
CIEDE2000
6 Eymard4F 6
4 4
2 2
0 0
0.4 0.8 1.2 1.6 2 0.4 0.8 1.2 1.6 2
Thickness (mm) Thickness (mm)
Figure 3: Deviation between measured and predicted reflectance factors (left graphic) and
transmittance factors (right graphic) expressed in CIEDE2000 color distance units, for samples of the
Estelite Universal Flow Medium material, A2 shade. (a) The sample with thickness 1.2 mm is used for
the calibration of the models. (b) The sample with thickness 2.0 mm is used for the calibration of the
models.
Notice that the color distance between the predictions and the measurements reaches 0 for the
thickness of the calibration sample except for the 2F-Rbw model. This is because the 2F-RT, dir2F-RT,
Rozé4F-RT and Eymard4F-RT models were calibrated using reversible protocols aimed at finding the
parameters that minimize the color deviation for this sample, while the 2F-Rbw model predicting the
reflectance and transmittance factors was calibrated using reflectance factor measurements performed
on drawdown cards (which minimizes the color deviation of predictions made on drawdown cards
rather than reflectance/transmittance factor predictions).
Reflectance Transmittance
10 10
2F-RT
8 dir2F-RT 8
2F-Rbw
Rozé4F
CIEDE2000
CIEDE2000
6 Eymard4F 6
4 4
2 2
0 0
0.4 0.8 1.2 1.6 2 0.4 0.8 1.2 1.6 2
Thickness (mm) Thickness (mm)
Reflectance Transmittance
10 10
2F-RT
8 dir2F-RT 8
2F-Rbw
Rozé4F
CIEDE2000
CIEDE2000
6 Eymard4F 6
4 4
2 2
0 0
0.4 0.8 1.2 1.6 2 0.4 0.8 1.2 1.6 2
Thickness (mm) Thickness (mm)
Figure 4: Deviation between measured and predicted reflectance factors (left graphic) and
transmittance factors (right graphic) expressed in CIEDE2000 color distance units, for samples of the
Estelite Universal Flow Medium material, OA2 shade. (a) The sample with thickness 1.2 mm is used
for the calibration of the models. (b) The sample with thickness 2.0 mm is used for the calibration of
the models.
Generally, the five optical models are more accurate for predicting the spectral reflectance factor of
the OA2 shade than the A2 shade. Indeed, OA2 shade, being opaquer and less translucent, is more in
line with the simplifying assumptions of two-flux and four-flux models.
For thicker samples, we expect the assumption of the Lambertian angular distribution of light at the
interface and then reaching the sensor to be more valid than for thin samples, since light is necessarily
more and more diffuse as it crosses the translucent layer. However, extracting the optical parameters
from the thickest sample does not improve the prediction accuracy of two-flux models: the color
distance increases rapidly when the thickness of the evaluated sample differs from the thickness of the
calibration sample. This shows that the two-flux model, as we implemented it, is not suited for
predicting the spectral reflectance factor of thin resin composites of dental biomaterials. Nonetheless,
the dir2F-RT model accurately predicts the spectral transmittance factor of most samples of A2 and
OA2 shades.
The four-flux models are generally more accurate than the two-flux models for predicting the
spectral reflectance factor of A2 and OA2 samples. The Eymard4F-RT model, using ri as an indirect
fitting parameter, is more precise than the Rozé4F-RT model in most cases, especially for thin samples,
which shows the importance of adapting the value of the internal reflectance at the interfaces.
Nonetheless, even four-flux models do not provide acceptable reflectance factor predictions for the
thinnest samples, which shows that work is still needed to address the physical issues posed by samples
with thicknesses lower than 0.4 mm.
3.2. Fitting the interfaces’ internal reflectance.
We have observed that the value of ri has a major influence in the accuracy of two-flux and four-
flux models. However, means to account for the specific angular light distribution at the interfaces are
scarce, as spectrophotometers used in conjunction with two-flux and four-flux models do not permit to
measure it. In a recent paper, Gautheron et al. [23] have shown the dependence of the internal
reflectance at the top (denoted ri) and bottom (denoted ri’) interfaces of layers of translucent materials
with respect to their optical properties (namely, its absorption and scattering coefficients, asymmetry
parameter of the scattering phase function and refractive index) and their thickness. In an attempt to
evaluate the internal reflectance at both top and bottom interfaces, we implemented an optimization
algorithm to fit the reflectance and transmittance factors predicted by 2F-Rbw model to measurements,
with ri and ri’ as fitting parameter. As the white background is highly scattering, we assume that the ri
value 0.5963 used to extract the absorption and scattering coefficients of the 2F-Rbw model is the most
valid in this case. Once the absorption K and scattering S parameters are extracted, the reflectance and
transmittance factors are fitted with ri and ri’ as free parameters. This experiment was repeated with
many other dental resin composites at our disposal. The extracted ri and ri’ parameters are presented on
Figure 5. Values of ri and ri’ show a strong dependence to the thickness of the sample from which they
are extracted. Furthermore, their distribution seems consistent with the modelling presented by
Gautheron et al. [23].
Figure 5: ri and ri’ parameters extracted from numerous translucent dental biomaterials.
Figure 6a (resp. Figure 6b) shows the improvement in prediction accuracy of the 2F-Rbw model
enabled by the fitting of ri and ri’. Note that Figure 6 should not be compared with Figure 3 and 4 but
it shows that a thickness dependent value for ri and ri’ improves the accuracy of the model.
Figure 6: New prediction accuracy (cyan curves) of the 2F-Rbw model with ri and ri’ fitted compared
to the previous predictive performance (blue curves). The calibration sample has a 1.2 mm thickness.
(a) A2 Material. (b) OA2 Material.
3.3. Prediction of layered samples.
Finally, we implemented the matrix formalism of the two-flux model, which enables to predict the
spectral reflectance factor of layered resin composites according to the 2F-RT, dir2F-RT, and 2F-Rbw
formalisms. The CIEDE2000 values assessing the deviations between the predictions and the
measurements are presented in Figure 7.
Reflectance factor
4
2F-RT
dir2F-RT
3.5
2F-Rbw
3
2.5
CIEDE2000
2
1.5
1
0.5
0
0.5 1 1.5 2 2.5 3 3.5 4
Total thickness of the stack (mm)
Figure 7: Deviation between measured and predicted reflectance factors expressed in CIEDE2000
color distance units, for A2/OA2 layered samples of the Estelite Universal Flow Medium material. The
A2 and OA2 samples with thickness 1.0 mm are used for the calibration of the 2F-Rbw model while A2
and OA2 samples with thickness 2.0 mm are used for the calibration of the 2F-RT and dir2F-RT models.
The A2 and OA2 samples with thickness 1.0 mm were used to calibrate the 2F-Rbw model, while
the A2 and OA2 samples with thickness 2.0 mm were used to calibrate the 2F-RT and dir2F-RT models.
These calibration samples enable the best predictive performance for each model on average.
In contrast with the previous observations on thinner samples, the 2F-Rbw is the most accurate
model. In this situation, the A2 samples are placed in optical contact against the OA2 sample, an
opaquer (although still translucent) white background. Thus, optical parameters of the A2 and OA2
samples extracted with the 2F-Rbw model are necessarily closer to this case than the optical parameters
extracted from reflectance and transmittance factor measurements, which partly explains the good
predictive performance of this model. However, the 2F-RT model being almost as accurate as the 2F-
Rbw model, and the dir2F-RT being the least accurate, indicates that the assumption of a Lambertian
angular light distribution at both interfaces of the stack generally applies, which justifies using ri =
0.5963 in this case.
4. Conclusions
We implemented five different flux transfer models (two-flux and four-flux models) and analyzed
their predictive performance for the reflectance and transmittance factors of samples with different, but
relatively low, thicknesses. While two-flux models generally fail to predict the reflectance factor below
the perceptibility threshold, we observe that the value of the internal reflectance of the bordering
interfaces, ri, plays a significant role in their respective prediction accuracy. Four-flux models,
especially Eymard’s model, were much more accurate than two-flux models but their prediction
accuracy for the thinnest samples must still be improved. These results indicate that when the
measurements are based on a geometry where the incident light or the captured light is directional
(directional-hemispherical or hemispherical-directional geometry), which is almost always the case in
practice, the light that is received by the sensor cannot be assumed perfectly diffuse (i.e., Lambertian)
at any depth within the layer, in particular at the bordering interfaces. Consequently, the internal
reflectance of the interface is different from the value 0.5963 that we would have with perfectly diffuse
light in materials of refractive index 1.5.
By fitting the internal reflectance of the top and bottom interfaces with an optimization algorithm
based on the two-flux model and repeating this operation with many different dental resin composites,
we clearly observe that the optimal internal reflectance values vary with the sample thickness. This
corroborates the idea that the angular distribution of light propagating into the material and falling on
the interfaces before reaching the sensor is not uniform (i.e., non-Lambertian).
However, when thicker layered translucent resin composites are considered, the 2F-RT and 2F-Rbw
models were found accurate enough at predicting their reflectance factor. The samples are thick enough
in this case to strongly scatter light and meet the assumption that the light captured by the sensor is
representative of the whole Lambertian light that propagates into the sample. It is also consistent with
already published studies which are focused on similar types of translucent dental materials for
thicknesses greater than 4 mm [24-26].
5. Acknowledgments
This work has been funded by a public grant from the French National Research Agency (ANR)
under the “France 2030” investment plan, which has the reference EUR MANUTECH SLEIGHT -
ANR-17-EURE-0026. It is also supported by LABEX PRIMES (ANR-11-LABX-0063) of Université
de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French
National Research Agency (ANR) and carried out within the framework of France Life Imaging (ANR-
11-INBS-0006).
6. References
[1] S. L. Jacques, Optical properties of biological tissues: a review, Phys. Med. Biol. 58 (2013): R37-
R61.
[2] X. Hu, Translucency estimation for thick pigmented maxillofacial elastomer, Journal of Dentistry
39S (2011): e2-e8.
[3] P. Kubelka, M. Munk, Ein Beitrag zur Optik der Farbanstriche, Zeitschrift für technische Physik
23 (1931): 593-601.
[4] P. Kubelka, New contributions to the optics of intensely light-scattering material, part I, J. Opt.
Soc. Am. A 38 (1948): 448-457.
[5] P. Kubelka, New contributions to the optics of intensely light-scattering material, part II: Non-
homogeneous layers, J. Opt. Soc. Am. A 44 (1954): 330-335.
[6] B. Maheu, J. N. Letoulouzan, G. Gouesbet, Four-flux models to solve the scattering transfer
equation in terms of Lorenz-Mie parameters, Appl. Opt. 23(19) (1984): 3353-3362.
[7] B. Maheu, G. Gouesbet G, Four-flux models to solve the scattering transfer equation: special cases,
Appl. Opt. 25(7) (1986): 1122-1128.
[8] R. S. Berns, M. Mohammadi, Single-Constant Simplification of Kubelka-Munk Turbid Media
Theory for Paint Systems – A Review, Color Res Appl 32(3) (2007): 201-207.
[9] Y. Zhao, R. S. Berns, Predicting the spectral reflectance factor of translucent paints using Kubelka-
Munk turbid media theory: Review and evaluation, Color Res Appl 34(6) (2009): 417-431.
[10] A. Moussa, Textile color formulation using linear programming based on Kubelka-Munk and
Duncan theories, Color Res Appl 46(5) (2021): 1046-1056.
[11] S. Chandrasekhar, Radiative Transfer, Dover Publication Inc., New York, 1960.
[12] A. Ishimaru, Wave propagation and scattering in random media, volume 2, Academic press, New
York, 1978.
[13] K. Stamnes, S. C. Tsay, W. Wiscombe, K. Jayaweera, Numerically stable algorithm for discrete-
ordinate-method radiative transfer in multiple scattering and emitting layer media, Appl. Opt. 27
(1988): 2502-2510.
[14] M. Elias, G. Elias, New and fast calculation for incoherent multiple scattering, J. Opt. Soc. Am.
A, 19 (2002): 894-905.
[15] S. A. Prahl, Light transport in tissues. Ph.D. thesis, University of Texas, USA, 1988.
[16] H. C. Van de Hulst H. C, Multiple Light Scattering, Academic Press, New York, 1980.
[17] Q. Fang, D. Boas, Monte Carlo Simulation of Photon Migration in 3D Turbid Media Accelerated
by Graphics Processing Units, Opt. Express 17 (2009): 20178-20190.
[18] S. S. Mikhail, S. S. Azer, W. M. Johnston, Accuracy of Kubelka-Munk reflectance theory for
dental resin composite material, Dent. Mat. 28 (2012): 729-735.
[19] V. Duveiller, L. Gevaux, R. Clerc, J-P. Salomon, M. Hébert, Reflectance and transmittance of
flowable dental resin composite predicted by the two-flux model: on the importance of analyzing
the effective measurement geometry, in: Proceedings of the 28th Color Imaging Conference,
Society for Imaging Science and Technology, 2020, pp. 313-320.
[20] C. Rozé, T. Girasole, A-G. Tafforin, Multilayer four-flux model of scattering, emitting and
absorbing media, Atmospheric Environment 35 (2001): 5125-5130.
[21] C. Rozé, T. Girasole, G. Gréhan, G. Gouesbet, B. Maheu, Average crossing parameter and forward
scattering ratio values in four-flux model for multiple scattering media, Optics Communications
194 (2001): 251-263.
[22] L. G. Henyey, J. L. Greenstein, Diffuse Radiation in the galaxy, The Astrophysical Journal 93
(1941): 70-83.
[23] A. Gautheron, R. Clerc, V. Duveiller, L. Simonot, B. Montcel, M. Hébert, Light scattering in
translucent layers: angular distribution and internal reflections at flat interfaces, in Proceedings of
the IS&T International Symposium on Electronic Imaging 2022, Society for Imaging Science and
Technology, pp.221-1–221-6.
[24] S. S. Mikhail, W. M. Johnston, Confirmation of theoretical colour predictions for layering dental
composite materials, Journal of Dentistry 42(4) (2014): 419-424.
[25] J. Kristiansen, M. Sakai, J. Da Silva, M. Gil, S. Ishikawa-Nagai, Assessment of a prototype
computer colour matching system to reproduce natural tooth colour on ceramic restorations,
Journal of Dentistry 39(Suppl. 3) (2011): e45-e51.
[26] J. Wang, J. Lin, M. Gil, A. Seliger, J. D. Da Silva, S. Ishikawa-Nagai, Assessing the accuracy of
computer color matching with a new dental porcelain shade system, Journal of Prosthetic Dentistry
111(3) (2014): 247-253.
[27] CIE: Absolute methods for reflection measurements, CIE Technical Report, 1979.
[28] W. M. Johnston, N. S. Hesse, B. K. Davis, R. R. Seghi, Analysis of edge-losses in reflectance
measurements of pigmented maxillofacial elastomer, Journal of Dental Research 75(2) (1996):
752-760.
[29] L. Gevaux, L. Simonot, R. Clerc, M. Gerardin, M. Hébert, Evaluating edge loss in the reflectance
measurement of translucent materials, Appl. Opt. 59(28) (2020): 8939-8950.
[30] D. B. Judd, Fresnel reflection of diffusely incident light, J. Natl. Bur. Standards 29 (1942): 329-
332.
[31] J. Eymard, R. Clerc, V. Duveiller, B. Commault, M. Hébert, Solar Energy Materials and Solar
Cells Characterization of UV – Vis – NIR optical constants of encapsulant for accurate
determination of absorption and backscattering losses in photovoltaics modules, Solar Energy
Materials and Solar Cells 240 (2022): 11717.
[32] CIE: Colorimetry, CIE Technical report, 3rd Ed., 1998.
[33] R. Ghinea, M. M. Pérez, L. J. Herrera, M. José Rivas, A. Yebra, R. D. Paravina, Color difference
thresholds in dental ceramics, Journal of Dentistry 38s (2010): e-57 – e-64.
[34] R. D. Paravina, R. Ghinea, L. J. Herrera, A. D. Bona, C. Igiel, M. Linninger, M. Sakai, H.
Takahashi, E. Tashkandi, M. M. Pérez, Color difference thresholds in dentistry, Journal of Esthetic
and Restorative Dentistry 27(S1) (2015): S1-S9.
[35] R. D. Paravina, M. M. Pérez, R. Ghinea, Acceptability and perceptibility thresholds in dentistry:
A comprehensive review of clinical and research applications, Journal of Esthetic and Restorative
Dentistry 31(2) (2019): 103-112.
[36] J. A. Medeiros, O. E. Pecho, M. M. Pérez, F. Carrillo-Pérez, L. J. Herrera, A. Della Bona, Influence
of background color on color perception in dentistry, Journal of Dentistry 108 (2021).