Generalization of Lambert''s reflectance model.
Lambert's model for body reflection is widely used in computer graphics. It is used extensively by rendering techniques such as radiosity and ray tracing. For several real-world objects, however, Lambert's model can prove to be a very inaccurate approximation to the body reflectance. While the brightness of a Lambertian surface is independent of viewing direction, that of a rough surface increases as the viewing direction approaches the light source direction. In this paper, a comprehensive model is developed that predicts body reflectance from rough surfaces. The surface is modeled as a collection of Lambertian facets. It is shown that such a surface is inherently non-Lambertian due to the foreshortening of the surface facets. Further, the model accounts for complex geometric and radiometric phenomena such as masking, shadowing, and interreflections between facets. Several experiments have been conducted on samples of rough diffuse surfaces, such as, plaster, sand, clay, and cloth. All these surface demonstrate significant deviation from Lambertian behavior. The reflectance measurements obtained are in strong agreement with the reflectance predicted by the model.
Introduction
An active area of research in computer graphics involves the
creation of realistic images. Images are rendered using one
of two well-known techniques, namely, ray tracing [34] or
radiosity [6]. The quality of a rendered image depends to a
great extent on the accuracy of the reflectance model used.
In the past decade, computer graphics has witnessed the application
of several physically-based reflectance models for
image rendering (see [7], [16], [9], [13]). Reflection from a
surface can be broadly classified into two categories: surface
reflectance which takes place at the interface between two
media with different refractive indices and body reflectance
which is due to subsurface scattering. Most of the previous
work on physically-based rendering has focused on accurate
modeling of surface reflectance. They predict ideal specular
reflection from smooth surfaces as well as wide directional
lobes from rougher surfaces [13]. In contrast, the body component
has most often been assumed to be Lambertian. A
Lambertian surface appears equally bright from all direc-
tions. This model was advanced by Lambert [18] more than
200 years ago and remains one of the most widely used models
in computer graphics.
For several real-world objects, however, the Lambertian
model can prove to be a poor and inadequate approximation
to body reflection. Figure 1(a) shows a real image of
a clay vase obtained using a CCD camera. The vase is illuminated
by a single distant light source in the same direction
as the sensor. Figure 1(b) shows a rendered image
of a vase with the same shape as the one shown in Figure
1(a). This image is rendered using Lambert's model, and the
same illumination direction as in the case of the real vase.
As expected, Lambert's model predicts that the brightness
(a) (b)

Figure

1: (a) Real image of a cylindrical clay vase. (b) Image
of the vase rendered using the Lambertian reflectance model. In
both cases, illumination is from the viewing direction.
of the cylindrical vase will decrease as we approach the occluding
boundaries on both sides. However, the real vase
is very flat in appearance with image brightness remaining
almost constant over the entire surface. The vase is clearly
not Lambertian 1 . This deviation from Lambertian behavior
can be significant for a variety of real-world materials,
such as, concrete, sand, and cloth. An accurate model that
describes body reflection from such commonplace surfaces
is imperative for realistic image rendering.
1 Note that the real vase does not have any significant specular
component, in which case, a vertical highlight would have appeared
in the middle of the vase.
What makes the vase shown in Figure 1(a) non-
Lambertian? We show that the primary cause for this deviation
is the roughness of the surface. Figure 2 illustrates
the relationship between magnification and reflectance (also
see [16]). The reflecting surface may be viewed as a collection
of planar facets. At high magnification, each picture
element (rendered pixel) includes a single facet. At lower
magnifications, each pixel can include a large number of
facets. Though the Lambertian assumption is often reason-able
when looking at a single planar facet, the reflectance is
not Lambertian when a collection of facets is imaged onto a
single pixel. This deviation is significant for very rough sur-
faces, and increases with the angle of incidence. In this pa-
per, we develop a comprehensive model that predicts body
reflectance from rough surfaces, and provide experimental
results that support the model. Lambert's model is an in-
stance, or limit, of the proposed model.
Reflectance and Scale
pixel
pixel
Specular facets (Torrance-Sparrow)
Lambertian facets (Oren-Nayar)

Figure

2: The roughness of a surface causes its reflectance properties
to vary with image magnification.
The topic of rough surfaces has been extensively studied
in the areas of applied physics, geophysics and engineer-
ing. The development of the first few models were primarily
motivated to describe the non-Lambertian behavior of the
moon. Some of the models are empirical such as Opik's
model [22] and its modification by Minnaert [19]. These
models do not have any physical foundation and have been
found to be incorrect. In contrast, Smith [29] and Buhl et al.
[3] attempted to develop theoretical models for reflectance
from rough surfaces. Smith modeled the rough surface as
a random process and assumed each point on the surface
to be Lambertian in reflectance. Smith's analysis, however,
was confined to the plane of incidence and is not easily extensible
to reflections outside this plane. Moreover, Smith's
model does not account for interreflection effects. Buhl et al.
[3] modeled the surface as a collection of spherical cavities.
They analyzed interreflections using this surface model, but
did not present a complete model that accounts for masking
and shadowing effects for arbitrary angles of reflection and
incidence. Subsequently, Hering and Smith [14] derived a
detailed thermal emission model for surfaces modeled as a
collection of V-cavities. However, all cavities are assumed to
be identical and aligned in the same direction, namely, perpendicular
to the source-viewer plane. Further, the model
is limited to the plane of incidence.
More recently, body reflection has emerged as a topic of
interest in the graphics community. Poulin and Fournier [26]
derived a reflectance function for anisotropic surfaces modeled
as a collection of parallel cylindrical sections. Addressing
a different cause for non-Lambertian reflectance from
the one discussed here, Hanrahan and Krueger [10] used linear
transport theory to analyze subsurface scattering from
a multi-layered surface. Other researchers in graphics have
numerically pre-computed fairly complex reflectance functions
and stored the results in the form of look-up tables or
coefficients of spherical harmonic expansion (for examples,
see [4] [16] [33]). This approach, though practical in many
instances, does not replace the need for accurate analytical
reflectance models.
The reflectance model developed here can be applied to
isotropic as well as anisotropic rough surfaces, and can handle
arbitrary source and viewer directions. Further, it takes
into account complex geometrical effects such as masking,
shadowing, and interreflections between points on the sur-
We begin by modeling the surface as a collection of
long symmetric V-cavities. Each V-cavity has two opposing
facets and each facet is assumed to be much larger than
the wavelength of incident light. This surface model was
used by Torrance and Sparrow [30] to describe incoherent
directional component of surface reflection from rough sur-
faces. Here, we assume the facets to be Lambertian 2 . First,
we develop a reflectance model for anisotropic surfaces with
one type (facet-slope) of V-cavities, with all cavities aligned
in the same direction on the surface plane. Next, this result
is used to develop a model for the more general case of
isotropic surfaces that have normal facet distributions with
zero mean and arbitrary standard deviation. The standard
deviation parameterizes the macroscopic roughness of the
surface. The fundamental result of our work is that the body
reflectance from rough surfaces is not uniform but increases
as the viewer moves toward the source direction. This deviation
from Lambert's law is not predicted by any previous
reflectance model.
We present several experimental results that demonstrate
the accuracy of our model. The experiments were
conducted on real samples such as sand, plaster, and cloth.
In all cases, reflectance predicted by the model was found
to be in strong agreement with measurements. The derived
model has been implemented as a shading function in RenderMan
[32]. We conclude by comparing real and rendered
images of a variety of objects. These results demonstrate
two points that are fundamental to computer graphics. (a)
Several real-world objects have body reflection components
that are significantly non-Lambertian. (b) The model presented
in this paper can be used to create realistic images
of a variety of real-world objects.
Surface Roughness Model
The effects of shadowing, masking and interreflection need
to be analyzed in order to obtain an accurate reflectance
model. To accomplish this, we use the roughness model
proposed by Torrance and Sparrow [30] that assumes the
surface to be composed of long symmetric V-cavities (see

Figure

3). Each cavity consists of two planar facets. The
width of each facet is assumed to be small compared to its
length. We assume each facet area da is small compared to
the area dA of the surface patch that is imaged by a single
sensor pixel. Hence, each pixel includes a very large number
of facets. Further, the facet area is large compared to the
wavelength - of incident light, and therefore geometrical
optics can be used to derive the reflectance model. The
above assumptions can be summarized as: - 2 - da - dA
We denote the slope and orientation of each facet in the
V-cavity model as ('a ; OE a ), where 'a is the polar angle and
This assumption does not limit the implications of the reflectance
model presented here. The non-Lambertian behavior reported here is
expected for a wide range of local body reflectance models (see [ 5 ] , for
example) since surface roughness is shown to play a dominant role.
da
dA
a
q a

Figure

3: Surface modeled as a collection of V-cavities.
OE a is the azimuth angle. Torrance and Sparrow have assumed
all facets to have equal area da. They use the distribution
a) to represent the number of facets per unit
surface area that have the normal - a = ('a ; OE a ). Here, we
use a probability distribution to represent the fraction of the
surface area that is occupied by facets with a given normal.
This is referred to as the slope-area distribution P ('a ; OE a ).
The facet-number distribution and the slope-area distribution
are related as follows:
The slope-area distribution is easier to use than the
number distribution in the following model derivation. For
since the distributions are rotationally symmetric
with respect to the global surface normal -

Figure

3).
3 Reflectance Model
In this section, we derive a reflectance model for rough diffuse
surfaces. For lack of space, only important results are
discussed. For details we refer the reader to Oren and Nayar
[23, 24]. During the derivation, we will draw on several
well-known radiometric definitions that are given in [20].
Consider a surface area dA that is imaged by a single
sensor element in the direction -
by a distant point light source in the direction -
The area dA is composed of a very large number of symmetric
V-cavities. Each V-cavity is composed of two facets
with the same slope but facing in opposite directions. Consider
the flux reflected by a facet with area da and normal
The projected area on the surface occupied by
the facet is da cos 'a (see Figure 3). Thus, while computing
the contribution of the facet to the radiance of the surface
patch, we need to use the projected area da cos 'a and not
the actual facet area da. This radiance contribution is what
we call the projected radiance of the facet:
(da cos 'a) cos 'r d!r
(2)
where, d!r is the solid angle subtended by the sensor optics.
For ease of description, we have dropped the source and
viewing directions from the notations for projected radiance
and flux. Now consider the slope-area distribution of facets
given by P ('a ; OE a ). The total radiance of the surface can be
obtained as the aggregate of Lrp ('a ; OE a) over all facets on
the surface:
Z -'a=0
3.1 Model for Uni-directional Single-Slope
Distribution
The first surface type we consider has all facets with the
same slope 'a . Further, all V-cavities are aligned in the same
direction; azimuth angles of all facets are either OE a or OE a +-.
Consider a Lambertian facet with albedo ae, that is fully illuminated
(no shadowing) and is completely visible (no mask-
ing) from the sensor direction. The radiance of the facet
is proportional to its irradiance and is equal to ae
The irradiance of the facet is E('a
s; - a?, where,
E0 is the irradiance when the facet is illuminated head-on
(i.e. -
denotes the dot product between
two vectors. Using the definition of radiance [20],
the flux reflected by the facet in the sensor direction is:
s; - a?! -
v; -a ?. Substituting this expression in
(2), we get:
The above expression clearly illustrates that the projected
radiance of a tilted Lambertian facet is not equal in all viewing
directions.
Geometric Attenuation Factor: If the surface is illuminated
and viewed from the normal direction
n), all facets are fully illuminated and visible. For larger
angles of incidence and reflection, however, facets are shadowed
and masked by adjacent facets (see Figure 4). Both
these geometrical phenomena reduce the projected radiance
of the facet. This reduction in brightness can be derived
using geometry and incorporated into a single term, called
the geometrical attenuation factor (GAF ), that lies between
zero and unity. Several derivations of the GAF have been
presented [30] [2] [23]. The final result can be compactly
represented as:
a?
The above GAF is valid for any facet normal, - a, not necessarily
the bisector of the angle between the source and the
sensor direction.
x
z
s
q a x
z
x
z
y
x
s
(a) Shadowing (b) Masking (c) Interreflection

Figure

4: Shadowing, masking and interreflection in a V-cavity
Projected Radiance and GAF : The projected radiance
of a Lambertian facet is obtained by multiplying the projected
radiance given by (4) with the GAF given by (5).
Note that the projected radiance is denoted as L 1
rp ; the superscript
is used to indicate that the radiance is due to direct
illumination by the source. In the following discussion, we
will use L 2
rp to denote radiance due to interreflections.
Interreflection Factor: We have the task of modeling interreflections
in the presence of masking and shadowing ef-
fects. In the case of Lambertian surfaces, the energy in an
incident light ray diminishes rapidly with each interreflection
bounce. Therefore, we model only two-bounce interreflections
and ignore subsequent bounces. Since the length l of
the V-cavity is much larger than its width w, i.e. l AE w, it
can be viewed as a one-dimensional shape with translational
symmetry. For such shapes, the two-bounce interreflection
component can be determined as an integral over the one-dimensional
cross-section of the shape [28]:
Z
r (y)dy (7)
where x and y are the shortest distances of facet points from
the intersection of the two facets (see Figure 4(c)). K 0 is the
kernel for the translational symmetry case and is derived in
[15] and [8] to be:
We know that the orientation of the considered facet is -
a) and the orientation of the adjacent facet is - a
-). The limits of the integral in the interreflection
equation are determined by the masking and shadowing of
these two facets. Let mv be the width of the facet which is
visible to the viewer, and m s be the width of the adjacent
facet that is illuminated. From the definitions of radiance
and projected radiance we get:
v?
da !-a; -
r (x) dx
Using the following change of variables:
w , the
radiance due to two-bounce interreflections given by (7) and
can be written as:
Using (8), the above integral is evaluated as:
--
where:
We refer to
right hand side of equation (11) as the interreflection factor
(IF). The total projected radiance of the facet is the sum
of two the components, the radiance due to direct source
illumination given by equation (6) and the above interreflection
component. Therefore, Lrp('a ; OE
The uni-directional single-slope surface considered
here has only two types of facets with normals ('a ; OE a)
and ('a ; OE a -). Hence, the radiance of the surface for any
given source and sensor directions is simply the average of
the projected radiances of the two facet types.
3.2 Model for Isotropic Single-Slope Dis-
tribution
All facets on this isotropic surface have the same slope 'a
but are uniformly distributed in OE a . As we did in the previous
section, we evaluate the projected radiance as the sum of
two components: projected radiance due to direct illumina-
tion,
rp ('a ), and projected radiance due to interreflection,
In the previous section, we calculated each of the
two components for a single facet with normal -
Therefore, the radiance of the isotropic surface is determined
as an integral of the projected radiance over OE a :
Given a source direction (' direction
first find the ranges of facet orientation OE a for
which the facets are masked, shadowed, masked and shad-
owed, and neither masked nor shadowed 3 . This requires
careful geometrical analysis. Then the above integral is decomposed
into parts corresponding to masking/shadowing
ranges. Each range is evaluated using the corresponding
radiance expressions (6) and (11).We refer the interested
reader to Oren and Nayar [23, 24] for details.
3.3 Model for Gaussian Slope-Area Distri-
bution
The surface considered above consists of V-cavities with a
single slope. Realistic surfaces can be modeled only
if the slope-area distribution P ('a ; OE a) includes a variety of
different facet slopes. If the surface roughness is isotropic,
the slope-area distribution can be described using a single
parameter namely 'a since the facets are uniformly distributed
in OE a . The two components of the radiance of any
isotropic surface can therefore be determined as:
Z -0
rp ('a) sin 'a d'a (13)
are the projected radiance components
obtained in the previous section. Here, we assume
the isotropic distribution to be Gaussian with mean - and
standard deviation oe, i.e. P ('a ; oe; -). Reasonably rough
surfaces can be described using a zero mean
distribution:
a =2oe 2
where c is the
normalization constant.
3.4 Functional Approximation
The reflectance model is to be obtained by evaluating integral
using the results of section 3.2 and the Gaussian
distribution, 0). The resulting integral cannot be
easily evaluated. Therefore, we pursued a functional approximation
to the integral that is accurate for arbitrary surface
roughness and angles of incidence and reflection. In deriving
this approximation, we carefully studied the functional
forms of L i
which were evaluated in the previous
step (the details can be found in Oren and Nayar [23,
24]). This enabled us to identify basis functions that can
be used in the approximation. Then, we conducted a large
set of numerical evaluations of the integral in (13) by varying
surface roughness oe, the angles of incidence
reflection ('r ; OE r ). These evaluations and the identified basis
functions were used to arrive at an accurate functional
approximation for surface radiance. This procedure was applied
independently to the source illumination component
as well as the interreflection component.
The final approximation results are given below. We define
The source illumination
component of radiance of a surface with roughness
oe is:
3 Imagine a V-cavity rotated about the global surface normal for
any given source and sensor direction. Various masking/shadowing
scenarios can be visualized.
where the coefficients are:
sin ff if cos (OE
sin
otherwise
The approximation to the interreflection component is:
The two components are combined to obtain the total surface
radiance:
Finally, the BRDF of the surface is obtained from its radiance
and irradiance as fr ('r
. It is important to note that the approximation
presented above obeys Helmholtz's reciprocity principle
(see [1]). Also note that the above model reduces to the Lambertian
model when Note that by substituting the
albedo as function of the wavelength, ae(-), the dependency
of the model on the wavelength comes out explicitly.
In the next section, we present several experimental results
that verify the above diffuse reflectance model. Here,
we give a brief illustration of the main characteristics of
the model. Figure 5 shows the reflectance predicted by the
model for a very rough surface with
The radiance Lr in the plane of incidence (OE
plotted as a function of the reflection angle 'r for incidence
angle Two curves are shown in the figure, both
obtained by the numerical evaluation of the integral in (13).
The first curve (solid line) includes both direct illumination
and interreflection components of radiance, while the
second (thin line) is only the direct illumination compo-
nent. Notice that these radiance plots deviate substantially
from Lambertian reflectance. Surface radiance increases as
the viewing direction approaches the source direction. The
curves can be divided into three sections. In the backward
(source) direction, the radiance is maximum and gets "cut-
" due to strong masking effects when 'r exceeds ' i . This
cut-off occurs exactly at and is independent of rough-
ness. In the middle section of the plot, radiance varies approximately
as a scaled tan 'r function with constant offset.
Finally, interreflections dominate in the forward direction
where most facets are self-shadowed and the visible facets
receive light primarily from adjacent facets. This is illustrated
by the difference between the two curves.
In

Figure

6(a), the effect of varying the incidence angle ' i
is shown. Here we have chosen to plot BRDF rather than
radiance to better illustrate the effect of varying ' i . It is
r
r
strong
masking
strong
interreflection
Lambertian
forward backward

Figure

5: Diffuse reflectance in the plane of incidence for a surface
with incidence angle ' . The
thin line is radiance due to direct illumination (without inter-
reflections).
f r
f
f r
f
f r
f
(a) (b)

Figure

(a) BRDF for different angles of incidence. (b)
Radiance outside the plane of incidence. In both plots,
interesting to note that the model predicts near-Lambertian
behavior for very small incidence angles (' i - 0). This results
from both facets of a V-cavity having nearly equal irradiance
for small angles of incidence. As the incidence angle
increases, the backscatter phenomenon begins to dominate.

Figure

6(b) shows the effect of placing the sensor outside
the plane of incidence. When the sensor-normal plane is
perpendicular to the source-normal plane, the rough surface
again exhibits near-Lambertian characteristics.
Qualitative Model
In this section, we propose a further simplification to the reflectance
model presented in the previous section. In order
to obtain this simplification, a slight sacrifice in accuracy
must be made. In return, some computations can be saved
during image rendering. The following simplified model was
arrived at by studying, through numerous simulations, the
relative significance of various terms in the functional approximation
given by (14). The simulations showed that
coefficient C3 makes a relatively small contribution to the
total radiance. A simpler model is thus obtained by discarding
C3 and ignoring interreflections:
ae
sin ff tan fi)
The two coefficients A and B are obtained directly from C1
and C2 , respectively. Note that the qualitative model also
reduces to the Lambertian model when
4 Experimental Verification
We have conducted several experiments to verify the accuracy
of the reflectance model. The experimental set-up ([23,
24]) used to measure the radiance of samples is shown in
figure 7.
camera
light source
sample

Figure

7: Sketch and photograph of the set-up used to measure
reflectance.

Figures

8 and 9 shows results obtained for samples of
wall plaster (A) and sand (B). The radiance of each sample
is plotted as a function of sensor direction 'r for different
angles of incidence ' i . These measurements are made in the
plane of incidence (OE 0). The measured brightness
values, shown as dots, are compared with those predicted
by the model plotted as solid lines. For these two
samples and B), oe and ae were selected empirically to
obtain the best match between measured and predicted re-
flectance. Here, we have used the numerical evaluation of the
model (equation 13). For both samples, radiance increases
as the viewing direction 'r approaches the source direction
reflection). This is in contrast to the behavior
of rough specular surfaces that reflect more in the forward
direction, or Lambertian surfaces where radiance does not
vary with viewing direction. For both samples, the model
predictions and experimental measurements match remarkably
well. In both cases, a small peak is noticed near the
source direction. This phenomenon is known as the opposition
effect or retroreflection. It is a sharp peak close to the
source direction and is caused by a different backscattering
mechanism from the one described by our model. (see [12,
17, 31, 21, 27, 11]).

Figures

show results for a sample C (foam) and
sample D (cloth) that has not only a body reflectance component
but also a significant surface reflection component.
In this case, the reflectance model used is a linear combination
of new model and the Torrance-Sparrow model [30]
that describes the incoherent directional component of surface
reflection and which is based on the same surface model
(long symmetric
ks L s
r , where
r and L s
r are the body and surface reflection components,
respectively. kb and ks are weighting coefficients for the two
components. For this experiment, we used the functional
approximation and the reflectance parameters oe, ae, kb , and
ks were estimated by fitting (using non-linear optimization)
the model to measured data. Additional experiments are
reported in Oren and Nayar [23].
Wall Plaster
r
Figure

8: Reflectance measurement and reflectance model (using
plots for wall plaster (sample A). Radiance is
plotted as a function of sensor direction ('r ) for different angles
of incidence ('
Sand
r
Figure

9: Reflectance measurement and reflectance model (using
plots for white sand (sample B).
Foam
r

Figure

10: Reflectance measurement and reflectance model
plots for foam (sample C).
Cloth
r

Figure

Reflectance measurement and reflectance model
plots for a cotton towel (sample
D).
5 Implications for Graphics
In this section, we describe the implications of the proposed
model for realistic rendering. Figure 12(a) shows a real image
of the rough cylindrical clay vase discussed in the in-
troduction. Figure 12(b) shows a rendered image of the
vase using the Lambertian model and its known geometry.
Clearly, this rendered image does not match the real image
of the vase. On the other hand, the appearance of the
rendered vase using the proposed reflectance model, shown
in

Figure

12(c), closely resembles the real vase. The model
parameters chosen empirically to
obtain the best fit to the measured brightness values. Figure
13(a) compares brightness values along the cross-section of
the three different vase images in Figure 12. It is interesting
to note that the brightness of the real vase remains nearly
constant over most of the cross-section and drops quickly to
zero very close to the limbs. The proposed model does very
well in predicting this behavior, while the Lambertian model
produces large brightness errors. Figure 13(b) shows similar
plots for illumination from 20 ffi to the right of the sensor. In
this case, brightness variation on the real vase is asymmet-
ric. Once again, the proposed model closely matches the real
image. However, the Lambertian model forces the brightness
close to the right limb of the vase to drop much faster
than in the real image. As a result, the brightness peak predicted
by the Lambertian model is significantly away from
the actual peak.
(a) Image (b) Lambertian (c) Model

Figure

12: Real image of a cylindrical clay vase compared with
images rendered using the Lambertian and proposed models. Illumination
is from the camera direction.
Lambertian
Model Measurements
Brightness
Brightness
(a) (b)

Figure

13: Comparison between image brightness along the
cross-section of the real vase, and vases rendered using the Lambertian
and proposed models. (a) '
The functional approximation, given by equation (14),
and the qualitative model, given by (17), are easily used
for realistic rendering. We have implemented the functional
approximation as a "shader" using the RenderMan shading
language [32]. Figure 14 shows spheres rendered using the
shader. In all four cases, the sphere is illuminated from
the viewer direction. In the first case, oe = 0, and hence the
sphere appears Lambertian. As the roughness increases, the
sphere begins to appear flatter. In the extreme roughness
case shown in Figure 14(d), the sphere appears like a flat
disc with near constant brightness. This phenomenon has
been widely observed and reported in the case of the full
moon ([25],[28]).
Finally, Figure 15 shows rendered images of a scene with
three matte objects, a vase, cylindrical block and a cube. In

Figure

15(a), all three objects have zero macroscopic rough-
ness, i.e. they are Lambertian. Illumination in this case is
from the viewer direction. Note that the vase and the cylinder
have strong brightness variations, and the three visible
faces of the cube have distinctly different brightness val-
ues. In

Figure

15(b), the scene is again illuminated from
the viewer direction, but the three objects have roughness
Consequently, the shading over the vase and the
cylinder is diminished considerably. Furthermore, the contrast
between the flat and curved sections of the cylindrical
block and also the contrast between the three faces of
the cube are reduced substantially. It is important to note
that the moderate shading is achieved without any ambient
component in the illumination, but rather from modeling of
roughness effects.
6

Summary

In conclusion, we have developed a comprehensive model for
body reflectance from surfaces with macroscopic roughness.
A model was first derived for anisotropic surfaces that have
facets with only one slope. This result was used to develop a
model for isotropic surfaces with Gaussian slope-area distri-
bution. We have also presented a qualitative model for diffuse
reflection that has a simple functional form. Numerous
experiments were conducted to verify the reflectance mechanism
described in this paper. Real and rendered images
of diffuse objects were compared to demonstrate that the
proposed model has important implications for computer
graphics.



--R

The Scattering of Electro-magnetic Waves from Rough Surfaces
Models of light reflection for computer synthesized pictures.
Reradiation and thermal emission from illuminated craters on the lunar surface.
Bidirectional reflection functions from surface bump maps.
Radiative Transfer.
The hemi-cube
A reflection model for computer graphics.
Mutual illumination.
Illumination and Color in Computer Generated Im- agery
Reflection from layered surfaces due to subsurface scattering.
The opposition effect of the moon: The contribution of coherent backscatter.
Photometric studies of complex surfaces
A comprehensive physical model for light reflection.
Apparent radiation properties of a rough surface.
Heat Transfer.
Anisotropic reflection model.
Retroreflectance from a dense distribution of spherical particles.
Photometria sive de mensure de gratibus luminis
The reciprocity principle in lunar photometry.
Geometrical Considerations and Nomenclature for Reflectance.
Photometric studies of diffusely reflecting surfaces with application to the brightness of the moon.
Photometric measures of the moon and the moon the earth-shine
Generalization of the lambertian model and implications for machine vision.
Generalization of lambert's reflectance model.
Photometric relief of the lunar surface.
A model for anisotropic reflection.
Digital correction of solar illumination and viewing angle artifacts in remotely sensed images.
Thermal Radiation Heat Transfer.
Lunar surface roughness: Shadowing and thermal emission.
Theory for off-specular reflection from rough surfaces
Backscattering enhancement of random discrete scatterers.
The RenderMan Companion.
Predicting reflectance functions from complex surfaces.
An improved illumination model for shaded display.
--TR
Bidirectional reflection functions from surface bump maps
Illumination and color in computer generated imagery
A model for anisotropic reflection
Shape from interreflections
A comprehensive physical model for light reflection
Predicting reflectance functions from complex surfaces
Reflection from layered surfaces due to subsurface scattering
The hemi-cube
A Reflectance Model for Computer Graphics
An improved illumination model for shaded display

--CTR
Ravi Ramamoorthi , Melissa Koudelka , Peter Belhumeur, A Fourier Theory for Cast Shadows, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.2, p.288-295, February 2005
Michael Ashikmin , Simon Premoe , Peter Shirley, A microfacet-based BRDF generator, Proceedings of the 27th annual conference on Computer graphics and interactive techniques, p.65-74, July 2000
P. F. Lafortune , Sing-Choong Foo , Kenneth E. Torrance , Donald P. Greenberg, Non-linear approximation of reflectance functions, Proceedings of the 24th annual conference on Computer graphics and interactive techniques, p.117-126, August 1997
Alexander Wilkie , Andrea Weidlich , Caroline Larboulette , Werner Purgathofer, A reflectance model for diffuse fluorescent surfaces, Proceedings of the 4th international conference on Computer graphics and interactive techniques in Australasia and Southeast Asia, November 29-December 02, 2006, Kuala Lumpur, Malaysia
Dave Edwards , Solomon Boulos , Jared Johnson , Peter Shirley , Michael Ashikhmin , Michael Stark , Chris Wyman, The halfway vector disk for BRDF modeling, ACM Transactions on Graphics (TOG), v.25 n.1, p.1-18, January 2006
G. Scott Owen , Ying Zhu , Jeff Chastine , Bryson R. Payne, Teaching programmable shaders: lightweight versus heavyweight approach, ACM SIGGRAPH 2005 Educators program, July 31-August
Florian Struck , Christian-A. Bohn , Sebastian Schmidt , Volker Helzle, Realistic shading of human skin in real time, Proceedings of the 3rd international conference on Computer graphics, virtual reality, visualisation and interaction in Africa, November 03-05, 2004, Stellenbosch, South Africa
Marc Levoy , Kari Pulli , Brian Curless , Szymon Rusinkiewicz , David Koller , Lucas Pereira , Matt Ginzton , Sean Anderson , James Davis , Jeremy Ginsberg , Jonathan Shade , Duane Fulk, The digital Michelangelo project: 3D scanning of large statues, Proceedings of the 27th annual conference on Computer graphics and interactive techniques, p.131-144, July 2000
Marco Pellegrini, Rendering equation revisited: how to avoid explicit visibility computations, Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms, p.725-733, January 17-19, 1999, Baltimore, Maryland, United States
Stphane Mrillou , Jean-Michel Dischler , Djamchid Ghazanfarpour, A BRDF Postprocess to Integrate Porosity on Rendered Surfaces, IEEE Transactions on Visualization and Computer Graphics, v.6 n.4, p.306-318, October 2000
Jason Lawrence , Szymon Rusinkiewicz , Ravi Ramamoorthi, Efficient BRDF importance sampling using a factored representation, ACM Transactions on Graphics (TOG), v.23 n.3, August 2004
Andrew Gardner , Chris Tchou , Tim Hawkins , Paul Debevec, Linear light source reflectometry, ACM Transactions on Graphics (TOG), v.22 n.3, July
Yizhou Yu , Paul Debevec , Jitendra Malik , Tim Hawkins, Inverse global illumination: recovering reflectance models of real scenes from photographs, Proceedings of the 26th annual conference on Computer graphics and interactive techniques, p.215-224, July 1999
Aner Ben-Artzi , Ryan Overbeck , Ravi Ramamoorthi, Real-time BRDF editing in complex lighting, ACM Transactions on Graphics (TOG), v.25 n.3, July 2006
Paul Debevec , Tim Hawkins , Chris Tchou , Haarm-Pieter Duiker , Westley Sarokin , Mark Sagar, Acquiring the reflectance field of a human face, Proceedings of the 27th annual conference on Computer graphics and interactive techniques, p.145-156, July 2000
Lifeng Wang , Wenle Wang , Julie Dorsey , Xu Yang , Baining Guo , Heung-Yeung Shum, Real-time rendering of plant leaves, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005
Naty Hoffman , Dan Baker , Jan Kautz, Conclusions/summary, ACM SIGGRAPH 2006 Courses, July 30-August 03, 2006, Boston, Massachusetts
Lifeng Wang , Wenle Wang , Julie Dorsey , Xu Yang , Baining Guo , Heung-Yeung Shum, Real-time rendering of plant leaves, ACM SIGGRAPH 2006 Courses, July 30-August 03, 2006, Boston, Massachusetts
Bruno Mercier , Daniel Meneveaux , Alain Fournier, A Framework for Automatically Recovering Object Shape, Reflectance and Light Sources from Calibrated Images, International Journal of Computer Vision, v.73 n.1, p.77-93, June      2007
Zhouchen Lin , Tien-Tsin Wong , Heung-Yeung Shum, Relighting with the Reflected Irradiance Field: Representation, Sampling and Reconstruction, International Journal of Computer Vision, v.49 n.2-3, p.229-246, September-October 2002
Donald P. Greenberg , Kenneth E. Torrance , Peter Shirley , James Arvo , Eric Lafortune , James A. Ferwerda , Bruce Walter , Ben Trumbore , Sumanta Pattanaik , Sing-Choong Foo, A framework for realistic image synthesis, Proceedings of the 24th annual conference on Computer graphics and interactive techniques, p.477-494, August 1997
moon reflectance;BRDF;reflection models;rough surfaces;lambert's model