REALISTIC TRANSMISSION MODEL OF ROUGH SURFACES

Huiying Xu and Yinlong Sun

Department of Computer Sciences, Purdue University, West Lafayette, Indiana 47907, USA

Keywords: Transmission model, BTDF, rough surface, Monte Carlo simulation, light scattering.

Abstract: Transparent and translucent objects involve both light reflection and transmission at surfaces. This paper

develops a realistic transmission model of rough surfaces using the statistical ray method, which is a

physically based approach that has been developed recently. The surface is assumed locally smooth and

statistical techniques can be applied to calculate light transmission through a local illumination area. We

have obtained an analytical expression for single scattering. The analytical model has been compared to our

Monte Carlo simulations as well as to the simulations by others, and good agreements have been achieved.

The presented model has a potential for realistic rendering of transparent and translucent objects.

1 INTRODUCTION

Light scattering by objects is generally characterized

by a bidirectional scattering distribution function

(BSDF) (Glassner, 1995)

(, ,)

(, , , ,)

(, ,)cos

oo o

iioo

iii i i

θϕλ

ρθ ϕ θ ϕ λ

θϕλ θ

, (1)

which is the ratio of the scattered radiance

the outgoing direction

(, )

to the irradiance

cos

iii

Ω in the direction (, )

(Figure 1) at

wavelength

. When referring to reflection or

transmission, a BSDF becomes a bidirectional

reflectance distribution function (BRDF) or a

bidirectional transmittance distribution function

(BTDF). This paper studies the case of transmission.

In computer graphics application, materials may

be classified into three major types: opaque,

transparent and translucent. An opaque object only

involves reflection, a transparent object involves

both reflection and transmission, and a translucent

object has volumetric scattering in addition to

reflection and transmission at the object surface.

Thus, a transmission model is needed for not only

transparent but also translucent objects. Such objects

include glass wares, plastics, ices, biological tissues,

marbles, waxes, and so on.

There has been extensive research on modelling

BRDFs in computer graphics, but studies on BTDFs

are limited. Different from at an opaque surface, a

scattering process at a surface of some transparent or

translucent material is generally a combination of

reflection and transmission events, and the number

of the events may be one (single scattering), two or

more (multiple scattering). Solving the case of single

scattering is a basis for solving the case of multiple

scattering.

Figure 1: Light scattering at a surface (transmission case).

This paper presents a realistic transmission

model of rough surfaces. The model is derived using

a physically based approach called the statistical ray

method that has been developed recently by Sun

(2007). The key assumption of the surface is that the

surface is sufficiently smooth locally and statistical

techniques can be applied to calculate light

transmission through a local illumination area. We

have obtained an analytical expression for single

scattering. The model has been compared to our

Monte Carlo simulations as well as to the

simulations by others, and good agreements have

been achieved.

Xu H. and Sun Y. (2007).

REALISTIC TRANSMISSION MODEL OF ROUGH SURFACES.

In Proceedings of the Second International Conference on Computer Graphics Theory and Applications - GM/R, pages 77-84

DOI: 10.5220/0002082600770084

 SciTePress

2 BACKGROUND

Existing BRDF models commonly consist of the

diffuse and specular terms. The diffuse component is

typically Lambertian, but the specular term differs in

various models. A simple approach describes the

specular component with an empirical function, such

as the models of Phong (1975), Ward (1992), and

Lafortune (1997).

Deriving accurate models needs physically based

approaches. One approach uses the Kirchhoff theory

with the tangent plane approximation of the surface

(Beckmann, 1963; He, 1991). Another approach is

based on the microfacet assumption of Torrance and

Sparrow (1967). In this approach, the specular term

is expressed as a product of the Fresnel coefficient,

masking and shadowing factor, and surface

orientation probability (Blinn, 1977; Cook and

Torrance, 1982). Ashikhmin et al. (2000) developed

an analytic model to remove the limitation of V-

shaped grooves needed for the traditional microfacet

model. Recently, Sun (2007) proposed a statistical

ray method for deriving illumination models of

rough surfaces. This method will be employed to

model light transmission in this paper.

To our best knowledge, two transmission models

exist in computer graphics. The first was proposed

by He (1993) based on the Kirchhoff theory, and the

second by Stam (2001) as an extension from Cook-

Torrance’s reflection model (1982). In practice, the

rendering of light transmission is rather simple,

typically based on a formula that extends Phong’s

reflection model to the case of transmission.

Beyond computer graphics, some research has

been conducted to numerically simulate transmission.

One example is the work of Nieto-Vesperinas et al.

(1990) where light transmission at rough surfaces

was computed using a Monte Carlo method.

Since transmission at a surface is a part of the

problem of object translucency, we briefly review

some research on translucency models. Hanrahan

and Krueger (1993) developed a pioneering model

of subsurface scattering using the linear transport

theory. Jensen and Christensen (1998) studied light

transport in participating media using Monte Carlo

bi-directional ray tracing and volumetric photon

mapping. Dorsey et al. (1999) simulated subsurface

scattering of weathered stones using Monte Carlo

ray tracing. Pharr and Hanrahan (2000) developed a

Monte Carlo approach to solve generic scattering

equations. Stam (2001) used the radiative transfer

equation to model subsurface scattering of human

skins. Koenderink and van Doorn (2001) studied

subsurface scattering with a diffusion approximation

of light transport theory. Jensen et al. (2001)

proposed an analytic model of BSSRDF, and later

Jensen et al. (2002) developed a two-pass technique

to efficiently render translucent objects. Recently,

Wang et al. (2005) presented a technique based on

pre-computed light transport to render translucent

objects, and Mertens et al. (2005) proposed an

efficient algorithm to render the local effect of

subsurface scattering. These studies focused on the

subsurface or volumetric scattering, and light

transmission at the surface was not considered.

3 ANALYTICAL MODELING

Light transmission processes at a rough surface can

be classified into single and multiple scattering. In

single scattering (ray 1 in Figure 2), a light ray is

scattered one time (this is in fact a refraction at the

local area). In multiple scattering (ray 2 or 3 in

Figure 2), there are multiple times of reflection and

transmission. The total BTDF may be expressed as

total single multiple

ρρ

=+ , (2)

where

single

and

multiple

are the contributions from

single and multiple scattering, respectively.

Figure 2: Light transmission processes of single scattering

(ray 1) and multiple scattering (rays 2 and 3).

Now we use the statistical ray method proposed

by Sun (2007) to calculate light transmission at a

rough surface. The assumptions and conditions of

our considered surface are similar to those used by

Sun (2007) where the focus was on reflection, but

now the focus is on transmission. For convenience,

the assumptions are listed below:

1. Any surface micro-area

has size much larger

than wavelength and is sufficiently smooth such

that it can be replaced with its local tangent plane.

2. Any local illumination area

for the definition

of BTDF (Figure 1) contains many surface micro-

areas

. As a result, it is valid to use the concept

of probability of micro-areas

within

Δ .

3. The surface properties remain the same in

These properties include the material aspect such

as the optical constants, and the geometric aspect

such as the statistics of the surface profile.

GRAPP 2007 - International Conference on Computer Graphics Theory and Applications

4. The surface profile is a height field. That is, for

any line parallel with the z-axis, the line will

intersect with the surface profile exactly one time.

5. A combined probability can be approximated as a

product of the individual probabilities (see below).

6. The correlation between the incident and outgoing

directions are ignored.

As additional conditions, we assume that the surface

is isotropic and has a Gaussian height probability

density and correlation function.

Since the surface is a Gaussian height field, the

probability density function of surface height is

( ) exp( 2 )

ζσ

πσ

=−, (3)

where

is the surface height, and

is the

standard deviation or RMS.

To describe surface roughness, we need to

consider the surface height correlation. A two-point

correlation function is generally defined as

() ( )( )Chh

=< + >rrrr, (4)

which involves the average of the product of heights

at points

r and

+rr on the 0z = plane. Since the

surface is homogeneous (Assumption 3), the

correlation is independent of

. Also, because the

surface is isotropic, we can write

() ()CCr=r . A

common form of

()Cr is Gaussian, i.e.

( ) exp( )Cr r

=− , (5)

where

is the correlation length. Now we define

the surface smoothness as

= . (6)

The smaller is

, the rougher the surface; vice versa.

Given surface profile

(, )hxy

= , the orientation

of a micro-area

is described by the partial

derivatives

(, )

ζζ

′′

(, )

hxy

∂

′

∂

(, )

hxy

∂

′

∂

. (7)

From Sun (2007), the probability for the orientation

of a micro-area

ζζ

′

222

223

tan

(, ) exp

44cos

xy x y

pdd

ττθ

ζζ ζ ζ

σσθ

⎛⎞

′′ ′ ′

=−

⎜⎟

⎝⎠

(8)

where

dΩ is differential solid angle of (, )

n ,

sin

nnnn

ddd

θϕ

Ω=

. (9)

Given a micro-area

(Figure 3), the incident

radiance

(, )

iii

and the transmitted radiance

(, )

oo o

are related as,

(, )cos

(,) (, )(,, )cos

ooo o

tiiiio i

FL d

θϕ β

αλ θϕ δ α

=Ωne e

(10)

where

is the transmission angle for the incident

angle

dΩ is the solid angle in the transmission

direction,

is the solid angle in the incident

direction, and

(,)

is the Fresnel coefficient of

transmission averaged over polarizations.

(, , )

ne e

is a Dirac delta function. That is, when

n ,

e , and

e are coplanar and

sin sinn

( n is the relative

index of refraction),

(, , ) 1

=ne e ; otherwise,

(, , ) 0

ne e . The radiant flux (, )

θϕ

Φ through a

micro-area

is given as

(, ) (, )cos

(,) (, )(, , )cos

oo ooo o

tiiiio i

LAd

FL Ad

δθϕ θϕ βδ

αλ θϕ δ αδ

Φ= Ω

Ωne e

(11)

Figure 3: Ray transmission at a micro-area.

Since a local illumination area

Δ over which

the BTDF is defined (Figure 1) contains many

micro-areas

, the total radiant flux over

contains contributions from all possible micro-areas,

(, ) ( ) (, )

(, )cos ( )(,)(,, )

oo oo

iii i t io

dVAF A

θϕ δ δ θϕ

ϕα δ αλδ δ

ΔΦ = Φ

=Ω

∑

ne e

(12)

where

()VA

is a visibility function describing the

probability of a micro-area

that is visible in both

directions

(, )

ii i

e and (, )

ooo

e . The radiance to

the transmission direction

(, )

ooo

is given as

(, )

|cos |

(, )cos (,) ( )(,, )

|cos |

ooo

iii it io

dF VA A

θϕ

ϕα αλ δδ δ

ΔΦ

ΔΩ

ΩΔ

∑

ne e

(13)

Because

is measured from the positive z-axis

(see Figure 1) and its value is within

[/2,]

, we

take the absolute value of its cosine value in Eq. (13).

Substituting Eq. (13) into Eq. (1), we obtain

single

(over ) (fixed )

(, )

(, )cos

cos ( , ) ( , , )

()

cos | cos |

oo o

iii i i

tio

ioo

ζζ

θϕ

θϕ θ

αλ δ δ

θθ

ΩΔ

∑∑

ne e

(14)

Here the summation over the local illumination area

has been decomposed into the summation over

all micro-areas with fixed height

and over

REALISTIC TRANSMISSION MODEL OF ROUGH SURFACES

different heights. Since the visibility function

()VA

at a fixed height remains the same for given incident

and outgoing directions, it has been put outside the

inner summation for a fixed height. Considering that

the projected area of

on the 0z = plane is

() cos

δθ

⊥

= , (15)

where

is the polar angle of the normal (, )

(see Figure 3), the portion of the total

projected areas

(fixed )

()(,,)

δδ

⊥

∑

ne e in

Δ is the

probability of a surface point with height in

differential interval

[, ]d

ζζ

and with orientation

in intervals

[, ]

ζζ

′′ ′

and [,

′

]

′′

. Thus,

(fixed )

()(,,) (,,)

io x y x

pddd

δζζζζζζ

⊥

′′ ′ ′

∑

ne e

(16)

From Assumption 5, the combined probability

density function can be decomposed as

(,,) ()(,)

yxy

ppp

ζζ ζ ζζ

′

′′′

. (17)

Applying Eqs. (16,17) into Eq. (14), we obtain

222

single

cos ( , ) exp( tan 4)

4 cos | cos | cos

()(, , )

io n

sF s

dp V

ααλ θ

πθ θ θ

ζζ ζ

−

⋅

∫

(18)

This equation may be further expressed as

222

single

cos ( , ) exp( tan 4)

4 cos | cos | cos

(, ) (,, )

io n

sF s

ααλ θ

πθ θ θ

χαβ ζ

−

⋅ ee

(19)

where the function

(, )

αβ

describes

ddΩΩ (see

Appendix), and

(,,) ()(,,)

io io

VdpV

ζζζζ

∫

ee ee (20)

is the averaged bistatic visibility function. A bistatic

visibility function simultaneously involves the

incident direction

e and the outgoing direction

e .

For light transmission, since

e points into the

original medium and

e into the new medium, the

correlation between the two directions can be

ignored, as stated in Assumption 6. Therefore,

(, , ) (, )(, )

io i o

VVV

ζθ ζθ

≈ee

, (21)

where

(,)V

is an individual visibility function that

describes the probability of being visible for a ray

starting at height

and with angle

(Figure 4),

and accordingly,

(, , ) (, )(, )

()(, )(, )

io i o

VVV

dp V V

ζζ

ζζθζθ

≈

∫

(22)

We further approximate Eq. (22) as

(, , ) (0, )(0, )

io i o

VVV

θθ

≈ee

. (23)

where

(0, )

and (0, )

are the individual

visibility functions for the incident and outgoing

directions when the ray starts from

= . From the

previous study (Sun, 2007),

()

tan

(0, ) exp exp 4tan

⎡

⎤

≈− −

⎢

⎥

⎣

⎦

,(24)

where

0.7k = . Thus, we finally obtain

222

single

cos ( , )exp( tan 4)

4 cos | cos | cos

(,)(0, )(0, )

io n

sF s

ααλ θ

πθ θ θ

χαβ θ θ

−

⋅

(25)

where

(, )

αβ

is given in the Appendix.

Figure 4: A ray starts at height

and with polar angle

Figure 5: BTDF for different n and

. Parameters are

° , 1/1.4n

for the first row, and 1.4n = for the

second row. From the left to right, the values of

are 6, 3,

and 1, respectively.

Figure 5 shows

single

for different values of

relative index of refraction (IOR)

n and smoothness

. The solid straight lines (in green) in the upper

hemisphere indicate the incident direction, and the

solid straight lines (in blue) in the lower hemisphere

indicate the transmission direction.

single

has a sharp

lobe and shows the off-specular effect. When

(the first row), as the outgoing direction changes

from

−° to 180

° ,

single

increases gradually

and reaches a maximum, then decreases rapidly.

GRAPP 2007 - International Conference on Computer Graphics Theory and Applications

Also, the direction that

single

has the maximum

shifts toward

180

=° with the decrease of

. In

contrast, for

1n > (the second row), when the

outgoing direction changes from

=− ° to

180

=°,

single

increases rapidly and reaches the

maximum, then decreases gradually. Moreover, the

direction for the maximum

single

shifts toward

=− ° with the decrease of

The plots in Figure 5 can be explained as below.

First, when the surface is smooth, most micro-areas

distribute around

=° and they contribute to

single

with

→

. Second, Fresnel’s transmission

coefficient has the maximum for incident angle

=°, and decreases with the increase of

Therefore, those micro-areas with orientation around

the incident direction have large Fresnel’s

transmission coefficients. These two factors compete

with each other. And also, for

1n < , the refraction

angle

is larger than the incident angle

, and

vice versa. These result in the plot shapes in Figure

5. With the decrease of

, the maximum distribution

of orientations of micro-areas tends to shift from

=° toward 90

=°, which results in a shift of

the direction for the maximum

single

Figure 6: BTDF for different n and

. Here 0

° , and

other parameters and notations are the same as Figure 5.

For the normal incidence,

single

for different

values of

n and

is shown in Figure 6. For 1n

the sharp lobe becomes wider with the decrease of

, as same as Figure 5. However, for the case 1n >

in both Figures 5 and 6, although the sharp lobes for

3s = are all wider than those for 6s = , the sharp

lobes for

1s = have different shapes. Consider the

rotational geometry,

single

for 1s = in Figure 6 is

actually a lobe with an indented peak. We can

understand this by the micro-area model. For rough

surfaces (

1s = ), most micro-areas distribute with

orientations

° , and therefore the transmitted

light by single scattering tends to travel along the

direction with

90 180

°° . This results in the

indentation of the lobe in Figure 6. However, the

probability of ray blocking is higher for the rays

propagating along this direction. This results in the

sharper shape for

in Figure 5.

4 NUMERICAL SIMULATION

In our Monte Carlo simulation, given a Gaussian

rough surface with its mean equal to zero and

standard deviation

, totally

light rays are shot

from the incident direction

e , each ray carrying a

weight

( 1, 2,...,lN

) that represents its radiance

flux intensity. Once a shot ray hits the surface

profile, it typically splits into a reflected and a

transmitted ray. When the total internal reflection

occurs, only a reflected ray is generated.

The surface height at which a shot ray intercepts

with the profile is determined by the probability

density function of surface height and a generated

random number (all the generated random numbers

in this paper are uniformly distributed between 0 and

1). The normal direction of this intersection point is

obtained by the orientation probability density

function with two generated random numbers.

We set the incident flux density to 1. Then the

weight of the lth shot ray is given as

( , )cos / cos

li n

θαθ

, (26)

where

is the surface height that the shot ray first

intercepts with,

is the incident angle in the local

area, and

cos

is involved because Eq. (16) just

describes the probability distribution of

⊥

at a

fixed height.

When a ray with weight

W hits the surface

profile, it splits into a reflected and transmitted ray,

and the weight of the reflected ray is

(,)

αλ

and

that of the transmitted ray is

(,)

αλ

. Therefore,

after each ray splitting, the generated rays will

decrease in intensities. Once the weight of a newly

generated ray is lower than the threshold, the

tracking process terminates. Otherwise, it will be

tracked continuously; whether it is blocked or not

depends on its propagation direction, visibility

function, and a generated random number.

The radiance to the transmission direction

(, )

ooo

e is obtained as

|cos |

∈ΔΩ

ΔΩ

∑

, (27)

REALISTIC TRANSMISSION MODEL OF ROUGH SURFACES

where

ΔΩ is the solid angle along (, )

ooo

e , and

∈ΔΩ

∑

calculates the sum of the weights of those

rays transmitted into

ΔΩ . Consider that the incident

irradiance is

cos

(since incident flux density is set

to 1), the BTDF can be calculated by

|cos |cos

ooi

ΔΩ

∑

. (28)

In discussion below we may replace

with

|cos |

based on two considerations. First, the

previous simulations by Nieto-Vesperinas et al.

(1990) calculated the transmitted light intensity,

which is proportional to

|cos |

. For convenience

of comparing the results, we need to use

|cos |

instead of

. Second, Eq. (28) contains

1/ | cos |

and

∈ΔΩ

∑

. When

→

|cos | 0

→

. However,

we cannot take

ΔΩ →

for the calculation of Eq.

(28). Therefore,

might diverge at 90

→°.

Figure 7: Comparison between our analytical model and

simulations. The curves with x marks are from the

analytical model, the dot curves from the simulations of

Nieto-Vesperinas et al. (1990), and the solid curves from

our simulations. Here,

1.411n = , 2.522s = , (a) 0

° , (b)

=°, (c) 40

=°, and (d) 60

=°.

Figure 7 compares our analytical model and

simulations. Nieto-Vesperinas et al. (1990)

considered perpendicular and parallel polarizations

separately. For comparison, we calculate the average

of the two polarizations. In our analytical model and

simulation, light intensity can be calculated by

|cos |

ρθ

Δ . Since we do not know the value of

Δ used for the simulation of Nieto-Vesperinas et

al., we find it by matching our analytical model with

their results for

° . In Figure 7, the comparison

shows a very good match.

Figure 8: Comparison between simulation and analytical

model. The solid curves are from our simulation, and the

curves with x marks from analytical model. Here,

° ,

1/1.4n =

, (a)

, (b)

3s =

, (c)

1s =

, and (d)

0.5s

Figure 9: Comparison between simulation and analytical

model. The solid curves are from our simulation, and the

curves with x marks from analytical model. Here,

° ,

1.4n =

, (a)

, (b)

, (c)

1s =

, and (d)

0.5s =

Figures 8 and 9 compare our simulation and the

analytical model for different values of

n and

For smooth and moderately smooth surfaces (

is 3

or 6), the analytical model agrees well with the

simulation. With small

, the difference between

the analytical model and simulation increases. This

is because our analytical model only considers single

scattering. For smooth surfaces, light transmission is

dominated by single scattering. Overall, the model

has a good match with the simulation. For rough

surfaces (

is small), multiple scattering plays an

important role and should be considered.

GRAPP 2007 - International Conference on Computer Graphics Theory and Applications

5 CONCLUSIONS

This paper presents a realistic transmission model of

rough surfaces. The model is derived based on the

statistical ray method. We have obtained an

analytical expression for single scattering. The

model has been compared to our Monte Carlo

simulations as well as to the simulations by others,

and good agreements have been achieved.

In future work, the model can be applied to

render realistic transmission effects. The model

could be taken into consideration to study object

translucency. On simulation to verify the analytical

model, we may generate 2D surfaces for given

and

, and compute the average of transmission

through the surfaces. The current model has not

considered multiple scattering, and both the model

and simulation have not considered polarization

effects. We will consider them in our further work.

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APPENDIX

Here we derive the relationship between the

differential solid angles

and

dΩ . Given an

unit sphere (Figure 10), the area

BCD corresponds

and the area

BCD

′

′′′

dΩ . The points

′

, and

′

are coplanar, and similarly the

points

C , D , C

′

, and D

′

. The planes

BA B

′

and

DCD C

′

intersects at the line POQ , and the angle

REALISTIC TRANSMISSION MODEL OF ROUGH SURFACES

between them is

. Therefore, the length of the

curve segment

D is

|||| sin

DAEd d

αγ

=⋅=⋅

. (A1)

Since

= , so we obtain

||||sindADAB dd

γα

Ω= ⋅ = ⋅

. (A2)

(a)

(b)

Figure 10: Relationship between

dΩ and

dΩ ( 1n > ).

The Snell’s law gives the following relations:

sin sinn

=⋅ (A3)

and

sin sin( ) sin sin( )dn n d

αα β ββ

′′

=+=⋅=⋅+

, (A4)

where

n is the relative index of refraction ( 1n > )

and

is defined as

ββ

′

=−

. (A5)

From Eq. (A4), we can obtain

sincos()cossin()

[sin cos( ) cos sin( )]

nd d

αααα

βββ

⋅+⋅

=⋅ ⋅ + ⋅

(A6)

We make the following approximations:

cos( ) 1, sin( ) ,

cos( ) 1, sin( ) .

ddd

αα

ββ

≈≈

(A7)

Substituting Eqs. (A3) and (A7) into Eq. (A6), we

obtain

cos

⋅

. (A8)

From Figure 10(b), we obtain

OB d

βα

′

′′

−∠ + . (A9)

Therefore, we obtain

cos

OB d

⎛⎞

′′

∠=−

⎜⎟

⋅

⎝⎠

. (A10)

The length of the curve segment

′′

cos

|| 1

cos

BAOB d

⎛⎞

′′ ′ ′

=∠ = −

⎜⎟

⋅

⎝⎠

. (A11)

The length of the curve segment

′′

||||

sin( ) sin( )

AD AE d

OQ d d

αβγ

′

′′′

=⋅

′

=∠ = −

(A12)

Therefore, we obtain

||||

cos

1sin()

cos

dABAD

βγα

′′ ′′

Ω= ⋅

⎛⎞

=− −

⎜⎟

⎝⎠

(A13)

Finally, we obtain

sin cos

(, ) 1

sin( ) cos

αα

χαβ

αβ β

−

⎛⎞

≡= −

⎜⎟

Ω−⋅

⎝⎠

(A14)

Although Eq. (A14) is derived for

1n > , it is easy to

prove that this expression also holds for

1n < .

GRAPP 2007 - International Conference on Computer Graphics Theory and Applications