The article is devoted to the modelling of light reflection from real surfaces. Most of the existing models are often created to solve a certain task and therefore can not be used in other realms. To create a model useful for different purposes, we propose a reflective surface representation as a scattering layer bounded by the diffuse bottom and a randomly rough Fresnel upper boundary. Such an approach allows one to account for light polarization in the scattering layer and slope correlation of the randomly rough boundary which paves the way for the observation of some physical effects that take place in nature (for instance, statistical lens emergence). The first results of the light reflection modelling were taken at such initial parameters as to compare to those obtained in other research. Former occurred to be qualitatively of the same form as the latter. The model needs to undergo further validation in numerous experiments to prove its serviceability for different types of reflective surfaces.
While solving a large range of applied problems (3D-visualization, remote sensing, radiative transfer in turbid media, etc.), one inevitably faces the task of the radiance factor modelling, which has been remaining a problem of extremely high importance since the middle of the last century.
Despite the abundance of existing empirical, semi-empirical and analytical models,
having been proposed through the decades (for example, [
We reckon that the sole way to create a multipurpose comprehensive model of light reflection is to repeat and reproduce at the most possible precision all the physical processes occurring in nature when light interacting with a surface.
Though most of the existing models consider reflection from a randomly rough surface only, light does not reflect from a surface itself, but it does both from the upper ? Publication is supported by RFBR grant 19-07-00455 material facets and the material volume. Light rays penetrate the near-surface layers and then they scatter on the material particles, after that the rays re-enter the outer medium, all the processes being strongly dependant on the state of light polarization.
It was shown, that account for light polarization may lead to more than 30%
difference in results of multiple reflections modelling beside those obtained in the traditional
(depolarized) way [
Another noteworthy feature is the slope correlation of the randomly rough Fresnel
boundary formed by material facets. Since the Gemini V Program [
When considering a solid object surface one has to reckon in slope correlation as well, for it always appears after the surface being treated with a tool.
Based on the above considerations we represent a real reflective surface as a scattering layer bounded by a diffuse bottom and correlated randomly rough Fresnel upper boundary. The result of modelling will be wherein sought in the form of radiance factor curve.
Thus, in order to reach the aim one has to: 1. define a method for correlated randomly rough surface construction; 2. elaborate radiation transfer algorithm for the scattering layer; 3. realize the algorithm in a program and analyse obtained results.
The following sections of the article are devoted to solving these tasks. 2
Modelling of The Randomly Rough Fresnel Surface with The
Slope Correlation Account
There are two ways to construct a randomly rough surface [
This method consists in [
General Provisions
The main idea of the spectral models construction is [
sin(t ) (d ); B(x) =
cos(t ) (d );
Light Reflection from Real Surfaces: Probabilistic Model of the Layer Radiance Factor 3 where (d ), (d ) are real orthogonal stochastic measures on the semi-axis [0; +1); (d ) = 2 (d ); (d ) = 2Re (d ); (d ) = 2Im (d ); (d ) is a stochastic spectral measure of the process x(t).
Let (~x) is a random real field with correlation function B(~x), and its cosine transform is +1 Z 0 +1 Z 0 +1 Z 0 +1 Z 0
Z
Dk B(~x) =
cos(~ ~x)p(~ )d~ ; where p(~ ) is spectral density; ~x; ~ 2 Rn. In order to simplify further equations, we assume
M (~x) = 0;
D (~x) = 1; M and D being averaging and variance operators respectively.
In case of finite spectre one may construct [
In practical tasks, the properties of the solution being investigated are often defined
with needed certainty by the correlation function of the initiate field [
It is known [
Taking into account the assumption of the isotropic surface, we pass from cosinetransform to Hankel transform
B(r) =
p( ) J0(r )d ; 1 2 where J0 is Bessel function of the first kind. Then spectral density p( ) is p( ) = K0 1[B(r)] = K0 1[exp( k0r)] = k0 ( 2 + k02)3=2 ; 2 = 2 + 2:
one can find C = k0. Further, solving the equation with respect to , we achieve the expression for modelling of the value:
Light Reflection from Real Surfaces: Probabilistic Model of the Layer Radiance Factor 5 where is uniformly distributed in (0; 1).
After having passed to Cartesian coordinate system, it is easy to obtain = k0 s 1 2 2 cos 2 ; s 1 2 2 sin 2 ; where and are independent and uniformly distributed in (0; 1).
Thus, formula (2) takes its final form [
M
X p M m=1
" s cos k0 2 ln 0
m 1 2 m 2 (x cos 2 m m + y sin 2 m) 2
# m00) ; (3) where is standard deviation. 3
Program Realization and Analysis of The Results 3.1
Realization of The Spectral Representation Method The method of a correlated randomly rough surface construction described above was realized in MATLAB environment. However, in order to make the model more flexible we construct the final surface (x; y) as a superposition of two surfaces: (x; y) = (x; yj!~1) + (x; yj!~2); where !~ is a point in the phase space = fM; ; k0g. Such a modification of the formula (3) allows us to construct surfaces of different kind, which, however, maintain their statistical properties.
We want to especially notice that due to the model and its flexibility we managed to observe the statistical lens effect yet mentioned above. For instance, a random surface realization with parameters: !~1 = f500; 11:25; 0:025g; !~2 = f100; 0:2; 3g may look as shown at the Fig. 1. These parameters corresponds to an excited water surface with small ripples.
When we place a source of parallel rays in the form of the ring over the recess, we will see that the most of the rays converge together at a certain distance from the surface after passing through it. If one put a screen at the place of the rays’ convergence, there will be an image of the luminous ring on the screen (Fig. 2). Since the base of the reflective surface model is a scattering layer, the main problem to solve is radiative transfer in the turbid medium. The general scheme of light distribution modelling has the following steps: 1. model the initial point in accordance to distribution density of the source; 2. model the free path ; 3. prove the ray remain within the scattering medium; 4. evaluate the next interaction point; 5. choose the type of interaction (absorption or scattering);
Light Reflection from Real Surfaces: Probabilistic Model of the Layer Radiance Factor 7
The solution of the problem is sought for in the form of the radiance values in the
points on a circle above the surface. In terms of the task, the use of local estimations of
the Monte-Carlo Method [
The basic principles used in the algorithm are the same as those in [
The light scattering on a particle is governed by the Henyey-Greenstein function in
an easy and fairly certain way in the case of depolarized light:
( ) =
where is the cosine of the scattering angle, g is the average cosine of the scattering
angle. The main advantage of the formula is its dependence on the only parameter —
the average cosine of scattering angle. In [
The expression for the kernel of the local estimation given in [
(~r R(^l0 j~r ~r 0)2 ^l; N^ ~r 0j) jN^ 0; ^l0j (~r; ~r 0) ^l) (~r; ^l; ^l0)R(^l0
N^ 0; ^l0 ^l); in the case of the scattering on a particle k(~r; ~r 0) = where is extinction coefficient; ^l0 is unit vector of the incident ray direction; ^l is the same for the scattered ray; ~r is collision point; N^ is the surface outward normal; R is the reference plane rotation matrix; is the visibility function.
The results of the light reflection modelling are shown at Fig. 3. The curve of
reflected radiance was obtained for very rough Fresnel upper boundary with refractive
index 1.52 (to repeat frosted glass). The curve is qualitatively of the same form as that
gained in other research (for example, [
Further development of the model may be carried out in the following directions: – account for size and form of particles in the scattering medium; – thorough validation of the model in experiments with various materials; – creation of a list of parameters specific for materials of different kind. The latter is especially important for practical application of the model.