A Reflectance Model for Computer
Graphics
ROBERT L. COOK
Lucasfilm Ltd.
andKENNETH E. TORRANCE
Cornell University

A new reflectance model for rendering computer synthesized images is presented. The model accounts
for the relative brightness of different materials and light sources in the same scene. It describes the
directional distribution of the reflected light and a color shift that occurs as the reflectance changes
with incidence angle. A method for obtaining the spectral energy distribution of the light reflectedfrom an object made of a specific real material is presented, and a procedure for accurately reproducing
the color associated with the spectral energy distribution is discussed. The model is applied to thesimulation of a metal and a plastic.
Categories and Subject Descriptors: 1.3.7 [Computer Graphics]: Three-Dimensional Graphics andRealism--color, shading, shadowing, and texture
General Terms: Algorithms
Additional Key Words and Phrases: image synthesis, reflectance

INTRODUCTION
T h e r e n d e r i n g of realistic i m a g e s i n c o m p u t e r g r a p h i c s r e q u i r e s a m o d e l of h o w
o b j e c t s reflect light. T h e r e f l e c t a n c e m o d e l m u s t d e s c r i b e b o t h t h e color a n d t h e
s p a t i a l d i s t r i b u t i o n of t h e r e f l e c t e d light. T h e m o d e l is i n d e p e n d e n t of t h e o t h e r
a s p e c t s of i m a g e s y n t h e s i s , s u c h as t h e s u r f a c e g e o m e t r y r e p r e s e n t a t i o n a n d t h e
h i d d e n surface algorithm.
M o s t real s u r f a c e s are n e i t h e r i d e a l s p e c u l a r (mirrorlike) r e f l e c t o r s n o r i d e a ldiffuse ( L a m b e r t i a n ) reflectors. P h o n g [14, 15] p r o p o s e d a r e f l e c t a n c e m o d e l forc o m p u t e r g r a p h i c s t h a t was a l i n e a r c o m b i n a t i o n of s p e c u l a r a n d diffuse reflection.
T h e s p e c u l a r c o m p o n e n t was s p r e a d o u t a r o u n d t h e s p e c u l a r d i r e c t i o n b y u s i n ga cosine f u n c t i o n r a i s e d to a power. S u b s e q u e n t l y , B l i n n [5, 6] u s e d s i m i l a r i d e a s
t o g e t h e r w i t h a s p e c u l a r r e f l e c t i o n m o d e l f r o m [23], w h i c h a c c o u n t s for t h e offs p e c u l a r p e a k s t h a t o c c u r w h e n t h e i n c i d e n t l i g h t is a t a g r a z i n g a n g l e r e l a t i v e toThis paper is a revision of a paper that appeared in Computer Graphics, vol. 15, no. 3, 1981, ACM.
This work was supported in part by the National Science Foundation under Grant MCS-7811984.
Authors' addresses: R. L. Cook, Lucasfflm Ltd., P.O. Box 2009, San Rafael, CA 94912; K. E. Torrance,
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853
Permission to copy without fee all or part of this material is granted provided that the copies are notmade or distributed for direct commercial advantage, the ACM copyright notice and the title of the
publication and its date appear, and notice is given that copying is by permission of the Association
for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specificpermission.
O 1982 ACM 0730-0301/82/0100-0007 $00.75
ACM Transactionson Graphics,Vol. 1, No. 1, January 1982,Pages 7-24.

8

R.L. Cook and K. E. Torrance

the surface normal. Whitted [24] extended these models by adding a term for
ideal specular reflection from perfectly smooth surfaces. All of these models are
based on geometrical optics (ray theory).
The foregoing models treat reflection as consisting of three components:
ambient, diffuse, and specular. The ambient component represents light that isassumed to be uniformly incident from the environment and that is reflected
equally in all directions by the surface. The diffuse and specular components areassociated with light from specific light sources. The diffuse component represents
light that is scattered equally in all directions. The specular component represents
highlights, light that is concentrated around the mirror direction. The specularcomponent was assumed to be the color of the light source; the Fresnel equation
was used to obtain the angular variation of the intensity, but not the color, of the
specular component. The ambient and diffuse components were assumed to be
the color of the material. The resulting models produce images that look realistic
for certain types of materials.
This paper presents a reflectance model for rough surfaces that is more generalthan previous models. It is based on geometrical optics and is applicable to abroad range of materials, surface conditions, and lighting situations. The basis of
this model is a reflectance definition that relates the brightness of an object tothe intensity and size of each light source that illuminates it. The model predicts
the directional distribution and spectral composition of the reflected light. A
procedure is described for calculating red, green, and blue (RGB) values from thespectral energy distribution. The new reflectance model is then applied to thesimulation of a metal and a plastic, with an explanation of why images renderedwith previous models often look plastic, and how this plastic appearance can be
avoided.
THE R E F L E C T A N C E M O D E L

Given a light source, a surface, and an observer, a reflectance model describes theintensity and spectral composition of the reflected light reaching the observer.
The intensity of the reflected light is determined by the intensity and size of thelight source and by the reflecting ability and surface properties of the material.
The spectral composition of the reflected light is determined by the spectral
composition of the light source and the wavelength-selective reflection of thesurface. In this section the appropriate reflectance definitions are introduced andcombined into a general reflectance model. Figure 1 contains a summary of thesymbols used in this model.
The geometry of reflection is shown in Figure 2. An observer is looking at apoint P on a surface. V is the unit vector in the direction of the viewer, N is theunit normal to the surface, and L is the unit vector in the direction of a specificlight source. H is a normalized vector in the direction of the angular bisector of
V and L, and is defined by

H-

V+L

length(V + L) 'which is the unit normal to a hypothetical surface that would reflect light
ACM Transactions on Graphics, Vol. 1, No. 1, January 1982.

A Reflectance Model for Computer Graphics
{X

0
D
d
dw~
Ei
F

f
G

H

Ii
Ii.

L
17.
k
L
m

N
n

Ra
R
Rd
R,
8

V
W

Angle between N and H
Angle between L and H or V and H
Wavelength
Facet slope distribution function
Fraction of reflectance that is diffuse
Solid angle of a beam of incident light
Energy of the incident light
Reflectance of a perfectly smooth surface
Unblocked fraction of the hemisphere
Geometrical attenuation factor
Unit angular bisector of V and L
Average intensity of the incident lightIntensity of the incident ambient lightIntensity of the reflected lightIntensity of the reflected ambient light
Extinction coefficient
Unit vector in the direction of a light
Root mean square slope of facets
Unit surface normal
Index of refraction
Ambient reflectance
Total bidirectional reflectance
Diffuse bidirectional reflectanceSpecular bidirectional reflectance
Fraction of reflectance that is specular
Unit vector in direction of the viewer
Relative weight of a facet slope

9Fig. 1. Summary of symbols.

N/
Fig. 2. The geometry of reflection.

specularly from the light source to the viewer, a is the angle between H and N,
and 0 is the angle between H and V, so that cos(0) = V . H = L . t t .
T h e energy of the incident light is expressed as energy per unit time and perunit area of the reflecting surface. T h e intensity of the incident light is similar,
but is expressed per unit projected area and, in addition, per unit solid angle [20,8]. (Solid angle is the projected area of the light source divided by the square of
ACM Transactions on Graphics, Vol. 1, No. 1, January 1982.

10

R.L. Cook and K. E. Torrance

the distance to the light source and can be treated as a constant for a distantlight source.) The energy in an incoming beam of light is
El = Ii(N.L) d~0i.
Except for mirrors or near-mirrors, the incoming beam is reflected over a wide
range of angles. For this reason, the reflected intensity in any given direction
depends on the incident energy, not just on the incident intensity. The ratio of
the reflected intensity in a given direction to the incident energy from anotherdirection (within a small solid angle) is called the b i d i r e c t i o n a l reflectance. This
reflectance is fundamental for the study of reflection (for additional discussion,see [20, 8]). For each light source, the bidirectional reflectance R is thus

/ ~ - IrEl"

The reflected intensity reaching the viewer from each light source is thenIr = R E i= R I i (N. L) d~i.

The bidirectional reflectance may be split into two components, specular anddiffuse. The specular component represents light that is reflected from the surfaceof the material. The diffuse component originates from internal scattering (inwhich the incident light penetrates beneath the surface of the material) or frommultiple surface reflections (which occur if the surface is sufficiently rough). The
specular and diffuse components can have different colors if the material is nothomogeneous. The bidirectional reflectance is thus
where

R=sRs+dRa,

s+d--1.

In addition to direct illumination by individual light sources, an object may beilluminated by background or ambient illumination. All light that is not direct
illumination from a specific light source is lumped together into ambient illumination. The amount of light reflected toward the viewer from any particular
direction of ambient illumination is small, but the effect is significant when
integrated over the entire hemisphere of illuminating angles. Consequently, it isconvenient to introduce an ambient (or hemispherical-directional) reflectance,
R,. This reflectance is an integral of the bidirectional reflectance R and is thus a
linear combination of Rs and Rd. For simplicity, we assume that R, is independentof viewing direction. In addition we assume that the ambient illumination is
uniformly incident. The reflected intensity due to ambient illumination is definedby
La = R . l i . f .

The term f is the fraction of the illuminating hemisphere that is not blocked by
nearby objects (such as a corner) [25]. It is given by
f = 1 ¢ (N. L) do,i,
qr J

where the integration is done over the unblocked part of the illuminating
hemisphere.
ACM Transactions on Graphics, Vol. 1, No. 1, January 1982.

A Reflectance Model for Computer Graphics

11

Th e total intensity of the light reaching the observer is the sum of the reflected
intensities from all light sources plus the reflected intensity from any ambient
illumination. Assuming that f = 1, the basic reflectance model used in this paper
becomes/ r ---- I i a R a + ~ Iit(N.Ll) dwit(sR~ + d R d ) .
lThis formulation accounts for the effect of light sources with different intensities
and different projected areas which may illuminate a scene. For example, anilluminating beam with the same intensity (Ii) and angle of illumination (N. L) as
another beam, but with twice the solid angle (do,i) of that beam, will make asurface appear twice as bright. An illuminating beam with twice the intensity of
another beam, but with the same angle of illumination and solid angle, will also
make a surface appear twice as bright.
This paper does not consider the reflection of light from other objects in theenvironment. This reflection can be calculated as in [24] or [6] if the surface isperfectly smooth, but even this pure specular reflection should be wavelength
dependent.
The above reflectance model implicitly depends on several variables. For
example, the intensities depend on wavelength, s and d depend on the material,
and the reflectances depend on these variables plus the reflection geometry andthe surface roughness. The next two sections consider the directional and wavelength dependence of the reflectance model.
DIRECTIONAL DISTRIBUTION OF THE REFLECTED LIGHT

Th e ambient and diffuse components reflect light equally in all directions. Thus
Ra and Ro do not depend on the location of the observer. On the other hand, thespecular component reflects more light in some directions than in others, so thatRs does depend on the location of the observer.
The angular spread of the specular component can be described by assumingthat the surface consists of microfacets, each of which reflects specularly [23].
Only facets whose normal is in the direction H contribute to the specularcomponent of reflection from L to V. The specular component is
R s --F
DG
7r (N. L) (N. V) "The Fresnel term F describes how light is reflected from each smooth microfacet.
It is a function of incidence angle and wavelength and is discussed in the next
section. The geometrical attenuation factor G accounts for the shadowing andmasking of one facet by another and is discussed in detail in [5, 6, 23]. Briefly,
it is2(N.H)(N.V)
G=min

1,

(V.H)

2(N.H)(N.L)~

'

~

J"

The facet slope distribution function D represents the fraction of the facets thatare oriented in the direction H. Various facet slope distribution functions have
ACM Transactions on Graphics, Vol. 1, No. 1, January 1982.

12

R.L. Cook and K. E. Torrance

been considered by Blinn [5, 6]. One of the formulations he described is theGaussian model [23]:D = ce -(~/m)2

where c is an arbitrary constant.
In addition to the ones mentioned by Blinn, other facet slope distribution
models are possible. In particular, models for the scattering of radar and infrared
radiation from surfaces are available and are applicable to visible wavelengths.
For example, Davies [9] described the spatial distribution of electromagnetic
radiation reflected from a rough surface made of a perfect electrical conductor.
Bennett and Porteus [3] extended these results to real metals, and Torrance andSparrow [22] showed that they apply to nonmetals as well. Beckmann [2]provided a comprehensive theory that encompasses all of these materials and isapplicable to a wide range of surface conditions ranging from smooth to very
rough. For rough surfaces, the Beckmann distribution function isD -1
m 2 c0s40~ e -[(tana)/rn]2

This distribution function is similar in shape to the three functions mentioned byBlinn. The advantage of the Beckmann function is that it gives the absolutemagnitude of the reflectance without introducing arbitrary constants; the disadvantage is that it requires more computation.
In all of the facet Slope distribution functions, the spread of the specularcomponent depends on the root mean square (rms) slope m. Small values of msignify gentle facet slopes and give a distribution that is highly directional aroundthe specular direction, as shown in Figure 3a for the Beckmann distribution
model and in Figure 3b for the Gaussian model. Large values of m imply steep
facet slopes and give a distribution that is spread out, as shown in Figures 3c and
3d for the Beckmann and Gaussian models, respectively. Note the similarity
between the two models.
The wavelength dependence of the reflectance is not affected by the surface
roughness except for surfaces that are almost completely smooth, which aredescribed by physical optics (wave theory) and which have a distribution function
D that is wavelength dependent. The Beckmann distribution model accounts forthis wavelength dependence and for the transition region between physical andgeometrical optics (i.e., between very smooth surfaces and rough surfaces). For
simplicity, we ignore the cases in which D is wavelength dependent. (For a further
discussion, see [2] and [9].)Some surfaces have two or more scales of roughness, or slope m, and can bemodeled by using two or more distribution functions [16]. In such cases, D isexpressed as a weighted sum of the distribution functions, each with a different
value of m:
D = Y, w j D ( m j ) ,J

where mj is the rms slope of the j t h distribution and wj is the weight of the j t hdistribution. The sum of these weights is 1.
ACM Transactions on Graphics, Vol. 1, No. 1, J a n u a r y 1982.

d
n ~

A

c~

~°II

~s
c~

14

R.L. Cook and K. E. Torrance

SPECTRAL COMPOSITION OF THE REFLECTED LIGHT

The ambient, diffuse, and specular reflectances all depend on wavelength. Ra, Ro,
and the F term of Rs may be obtained from the appropriate reflectance spectra
for the material. A nonhomogeneous material may have different reflectancespectra for each of the three reflectances, though Ra is restricted to being a linear
combination of Rs and Rd.
Reflectance spectra have been measured for thousands of materials and collected in [10, 17-19]. T he reflectance data are usually for illumination at normal
incidence. These values are normally measured for polished surfaces and must be
multiplied by 1/qr to obtain the bidirectional reflectance for a rough surface [20].
Most materials are measured at only a few wavelengths in the visible range(typically around 10 to 15), so that values for intermediate wavelengths must be
interpolated (a simple linear interpolation seems to be sufficient). T he reflectancespectrum of a copper mirror for normal incidence is shown for visible wavelengths
in Figure 4a. In choosing a reflectance spectrum, careful consideration must be
given to the conditions under which the measurements were made. For example,
some metals develop an oxide layer with time which can drastically alter thecolor [1].
Th e spectral energy distribution of the reflected light is found by multiplying
the spectral energy distribution of the incident light by the reflectance spectrumof the surface. An example of this is shown in Figure 4b. T he spectral energy
distributions of the sun and a number of CIE standard illuminants are available
in [7]. Th e spectral energy distribution of CIE standard illuminant D6500, whichapproximates sunlight on a cloudy day, is the top curve shown in Figure 4b. T helower curve shows the corresponding spectral energy distribution of light reflectedfrom a copper mirror illuminated by CIE standard illuminant D6500 at normal
incidence. It is obtained by multiplying the top curve by the reflectance spectrum
in Figure 4a.
In general, Rd and F vary with the geometry of reflection. For convenience, we
subsequently take Rd to be the bidirectional reflectance for illumination in a
direction normal to the reflecting surface. This is reasonable because the reflectance varies only slightly for incidence angles within about 70 ° of the surface
normal [21]. We specifically allow for the directional dependence of F, however,as this leads to a color shift when the directions of incidence and reflection arenear grazing.
Th e reflectance F may be obtained theoretically from the Fresnel equation[21]. This equation expresses the reflectance of a perfectly smooth, mirrorlike
surface in terms of the index of refraction (n) and the extinction coefficient (k) of
the surface and the angle of illumination (0). In general, both n and k vary withwavelength, but their values are frequently not known. On the other hand,experimentally measured values of the reflectance at normal incidence are frequently known.
To obtain the spectral and angular variation of F, we have adopted a practicalcompromise. If n and k are known, we use the Fresnel equation. If not, but thenormal reflectance is known, we fit the Fresnel equation to the measured normal

ACM Transactions on Graphics, Vol. 1, No. 1, January 1982.

A Reflectance Model for Computer Graphics

15

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1

i

I

i

J

0.4

0.5

0.6

0.7

0.8

(a)

Energy

0.4

0.5

0.6

0.7

0.8

(b)

Fig. 4. (a) Reflectance of a copper mirror for n o r m a l incidence. W a v e l e n g t h is in micrometers.
(b) T o p curve: Spectral energy distribution of CIE s t a n d a r d i l l u m i n a n t D6500. B o t t o m curve:
Spectral energy distribution of light reflected from a copper mirror illuminated by D6500.

ACM Transactions on Graphics, Vol. 1, No. 1, January 1982.

16

R.L. Cook and K. E. Torrance

reflectance for a polished surface. For nonmetals, for which k = 0, this immediately gives us an estimate of the index of refraction n. For metals, for which k is
generally not 0, we set k equal to 0 and get an effective value for n from thenormal reflectance. T he angular dependence of F is then available from theFresnel equation. T he foregoing procedure yields the correct value of F for normal
incidence and a good estimate of its angular dependence, which is only weakly
dependent on the extinction coefficient k.
To illustrate this procedure, the Fresnel equation for unpolarized incident lightand k = 0 is
1 (g-

c) f

[c(g + c) -1]

F - 2 (g--+ c) 2 ~ 1 + [c(g - c) --+--~112j 'wherec = cos(0) = V . H
g2=n2+

c 2 _ 1.

(Note that a similar expression in [5] is missing the i factor.) At normal incidence,0 = 0; so c = 1, g = n, and2

F0 = ( n + l J "Solving for n gives the equation

1+
n

-

-

0

-

1-~0"
Values of n determined in this way are then substituted into the original Fresnel
equation to obtain the reflectance F at other angles of incidence. T he procedure
may be repeated at other wavelengths to obtain the spectral and directional
dependence of the reflectance.
Th e dependence of the reflectance on wavelength and the angle of incidence
implies that the color of the reflected light changes with the incidence angle.
Reflectance spectra for copper are shown in Figure 5a. As the incidence angle(0) approaches ~r/2, the color of the reflected light approaches the color of thelight source (since the reflectance F approaches unity). T he colors corresponding
to the reflection of white light (CIE standard illuminant D6500) from copper are
shown as a function of 0 in Figure 5b. It is evident that the color shift from theFresnel equations only becomes important as 8 approaches ~/2 (i.e., as the angle
between V and L approaches ~r).
Calculation of the color shift is computationally expensive. It can be simplified
in one of two ways: by creating lookup tables or by using the following approximation. Values of F are first calculated for a value of n corresponding to theaverage normal reflectance. These values are then used to interpolate between
the color of the material at 0 =- 0 and the color at 0 = 7r/2, which is the color of
the light source because F,~/2 is 1.0 at every wavelength. For example, let the red
ACM Transactions on Graphics, Vol. 1, No. 1, January 1982.

A Reflectance Model for Computer Graphics

17

0

(a)

0=0

0 = ¢r/2(b)0=0

0 = z'/2(c)Fig. 5. (a) Reflectance of a copper mirror as a function of wavelength andincidence angle. (b) T h e color of copper as a function of incidence angle.
(c) T h e color of copper as approximated by the m e t h o d discussed in thispaper.
ACM Transactions on Graphics, Vol. 1, No. 1, January 1982.

18

R.L. Cook and K. E. Torrancecomponent of the color at normal incidence be Redo and let the red componentof the color of the incident light be Red./2. Then the red component of the color
at other angles is
Redo = Redo + (Red~/2 - Redo)max(0, Fo - Fo)F,,/2 Fo-

The green and blue components are interpolated similarly. Figure 5c shows theeffect of using the approximate procedure to estimate the color of copper as afunction of incidence angle. The approximate procedure yields results that are
similar to those from the complete (but more expensive) procedure (Figure 5b).
The foregoing approximation must always be used if the spectral energy distribution of the reflected light is not known, in which case all of the RGB values are
estimates.
DETERMINING THE RGB VALUES

For a computer synthesized scene to be realistic, the color sensation of anobserver watching the synthesized scene on a color television monitor must be
approximately equivalent to the color sensation of an observer watching acorresponding scene in the real world. To produce this equivalent color sensation,
the laws of trichromatic color reproduction are used to convert the spectralenergy distribution of the reflected light to the appropriate RGB values for theparticular monitor being used.
Every color sensation can be uniquely described by its location in a threedimensional color space. One such color space is called the XYZ space. A pointin this space is specified by three coordinates, the color's XYZ tristimulus values.
Each spectral energy distribution is associated with a point in the XYZ color
space and thus with tristimulus values. If two spectral energy distributions areassociated with the same tristimulus values, they produce the same color sensation and are called metamers. The red, green, and blue phosphors of a monitor
can be illuminated in proportions to produce a set of spectral energy distributions
which define a:, region of XYZ space called the gamut of the monitor. The goal,
then, is to find fhe proportions of phosphor illumination t h a t produce a spectralenergy distribution that is a metamer of the spectral energy distribution of thereflected light.
These proportions are determined by calculating the XYZ tristimulus values
that are associated with the spectral energy distribution of the reflected light andthen calculating the RGB values that produce a spectral energy distribution withthese tristimulus values. To do this, the spectral energy distribution of thereflected light is multiplied by the XYZ matching functions (obtained from [7])at every wavelength. The resulting spectra are then integrated to obtain the XYZ
tristimulus values. These XYZ values, which can be precomputed for eachcombination of light source type and surface material, are converted by a matrix
multiplication to RGB linear luminance values for a particular set of phosphorsand monitor white point. The linear luminances are then converted to RGB
voltages, taking into account the nonlinearities of the monitor and the effects ofviewing conditions. For a more complete description of this procedure, see [13].
ACM Transactions on Graphics, Vol. 1, No. 1, J a n u a r y 1982.

A Reflectance Model for Computer Graphics

19

T h e monitor has a m a x i m u m luminance at which it can reproduce a given
chromaticity. Any XYZ values t h a t represent luminances greater t h a n thism a x i m u m are outside the gamut of the monitor. T o avoid this problem, all XYZvalues in the scene are scaled equally so t h a t t h e y all lie inside the monitor gamut
and usually so t h a t the color with the greatest luminance is r e p r o d u c e d on themonitor at the m a x i m u m luminance possible for its chromaticity. But even withscaling, some spectral energy distributions are associated with tristimulus values
t h a t lie outside the gamut of the monitor. Because such a color cannot be
reproduced on the monitor at any luminance, it must be approximated by asimilar color t h a t lies inside the monitor gamut. This color m a y be chosen in
m a n y different ways; for this paper, we have decided it is appropriate to maintainthe same hue, desaturating the color as necessary. T o do this, the tristimulus
XYZ values are converted to a color space in which locations are specified bydominant wavelength and purity. T h e purity is then reduced while the dominant
wavelength (and thus roughly the hue) is held constant until the color lies inside
the monitor gamut. T h e resulting color is then converted back to XYZ space.
{For a discussion of dominant wavelength and purity, see [12].)APPLICATIONSThis section discusses the application of the reflectance model to two particularclasses of materials, metals and plastics. T h e main consideration is the homogeneity of the material. Substances t h a t are composed of different materials, sucht h a t there is one material at the surface and a n o t h e r b e n e a t h the surface, arenonhomogeneous and m a y have specular and diffuse components t h a t differ incolor.
A typical plastic has a substrate t h a t is transparent or white, with e m b e d d e dpigment particles [11]. T h u s the light reflected directly from the surface is only
slightly altered in color from the light source. Any color alterations are a result of
the reflectance of the surface material. Light t h a t penetrates into the materialinteracts with the pigments. Internal reflections thus give rise to a colored,
uniformly distributed diffuse reflection.
A plastic m a y thus be simulated by using a colored diffuse c o m p o n e n t and awhite specular component. This is just the model used by P h o n g and Blinn, and
it is why so m a n y computer graphics images t h a t have a significant specularc o m p o n e n t look like plastic. Figure 6a shows a simulated copper-colored plastic
vase. This figure was generated with the following parameters.
2 lights:

Ii= CIE standard illuminant D6500

dwi = 0.0001 and 0.0002
Specular:

Diffuse:Ambient:s = 0.1
F = the reflectance of a vinyl mirror
D = B e c k m a n n function with m = 0.15
d = 0.9
Rd = the bidirectional reflectanceof copper for normal incidence/ia = 0.01 Ii
Ra = ~Rd
ACM Transactions on Graphics, Vol. l, No. 1, J a n u a r y 1982.

20

R . L . Cook and K. E. Torrance~i I¸¸~(a)(b)
Fig. 6. (a) A c o p p e r - c o l o r e d plastic vase. (b) A c o p p e r vase.
ACM Transactions on GraphiCs, Vol. 1, No. 1, January 1982.

iA Reflectance Model for Computer Graphics

21

TABLE I
MaterialsdmCarbonRubberObsidian
L u n a r dust
Olive drab
Rust

0.3
0.4
0.8
0.0
0.3
0.2

0.7
0.6
0.2
1.0
0.7
0.8

0.40
0.30
0.15
N o t used
0.50
0.35

Metals conduct electricity. An impinging electromagnetic wave can stimulate
the motion of electrons near the surface, which in turn leads to reemission(reflection) of a wave. T h e r e is little d e p t h penetration, and the depth penetration
decreases with increasing values of the extinction coefficient k. As a result,reflection from a metal occurs essentially at the surface [17]. T h u s internalreflections are not present to contribute to a diffuse component, which can be
i m p o r t a n t for a nonmetal. W h e n the rms roughness slope m is small, multiple
surface reflections m a y also be ignored and the entire diffuse c o m p o n e n t disappears. Figure 6b shows a simulated copper vase. This figure was generated withthe following parameters.
2 lights:Ii= C I E standard illuminant D6500
and 0.0002s= 1.0
F = the reflectance of a copper mirror
D = B e c k m a n n function with
ml = 0.4
wl = 0.4
m2 = 0.2
w2 = 0.6
d = 0.0
Rd = the bidirectional reflectanceof copper for normal incidence
du~i -- 0 . 0 0 0 1

Specular:

Diffuse:Ambient:

Iia= 0.01 Ii

Ra = 7rRd
Note t h a t two values for the rms slope are employed to generate a realistic r o u g hsurface finish. T h e specular reflectance c o m p o n e n t has a copper color. T h e copper
vase in Figure 6b does not display the plastic appearance of the vase in Figure 6a,
showing t h a t a correct t r e a t m e n t of the color of the specular c o m p o n e n t is needed
to obtain a realistic nonplastic appearance.
Figure 7 shows vases made of a variety of materials. In every case, the specularand diffuse c o m p o n e n t s have the same color (i.e., Rd = Fo/Tr). T h e lighting
conditions for all o f the vases are identical to the lighting conditions for Figures
6a and 6b. T h e six metals were generated with the same p a r a m e t e r s used for
Figure 6b, except for the reflectance spectra. T h e six nonmetals were generated
with the p a r a m e t e r s shown in Table I.
ACM Transactions on Graphics, Vol. 1, No. 1, January 1982.

22

R.L. Cook and K. E. Torrance

Fig. 7.

A variety of vases.

Fig. 8.

A watch.

Figure 8 shows a w a t c h m a d e with a variety of m a t e r i a l s and surface conditions.
I t is illuminated b y a single light source. T h e outer b a n d of the w a t c h is m a d e ofgold, and the inner b a n d is m a d e of stainless steel. T h e p a t t e r n on the links of theouter b a n d was m a d e b y using a r o u g h e r surface for the interior t h a n for theborder. T h e light-emitting diodes (LEDs) are s t a n d a r d red 640 n a n o m e t e r L E D s ,and their color was a p p r o x i m a t e d b y using a color with the s a m e d o m i n a n t
wavelength.
ACM Transactions on Graphics, Vol. 1, No. 1, January 1982.

A Reflectance Model for Computer Graphics

23

CONCLUSIONS

T h e following conclusions can be stated.
1. Th e specular component is usually the color of the material, not the color of
the light source. The ambient, diffuse, and specular components may havedifferent colors if the material is not homogeneous.
2. Th e concept of bidirectional reflectance is necessary to simulate different lightsources and materials in the same scene.
3. Th e facet slope distribution models used by Blinn are easy to calculate andare very similar to others in the optics literature. More than one facet slope
distribution function can be combined to represent a surface.
4. The Fresnel equation predicts a color shift of the specular component at
grazing angles. Calculating this color shift is computationally expensive unlessan approximation procedure or a lookup table is used.
5. The spectral energy distribution of light reflected from a specific material canbe obtained by using the reflectance model together with the spectral energy
distribution of the light source and the reflectance spectrum of the material.
The laws of trichromatic color reproduction can be used to convert thisspectral energy distribution to the appropriate RGB values for the particularmonitor being used.
6. Certain types of materials, notably some painted objects and plastics, havespecular and diffuse components that do not have the same color. Metals havea specular component with a color determined by the light source and thereflectance of the metal. The diffuse component is often negligible for metals.
ACKNOWLEDGMENTS

The work was performed at the Program of Computer Graphics of Cornell
University. The authors thank Gary Meyer for the invaluable contributions hemade to this paper. His color software and photometric measurements of themonitors made it possible to accurately convert spectral energy distributions into
RGB values. The authors also thank Dr. Donald Greenberg for his helpful
discussions and valuable suggestions at every stage of the research. T he watch(Figure 8) was the result of a joint effort with Stuart Sechrest.
REFERENCES
1. BARKMAN, E. F.

Specular and diffuse reflectance measurements of aluminum surfaces. In

Appearance of Metallic Surfaces, ASTM Special Tech. Publ. 478, American Society for Testingand Materials, Philadelphia, 1970, pp. 46-58.
2. BECKMANN, P. AND SPIZZICHINO, A. The Scattering of Electromagnetic Waves from RoughSurfaces. MacMillan, New York, 1963, pp. 1-33, 70-98.
3. BENNETT, H. n., AND PORTEUS, J . O . Relation between surface roughness and specular reflectance at normal incidence. J. Opt. Soe. Am., 51, 2 (1961), 123-129.
4. BLINN, Z. F., AND NEWELL, M. E. Texture and reflection in computer generated images.
Commun. ACM 19, 10 (Oct. 1976), 542-547.
5. BLINN, J . F . Models of light reflection for computer synthesized pictures. Computer Gr. 11, 2(1977), 192-198.
6. BLINN, J . F . "Coniputer Display of Curved Surfaces." Ph.D. dissertation, Univ. of Utah, Salt
Lake City, 1978.
ACM Transactions on Graphics, Vol. 1, No. 1, January 1982.

24

R.L. Cook and K. E. Torrance

7. CIE INTERNATIONAL COMMISSION ON ILLUMINATION Official recommendations of the international commission on illumination. Colorimetry (E-1.3.1), Publ. CIE 15, Bureau Central de la CIE,
Paris, 1970.
8. COOK,R.L. "A Reflection Model for Realistic Image Synthesis." Master's thesis, Cornell Univ.,
Ithaca, 1981.
9. DAVIES, H. The reflection of electromagnetic waves from a rough surface. Proceedings of theInstitution of Electrical Engineers 101 (1954), 209-214.
10. GUBAREFF,G. G., JANSSEN, J. E., AND TORBORG, R . H . Thermal Radiation Properties Survey:
A Review of the Literature. Honeywell Research Center, Minneapolis, 1960.
11. HUNTER, R . S . The Measurement of Appearance. Wiley, New York, 1975, pp. 26-30.
12. JUDD, D. B. AND WYSZECKI, G. Color in Business, Science, and Industry. Wiley, New York,
1975, pp. 170-172.
13. MEYER, G. W. AND GREENBERG, D.P. Perceptual color spaces for computer graphics. Computer
Gr. 14, 3 (1980), 254-261.
14. PHONG, B.T. "Illumination for Computer Generated Images." Ph.D. dissertation, Univ. of Utah,
Salt Lake City, 1973.
15. PHONG, B . T . Illumination for computer generated pictures. Commun. ACM 18, 6 (June 1975),
311-317.
16. PORTEUS, J.O. Relation between the height distribution of a rough surface and the reflectanceat normal incidence. J. Opt. Soc. Am. 53, 12 (1963), 1394-1402.
17. Purdue University, Thermophysical Properties of Matter, vol. 7: Thermal Radiative Propertiesof Metals, Plenum, New York, 1970.
18. Purdue University, Thermophysical Properties of Matter, vol. $: Thermal Radiative Propertiesof Nonmetallic Solids, Plenum, New York, 1970.
19. Purdue University, Thermophysical Properties of Matter, vol. 9: Thermal Radiative Properties
of Coatings, Plenum, New York, 1970.
20. SIEGEL, R. AND HOWELL, J . R . Thermal Radiation Heat Transfer. McGraw-Hill, New York,
1980, pp. 64-73.
21. SPARROW,E. M. AND CESS, R.D. Radiation Heat Transfer. McGraw-Hill, New York, 1978, pp.
64-68.
22. TORRANCE, K. E. AND SPARROW, E . M . Biangular reflectance of an electric nonconductor as afunction of wavelength and surface roughness. Journal of Heat Transfer 87, series C (1965), 283292.
23. TORRANCE, K. E. AND SPARROW, E. M. Theory for off-specular reflection from roughened
surfaces. J. Opt. Soc. Am. 57, (Sept. 1967), 1105-1114.
24. WHITTED, T. An improved illumination model for shaded display. Commun. ACM 23, 6 (June1980), 343-349.
25. WHITTED, T. Private communication.

Received August 1981; revised October 1981; accepted October 1981

ACM Transactions on Graphics, Vol. 1, No. 1, January 1982.