Rough surfaces are
all around us. When we generate realistic images, the task of visualizing them
arises. Fig. 1 shows examples of such visualizations created by us: frosted
glass with objects visible through it, and a rough car interior panel.
Fortunately, in many lighting simulations and optical design tasks it is
sufficient and more effective to replace real geometry of rough surface by a
smooth surface with certain optical characteristics. The definition of
scattering properties for a smooth boundary between two media is a simple task
and the light scattering can be easily simulated using Snell’s law of
refraction and reflection. However, in case of rough boundary the definition of
light scattering is more complex and can be expressed via Bidirectional
Scattering Distribution Function (BSDF). BSDF determines output angular light
transformation (refraction and reflection) in the dependence of input light
conditions, angles of the input light.
Figure
1: Examples of rough surface visualization
In simple cases,
when light scattering by whole plate is only important, the direct BSDF
measurements may be sufficient. The ordinary way of BSDF measurements for the
rough surface is presented on Fig. 2a. The sample, one side of which is rough,
is illuminated with a parallel light beam under the specific incident
directions, then an angular light distribution of transmitted light (BTDF 
BiDirectional Transmittance Distribution Function) and reflected light (BRDF –
BiDirectional Reflectance Distribution Function) is measured. In other words, such
BSDF measured for the whole sample works in cases when we can ignore object
thickness. The examples of such ordinary BSDF applications (Fig. 2b) may be various
diffuse films, thin plates, layers.
Figure 2: BSDF application in the simplified “onesheet” and
more “solid” models
However, in plenty
of cases the direct usage of measured BSDF is impossible. As an example we can
consider a light guiding plate with rough surface (Fig. 2c). Correct simulation
of light propagation in this optical system requires BSDF from each side of the
rough surface that includes BSDF from the material side. The BSDF measurement from
material side is impossible or very expensive because we cannot place the light
source and detector inside of the material. Another problem is the significant
inaccuracy of BSDF measurements for big illumination angles because of light
leakage inside of measured samples, shadowing of sample illumination and some
other reasons.
The mentioned
problems related to BSDF measurement result in the development of many
approaches and methods for BSDF reconstruction. One of the main purposes of
this paper is an analysis and verification of the popular methods of BSDF
reconstruction. The paper contains an overview of most prominent approaches and
their comparison done on the base of real measured samples with rough surfaces.
Generally, methods
of BSDF reconstruction can be divided into two main groups:
1.
Analytical methods.
The analytical methods are based on the
theory of physics (optics) or empirical formulas. The methods represent rough
surface models published by Ward, CookTorrance, Phong, etc. The main
advantages of analytical approaches are high efficiency because analytical solutions
are fast to calculate. This is important because optimization procedure is typically
used to get parameters of analytical functions describing required BSDF shape.
The disadvantage of the approaches is their approximation. They use approximate
algorithms to describe complex optical effects like a masking or shadowing of
the incident light illuminating of the rough surface (Fig. 3a, 3b) and a interreflection
of light on rough profile (Fig. 3c). During BSDF reconstruction this can
introduce noticeable inaccuracy for surfaces with big roughness.
2.
Numerical methods.
These approaches are based on simulation
of light propagation through models of rough surfaces. In the given paper two
main numerical approaches are considered which are based on distribution of microfacet
normals or heights. These approaches are more correct than analytical ones from
viewpoint of optical theory but require noticeable calculation resources.
Figure 3: Maskingshadowing and multiple reflections of light
on rough surface
The described
classification is not the only one. For example, the methods for BSDF
reconstruction can be also divided depending on what optics, geometrical (ray)
or wave, is applied. In our work we investigated all these groups of methods.
Most of the methods
for defining a light scattering (BSDF) through a rough surface are based on the
“microfacet” model. In the case of the “microfacet” model rough surface with
complex geometry is presented with a set of flat smooth surfaces (micro facets),
see Fig. 4. When boundary (microfacet) is smooth the transmission, reflection
can be easily simulated using Snell’s and Fresnel laws of refraction and
reflection. So it is possible to calculate a general light scattering through
the rough surface knowing general density distribution of microfacet slopes or
their normals. One of the earlier attempts to model light reflection from a rough
surface is described in [1]. It was restricted with reflection light component
only but it was a basis for developing one of the more wellknown microfacet
models introduced by Cook and Torrance [2]. A lot of different modifications of
the microfacet model have been developed at that time [35]. The next
developments are related to extensions of reflection “microfacet” models with the
support of anisotropy, sampling with correct weights, application of Backmann
distribution [6, 7], and development of alternative sampling methods with
fitted separate approximations [8].
Figure 4: Microfacet presentation of a rough surface
Shlick [9]
develops more simple approximation to the CookTorrance model with the help of
rational approximations with the application of the Fresnel formula widely
adopted nowadays. Ashikhmin and Shirley [10] introduced an anisotropic
reflection model on the base of Phong microfacet distribution including correct
importance sampling. Then an energyconserving reflection model [11] is
introduced. It is derived from arbitrary microfacet distributions, though this
formulation involves numerically estimating integrals without closedform
solutions.
Microfacet models
are widely used in computer graphics; experimental data appear for verification
of scattering models. For example, different models of BRDF reconstruction
(“Ward”, “WardDuer”, “BlinPhong”, “CookTorrance”, “Lafortune et al”, “He et
al”, “AshikhminShirley”) are compared with real measurements in [12]. A set of
development is related to the derivation of the refraction part of scattered
light [13]. There are investigations that take into account thin effects such
as the shadowing masking, multiinterreflections on elements of rough surface,
using of importance sampling [2, 14, 15].
The reflection
models based on wave optics are proposed in [16]. The method can simulate a
wider range of surface effects than microfacet models. However, wave approaches
are much more expensive to calculate and as a rule very approximate in support of
thin effects as a multi reflection on profile with rough surfaces.
The numerical
simulations of transmissions models are performed in [1720, 37]. The “GGX”
microfacet model was introduced in [21]. It is an improved variant of the CookTorrance
microfacet model supporting reflection as well as refraction and
shadowingmasking. The [21] work contains numerical data comparison of different
analytical models and demonstrates a lot of advantages relative to other
analytical methods of BSDF reconstruction of a rough surface. The “GGX” model
is considered one of the most accurate, flexible and wideused analytical approaches.
It supports both reflection and refraction components, maskingshadowing, and
importance sampling and shows more accurate output in relation to the CookTorrance
model [21].
So, the “GGX”
model is selected for examination in our paper as representative of the
analytical group of methods. Typically, an analytical model is represented with
two base functions. The first function, denoted as
D(m),
is a microfacet
distribution function. It describes the statistical distribution of surface normal
m
over microsurfaces. The second bidirectional function, denoted as
G(i,
o, m),
describes what fraction of the microsurface with normal
m
is
visible in both directions
i
and
o
(Fig. 5). Typically, the
shadowingmasking function has relatively little influence on the shape of the
BSDF except for near grazing angles or for very rough surfaces but is needed to
maintain energy conservation.
Figure 5: Micro vs. macro surface
The bidirectional
function
G(i, o, m )
can be approximated with two omnidirectional
shadowing terms:

(1)

where
G1
is
derived from the microfacet distribution D as described in [14, 15].
We used the “GGX”
model with the following microfacet distribution and masking shadowing function
D(m),
parameter
specifies surface roughness:

(2)

and omnidirectional
maskingshadowing function:

(3)

where
is the angle between
m
and
n,
between
v
and
n, and
is the positive characteristic function
(which equals one if
> 0 and zero if
≤ 0). v equals either to
i
or
o
vectors (Fig. 5). Note the function is rather similar to the wellknown
Beckmann distribution used in the CookTorrance model.
So the process of
BSDF reconstruction consists of the definition of the parameter
 degree of surface microroughness for
which generated BSDF gives more close results to measurement data. It will be
considered in the next chapters in more detail.
Nowadays with increasing
of computer’s power new approaches for BSDF reconstruction have been developed
in [22, 23, 25, 26]. Part of them is based on pure numerical methods in which a
BSDF is calculated by ray tracing simulation through an explicit geometry model
of rough surface. The method based on the normal density distribution of rough
surfaces is proposed in [26, 27]. In this method the microrelief is simulated
with the help of distribution of normals represented with an analytical
function having a set of parameters defined with the help of the optimization
process. The process of BSDF calculation is presented in Fig. 6.
Figure 6. “Normals” method of BSDF reconstruction
The approach is
maximally natural and transparent. To calculate BSDF the flat boundary
presenting a rough surface is illuminated with parallel light from both sides
of the boundary. Typically, stochastic (Monte Carlo) ray tracing is used. Each
time ray hits the boundary, normal is defined with a probability according to
analytical function – normal density distribution. Then ray reflection,
refraction is defined according to Snell’s law. The transformed light is
registered with detectors which finally form resultant BSDF. The main problem
here is the definition of analytical function specifying normal density
distribution and its parameters. In [27] it was proposed to use two analytical
functions like Gauss and Cauchy
(Fig. 7):

(4)


(5)

where
Θ
is an angular variable specifying angle of
surface normal,
is zero angle specifying the position of
function maximum and it should be equal to zero for most of cases for rough
surfaces with roughness distribution close to normal. So, these functions are
used in our work. The two main parameters
δ
and
n
specify shape of function of normal
density distribution and can be defined with the help of the optimization
process, which is presented in the scheme in Fig. 7 and consists of several
main steps:
1.
The first step includes
an input of measured BSDF and other sample parameters affecting light
propagation like refractive index and thickness.
2.
An objective function
for optimization and parameters of illumination and observation to be used in
light simulation are defined at the second step. The measured BSDF of whole
sample can be used directly as objective function, sometimes it is recalculated
to ordinary angular intensity distribution for simplification. The detector parameters
(angular, spatial resolution, distance to measured surface) during simulation
are chosen maximally close to parameters of real detectors used in
measurements. As a rule only small illumination angles close to normal of measured
sample are used during optimization. It is done because the accuracy of
measurements decreases significantly for incident angles far from normal
direction.
3.
During the third step
an explicit model of the sample with rough surface is generated for normal
density distribution for some initial parameters
δ
and
n.
4.
Most of modern light simulation
software can simulate light propagation through boundary between two dielectric
media specified with normal density distribution. So there is no problem to
calculate angular light distribution for sample model defined in the previous
optimization step.
5.
An optimization
criterion is defined as root mean square deviation (RMSD) between measured and
simulated angular intensity distributions.
6.
An optimization
criterion (RMSD) calculated on the previous step together with current
δ,
n
parameters transfer to
optimizer. An external optimizer of SCIPY library with “Simplex” algorithm was
used in our work.
7.
The optimizer makes decision
to continue optimization process (7.1 in Fig. 7) or to interrupt it (7.2 in
Fig. 7) in case the optimization goal is achieved or due to another reason (for
example, maximal number of optimization steps is achieved). If the goal is
achieved a final model of rough surface based on optimized normal density
distribution is generated. In case of simulation there is no problem to place
detectors and light sources anywhere including inside of sample material and
calculate light scattering from both sides of rough surface, i.e. to calculate
BSDF of rough surface.
Figure 7. Optimization procedure for “Normals” method of BSDF
reconstruction
The optimization procedure
(Fig. 7) was used to reconstruct BSDF with “Normal” numerical method. More
details are presented in [27]. The investigations show the “Normal” method is
very effective from viewpoint of calculation speed and fast convergence in
optimization procedure during BSDF reconstruction. However, it has evident
drawbacks too, namely, it does not support interreflections and maskingshadowing.
Another numerical
approach is based on height density distribution and described in [28]. There
is some similarity of the ”Heights” and the “Normals” methods. However, an analytical
function is used here for another goal: to define 2D height distribution H(x,
y).

(6)

Figure 8. Definition of “Height” distribution
It shows regular
grid of points with uniform steps along x and y axes (Fig. 8). Each point in
the grid presents a node of microprofile. To define profile height in each node
with (x, y) coordinates analytical probability function of one or several
parameters can be used. In other words, height in each node is defined
according to a probability defined according to normal (Gauss) or some another
analytical function specifying height density distribution. In our work two
analytical functions were used for “Heights” approach – Gauss and Cauchy:

(7)


(8)

Note that formulas
(7), (8) are similar to (4), (5) used for “Normal” approach but use z
coordinate instead of angular
variable, which is defined in the range
[0, H_{max}] and specifies heights distribution.
Both functions
depend on four parameters (
σ,
_{}
H_{max}
,
n
and z_{0}).
It is rather substantial number of parameters which
can complicate process of optimization convergence. However experiments show that
in most of the cases the only
σ
(sigma)
is sufficient,
“n” (degree) can slightly improve convergence in some
cases.
H_{max}
can be set to 1 in most of the cases if to set
step between nodes of profile grid around the same unit value. z_{0}
is
supposed to be zero (density of heights is symmetrical relatively to
H_{max}).
z is in range [0,
H_{max}]. So
defines distribution of height density.
According to
formula (68) height distribution of microprofile can be defined and used for
profile geometry generation (Fig. 9).
Figure 9. A schematic appearance of microprofile based on
analytical heights distribution: (a) – perspective view; (b) – top view
The definition of
optimal parameters specifying height distribution is fulfilled with optimization
procedure similar to “Normals” methods (Fig. 7). The only difference is an explicit
model of sample where rough surface is simulated with a geometry based on heights
distribution instead of simplified normal density distribution. At present,
most of the modern optical software, such as SPEOS, LightTools, Lumicept [35], for
example, allow calculating such microgeometry, so BSDF can be calculated
without a problem.
Note that the main
advantage of the “Height” method is support of all effects: interreflections
and shadowingmasking. The process of BSDF reconstruction with this method is
more complex relatively to “Normals” because the number of parameters used in
reconstruction is increased. In the “Heights” method the H_{max}
parameter defining the maximal scale of micro profile is added to parameters
specifying the shape of the height distribution function, namely parameters
δ
and n (formulas (4) and (5)).
Most of the methods
described in the previous chapters use ray optics for simulation of light
propagation. However, an application of geometrical optics can be inaccurate. A
rough surface is considered as a combination of microelements and their size varies
from great to small values, up to sizes comparable with wavelength. Application
of geometrical optics theory can result in the noticeable inaccuracy of
reconstructed BSDF. Another problem of the geometrical approach is the parasitic
influence of measurement noise in case of measured height distribution.
The main problem
of wave methods is their extreme complexity. The precise wave methods cannot be
applied practically due to the complexity of microsurface geometry. So an approximate
wave solution should be used for BSDF reconstruction. As an example, one of
them is described in [16] but it is related to the reflection model only. The
more wellknown and more usable method to reconstruct BSDF for rough surfaces
is based on the Kirchhoff approximation. The method is built on a simple FFT
(Fast Fourier transform) based procedure. A more detailed description can be
found in [2934]. The BSDF reconstruction based on the Kirchhoff approximation
is developed for both reflection and transmission components and was examined
in our work.
The Kirchhoff method should be applied to the surfaces
containing smooth roughness (without breaks) or consisting of enough great
facets. The local condition of applicability looks like this:

(9)

where
λ
is
a wavelength
(in the medium where scattered light propagates),
R
is a
"typical" curvature radius of roughness,
is a local angle of incidence.
The wavebased
approach does not consider the multiple reflections. The limitation can be
expressed in form:

(10)

where
is a characteristic roughness
length,
is RMS of height
deviation from a flat surface.
The
method also does not take into account shadowing and masking (shadowing is for
occlusion of illumination direction, masking is for occlusion of observation of
one). This limitation can be expressed as:

(11)

where
is an illumination/observation
angle that is counted from a normal to a flat surface. In the wave model
scattering is calculated for the infinite periodic surface. If there is no
seamless conjugation between opposite sample edges, an artifact scattering by
periodic conjugation can arise. It is negligible for a large relief sample but
can be quite serious for a small one. The calculations are done for
nonpolarized illumination.
Before describing
the samples to be used in the investigation let’s consider a profile of rough
surface, Fig. 10.
Figure 10. Parameters of microroughness
Several wideused
parameters describe a profile rough surface. These parameters will be used for the
description of measured samples of rough surfaces, so shortly consider them. The
first parameter Ra is the most common one and is calculated using the formula (12):

(12)

is an arithmetical mean deviation of the
assessed profile. The next parameter
is a root mean square

(13)

The next two
parameters specify average values of valleys (heights below the mean line) and
ledges (heights above the mean line) over the assessed profile, Rv and Rp
correspondently:
;

(14)

The next parameter is the most trivial. Rz is maximal
profile height and is calculated using parameters from (14):

(15)

One more wellknown
parameter is RzJIS or Rz5. It is related to the Japanese industrial format. It
is
based on the five highest peaks and lowest
valleys over the entire sampling length (l
in Fig. 10).

(16)

And the last two
advanced parameters present Rsk – skewness and Rku – Kurtosis:
;

(17)

The eight samples
made of acryl with refractive index = 1.49 are selected for investigation. One
surface in each sample has roughness and another is smooth. Two types of
measurements are fulfilled for all samples:
1.
A height distribution was
measured with the precise Taylor Hobson’s profilometer.
2.
Light transmission distribution
was measured with goniophotometer GP200
[35, 36] by Murakami Color Research Laboratory, see Fig.
11.
Figure 11. Input measured data of investigated samples
The parameters of
all eight profiles have been calculated based on measured height distributions
and using formulas (12)(17). These parameters are combined in Table 1. The #1#8
in the first line are sample identifiers. Additionally, the second and third rows
of the table 1 present size of the measured fragment on the sample and
resolution of measurements – number of measured profile points along with both
x and y directions. The step between measurement points was constant.
Table 1
.
Profile parameters of measured samples
Param./Samp..

#1

#2

#3

#4

#5

#6

#7

#8

Size (mm
x mm)

0.37x0.37

0.95x0.95

0.95x0.95

0.95x0.95

0.37x0.37

0.95x0.95

0.95x0.95

0.95x0.95

Resolution

1024x1024

1333x1333

1330x1330

1332x1332

1024x1024

1332x1332

1`3x1333

667x677

Ra (μm)

0.178

0.456

0.668

0.738

1.170

2.038

2.596

10.724

Rq (μ m)

0.232

0.581

0.866

0.956

1.466

2.669

3.308

13.456

Rv

1.754

2.908

5.870

5.400

5.726

13.731

16.889

33.913

Rp

0.594

1.613

3.380

2.369

3.588

7.817

7.847

40.7834

Rz

2.340

4.521

9.251

7.769

9.314

21.548

24.736

74.697

Rz5

2.329

4.519

9.235

7.762

9.313

21.542

24.727

74.582

Rsk

0.786

0.700

0.839

0.928

0.560

0.643

0.658

0.012

Rku

5.156

3.654

4.482

4.492

3.275

4.279

3.920

2.813

The images of
investigated profiles are presented in Fig. 12. For convenience the profiles in
Fig. 12 are placed in order of their root mean square (Rq) increasing from the
left to the right and from up to down.
Figure 12. The appearance of measured profiles
GP200
goniophotometer [35, 36] was selected for measurements because it has very
advanced characteristics, such as angular resolution = 0.6°,
very small angular step = 0.1°
and wide range of observation directions =
±
90°.
The high angular resolution is very important in case of our investigation
because part of measured samples has very small roughness, comparable with
wavelength, so angular transmission is supposed to have a very narrow shape.
The measurements of transmission are done for five angles of incident light
direction = 0°, 15°, 30°, 45° and 60°
in the single plane of light incidence. The
goniometer GP200 outputs data in the relative shape calibrated to measurements
without sample. So, for the correlation of measured and simulated data, the
same calibration process is fulfilled in simulation, as it is explained in
[31].
The first two
methods selected for verification are based on the measured heights
distribution and fulfilled in the Lumicept lighting simulation software [35]. The
software has special instruments for direct simulation of rough microgeometry
on the base of numerical height distribution, apart from it has physically
accurate MonteCarlo ray tracing and BSDF generator allowing to calculate BSDF
based on ray as well as wave optics (Kirchhoff approximation).
The first method
is based on ray optics. It is denoted as “Measured_Heights (ray)”. In
the given method explicit geometry is created as the boundary between air
medium with refractive index = 1 and dielectric medium with refractive index =
1.49 (the refractive index of sample material). The boundary is illuminated
under different incident directions from 0°
to 85°
with
parallel light from both sides: from the air and dielectric. Ray propagation through
the rough surface is based on Fresnel, Snell laws. The detectors are placed
above and below the boundary of the rough surface and detect transmitted and
reflected light. Then BSDF is generated based on calculated data. It supports
all complex effects such as interreflections on microrelief, masking and
shadowing explained in the second chapter. The main restriction of the method
is an applicability of ray optics. It can be inaccurate for the sample with
small roughness (with sizes close to the wavelength). Another possible drawback
of the method is also related to ray optics. It is the high sensitivity of
generated BSDF to the quality of measurements. The different steps between
measured nodes or noise can result in a noticeable difference in BSDF shape.
The second method
is denoted as “Measured_Heights (wave)”. It uses measured sample profile,
i.e. height distribution, too. However, light propagation is realized here
analytically based on Kirchhoff approximation. The disadvantages of the
approach are listed in chapter 2.3 above.
It should be
pointed out that measured profiles are not used directly in this investigation.
To minimize possible errors related to the quality of height distribution
measurements, application of ray or wave optics, and other possible reasons
optimization procedure is run for each profile. It is explained in [26] and
similar to optimization procedure presented for “Normals” approach in Fig. 7.
The purpose of the optimization is to obtain the transmitted light distribution
maximally close to the measured one. And parameters of optimization are scaling
and filtration of microrelief. The scaling allows to increase/reduce
microroughness and filtration allows reducing measurement noise. These ways of
profile modification are presented in Figure 13.
Figure
13:
Microrelief modification
The verification
of the next three methods is the main goal of this work. They do not require
measurements of height distribution which can be expensive or simply not
available. The method denoted as “GGX” was selected as the best representative
of analytical approach. A utility was created to generate BSDF based on the
analytical formulas (2) and (3). To define an optimal parameter of roughness (
according to formulas (2) and (3)) an
optimization procedure, similar to the one presented in Fig. 7 was executed.
The optimization goal was to obtain light transmission distribution maximally
close to measurements.
The generation of
BSDF with two numerical approaches denoted as “Normals” and “Heights”
is very similar to “GGX” and is explained in chapter 2.1 and in [31, 32]
in more details. The parameters to reconstruct normal and heights distribution
are defined with an optimization procedure with an objective function to obtain
maximal closeness to measured transmission.
So, finally, we
have five methods to be verified: two based on measured profile and angular
sample “Measured_ Heights (ray)”, “Measured_ Heights (wave)” and
three methods based only on angular sample transmission “GGX”, “Normals”
and “Heights”.
Comparison of
measured versus simulated light transmission through a plate with rough surface
is used for verification of different BSDF reconstruction methods. However, such
comparison can be not sufficient. The BSDF of rough surface can have complex
shape and even small inaccuracy in its generation can result in defects visible
in the image, appearance of some artifacts. Especially it can be noticeable if
BSDF is attached to complex curved objects which are illuminated under grazing
angles. So, it is also preferable to verify how BSDF samples are visualized
under some realistic conditions.
A special model
aimed at visualization was prepared, see Fig. 14. The scene presents a virtual model
of a special measuring box JUDGEII by XRite [39]. It has surfaces close to
diffuse and several luminescent tube lamps emulating daylight. The several
objects: a plate, a sphere and a torus are placed into the measuring box. The
reconstructed BSDF is attached to the external surface of the test objects.
Internal surfaces are simulated as ideally smooth and have perfect Fresnel
properties. The medium of all objects has the refractive index = 1.49, which
corresponds to measured samples.
Figure
14:
Scheme for visualization
The scene is
observed at a finite distance with a special sensor emulating the human eye or
camera. The image is generated with the help of simple forward Monte Carlo ray
tracing technique in Lumicept [35]. Although it is not the most effective tool
nowadays from viewpoint of efficiency and calculation speed, and generated
images, as a rule, contain noise, however it is a more reliable and safe tool
because of its simplicity.
The results of the
simulation are presented in two variants:
1.
As graphs with angular
distribution of transmitted light intensity. A special scene to simulate the
characteristic as precisely as possible has been prepared, which is maximally
close to the measurement scheme of GP200 goniophotometer [31]. The simulation
was done for normal incident direction of parallel light in one plane
corresponding to the plane of light incidence (“sigma” = 0deg). All six graphs
(one measured with GP200 + all five reconstructed methods) are combined into
the single graph picture.
2.
As images generated as
it is specified in section 3.3. The images are generated with the help of
simple forward Monte Carlo ray tracing renderer in Lumicept simulation system [35].
The simulation is fulfilled for all five methods of BSDF reconstruction
explained in section 3.2.
Figures 15 and 16
present graphs of angular intensity distribution of transmitted light for
normal incident direction (sigma = 0deg). More results for other incident
directions are published in [40].
Figure
15:
Angular intensity distribution of transmitted light for samples #1, #2, #3, and
#4.
Figure
16:
Angular intensity distribution of transmitted light for samples #5, #6, #7, and
#8
Fig. 17 presents
images generated for samples #1#4. Each row in the figure presents the same
simulated sample and rows presents different approached of BSDF reconstruction.
Fig. 18 presents samples #5#8.
Figure
17:
Images for samples #1#4.
Figure
18:
Images for samples #5#8.
The general
numerical difference (error) between measured and simulated angular
intensity distribution of transmitted light is estimated as root mean square
deviation (RMSD) reduced to the maximal value of measured intensity in relative
shape (*100%):

(18)

where
𝐼_{𝑚}
is the measured intensity and
𝐼_{𝑠}
is calculated. Index
i
means the
value of the intensity defined for a specific direction of illumination and
observation. All observation directions (in the range of ±90deg with step = 0.1deg)
and all illumination directions (sigma = 0, 15, 30, 45 and 60deg) are used in the
calculation of the difference.
The value of “error” for all samples and all BSDF
reconstruction methods is combined in Table 2. The best result (the lowest
error) is highlighted (bolded) for each sample.
Table 2.
Error
for different BSDF reconstruction methods
Sample
name

Measured_Heights
(ray)

Measured_heights
(wave)

Analytical
(“GGX”)

Numerical
(“Normals”)

Numerical
(“Heights”)

Sample
#1
(Rq
= 0.23μ m)

1.85%

1.83%

3.07%

2.94%

3.15%

Sample
#2
(Rq
= 0.58μm)

1.45%

1.64%

1.91%

1.81%

1.90%

Sample
#3
(Rq
= 0.87μm)

1.29%

1.33%

2.02%

1.57%

2.37%

Sample
#4
(Rq
= 0.96μm)

0.85%

1.07%

2.07%

1.79%

1.96%

Sample
#5
(Rq
= 1.47μm)

1.51%

3.75%

2.06%

1.69%

2.04%

Sample
#6
(Rq = 2.67μm)

2.80%

4.60%

3.03%

1.27%

2.68%

Sample
#7
(Rq
= 3.31μm)

2.93%

4.14%

3.19%

1.68%

2.75%

Sample
#8
(Rq = 13.46μm)

5.56%

27.22%

3.86%

5.23%

3.82%

As we can see from
the results in Table 2 the most of the investigated methods work well. The
exception is “Measured_Heights (wave)” based on Kirchhoff approximation, where
we see a noticeable error for samples #5#8 with Rq > 1μm.
This is also clearly seen on graphs
(Fig. 16). It is quite predictable analyzing the restrictions of the Kirchhoff
approximation based method listed in chapter 2.3. So, the “Measured_Heights
(wave)” method cannot be recommended for samples with roughness Rq > 1μm.
From the other side, analyzing graphs
for samples #1#3 (Fig. 15), the wave optics approach gives more close results
in the shape of angular distribution of transmitted light. Moreover the wave
approach almost does not require optimization of measured height distribution
unlike the “Measured_Heights (ray)” based on ray optics. It can be explained
with the big sensitivity of the ray approach to the quality of microrelief
measurements (noise, step between measured nodes).
The analytical
“GGX” method (the improved CookTorrance model) works reasonably for all
samples. In the case of “GGX” the noticeable inaccuracy appears only for
samples with big roughness. So, considering its simplicity because only one
parameter manages BSDF shape and analytical type of calculation, the method can
be recommended for modeling rough surfaces with average microroughness.
Comparison of
methods based on measured height distribution (“Measured_Heights (ray)” and
“Measured_Heights (wave)”) for samples with small microroughness versus all
other methods demonstrates that agreement between measured and simulated
intensity is better for methods which use measured geometry of microroughness, especially
for big illumination angles. Both numerical methods show good agreement with
measurements practically for all examined samples. The numerical “Normals”
method is slightly better than numerical “Heights” in the area of general
transmission estimation, has better convergence during optimization, and is
simpler in calculation. Surprisingly, “Measured_Heights (ray)” works not well
for sample #8 (Fig. 16, Table 2). One of the possible reasons is too small
measurement area of height distribution, so it is just not representative.
In general the
methods based on measured height distribution are supposed to be more precise
because the real profile geometry is used during ray transformation. In the
case of the “Measured_Heights (ray)” method the interreflection, shading, and
masking effects are supported in the whole volume because it uses the Monte Carlo
ray tracing. This method can suffer from the restrictions of ray optics,
inaccuracy of measurements of height distribution or measured fragments are not
representative. However, these drawbacks are overcome with modification of
microrelief during optimization procedure, at least partially. Thus the “Measured_Heights
(ray)” method can be considered as reference (“etalon”) one in visual comparison
with other methods, so all other methods are compared to it.
During visual
comparison (Fig. 17 and 18) we see the images of the flat plane with average
illumination and observation inclination are similar to each other. The
situation with the curved object is more complicated. The images generated with
the analytical “GGX” method are similar to the reference (etalon) image in the
case of small and average microroughness. The effect of “a dark ring in the
sphere” is absent for any images created with the “GGX” BSDF, however the
curved objects look darker for samples with big roughness. Likely the energy
conversation works not so well in approximations of analytical methods for
samples with noticeable microroughness. The numerical “Normals” approach
generates quite good images for samples #1#4 (Fig. 17) but images for rougher
samples #58 (Fig. 18) have noticeable artifacts like bright edge ring in the
sphere. The reason for the effect is evident: the approach does not support
interreflection, masking, and shadowing. From the viewpoint of visual
appearance the numerical “Height” method shows the best results (most close to
the etalon images) for all samples.
Summarizing all
simulated data we can recommend the numerical “Heights” distribution method as
more accurate in case the precise simulation is required and there are no
measurements of microrelief geometry. In case of rather small roughness the
analytical “GGX” or the numerical “Normals” can be sufficient.
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