MODELS OF LIQrHT REFLECTION

FOR CO1PUTER SYNTHESIZED PICTURES

James F. Blinn

University of Utah

ABSTRACT

In the production of computer generatedpictures of three dimensional objects, one stageof the calculation is the determination of theintensity of a given object once its visibilityhas been established. This is typically doneby modelling the surface as a perfect diffuser,sometimes with a specular component added for thesimulation of hilights. This paper presents amore accurate function for the generation ofhilights which is based on some experimentalmeasurements of how light reflects from realsurfaces. It differs from previous models in thatthe intensity of the hilight changes with thedirection of the light source. Also the positionand shape of the hilights is somewhat differentfrom that generated by simpler models. Finally,the hilight function generates different resultswhen simulating metallic vs. nonmetallic surfaces.Many of the effects so generated are somewhatsubtle and are apparent only during moviesequences. Some representative still frames fromsuch movies are included.

Key Words and Phrases: computer graphics,graphic display, shading, hidden surface removal.

CR Categories: 3.17, 5.12, 8.2

INTRODUCTION

In producing computer generated pictures ofthree dimensional objects, two types of calculat-ion must be performed. The first, and mostpopularly discussed, is the hidden surface problem;determining which object is visible where on thescreen and what is the normal vector to the objectat that point. The second is the intensitycalculation; given the normal vector and theposition of the light sources, what is the properintensity for the corresponding spot on the picture.Very simple models are typically used whichsimulate ideal diffuse reflectors. This uses the,so called, Lambert's law which states that thesurface will diffuse incident light equally in alldirections. Differences in intensity.are thencaused by the different amounts of incident lightper unit area intercepted by portions of thesurface at various angles to the light source.This will be proportional to the cosine of theangle between the normal to the surface, N, andthe vector to the light source, L. This cosine

is evaluated by computing the dot product of thetwo vectors after normallizing them to a lengthof 1. If this dot product is negative it indicatesthat the viewer is on the opposite side of thesurface from the light source. The intensityshould then be set to zero.

In addition, some constant value is usuallyadded to the intensity to simulate the effects ofambient light on the surface. This assumes thata small amount of light falls on the surfaceuniformly from all directions in addition to themain point light source. The integral of thisambient light from all directions yields aconstant value for any normal direction. The netfunction is:

This model is simple to compute and quite adequatefor many applications.

SIMPLE HILIGHT MODELS

A more realistic lighting model was intro-duces by Phong [2] as part of a technique forimproving the appearance of images of curvedsurfaces. The function makes use of the factthat, for any real surface, more light is reflect-ed in a direction making an equal angle of inci-dence with reflectance. The additional lightreflected in this direction is referred to as thespecular component. If the surface was a perfectmirror light would only reach the eye if thesurface normal, N, pointed halfway between thesource direction, L, and the eye direction, E.We will name this direction of maximum hilights H,where

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For less than perfect mirrors, the specular compo-nent falls off slowly as the normal direction movesaway from the specular direction. The cosine ofthe angle between H and N is used as a measure ofthe distance a particular surface is away from themaximum specular direction. The degree of sharp-nesssof the highlights is adjusted by taking thiscosine to some power, typically 50 or 60. Thenet Phong shading function is then:

d = max(0,N.L)s = (N*H)=’i=p,+dpd+sp s

wherei = percieved intensityp,= proportion of specular reflections = amount of specular reflectioncl= measure of shininess of surfaceother values as defined above

In addition, when simulating colored surfaces,there is a different intensity value for eachprimary. These. should be calculated by scalingonly the diffuse and ambient components by thecolor of the object. The highlights then appeardesaturated or white.

TORRANCE- SPARROW MODELThe reflection of light from real surfaces

has been the subject of much theoretical andexperimental work by physicists Is], [6] andillumination engineers [4]. The experimentalresults generally match the Phong shading functionbut some differences do arise. The main one isthe fact that the the specular bump,represented by the parameter p, above, varies withthe direction of the light source. Also thedirection of peak specular reflection is notalways exactly along H. In 1967 Torrance andSparrow [7 ] derived a theoretical model toexplain these effects. The match between theirtheoretically predicted functions and experimen-tally measured data is auite imoressive. In thissection we derive the Torrance-Sparrow highlightfunction in terms of the vectors N, L, H and E,all of which are assumed to be nonnallized.

The surface being simulated is assumed to becomposed of a collection of mirror like microfacets. These are oriented in random directionsall over the surface. The specular component ofthe reflected light is assumed to come ‘fromreflection from those facets oriented in thedirection of H. The diffuse component comesmultiple reflections between facets and frominternal scattering. The specular reflectionthen a combination of four factors:

from

is

D is the distribution function of the directionsof the micro facets on the surface. G is theamount by which the facets shadow and mask eachother. F is the Fresnel reflection law. Each ofthese factors will now be examined in turn.

The light reflected specularly in any givendirection can come only from the facets orientedto reflect the light in that direction. That is,the facets whose local normal vectors point in thedirection of H. The first term in the specularreflectance is the evaluation of the distributionof the number of facets pointing in that.direction.The distribution used by Torrance and Sparrow wasa simple Gaussian:

DI = ,-bCd2

Dz is the proportionate number of facets orientedat an angle CI from the average normal to thesurface. The factor cz is the standard deviationfor the distribution and is a property of thesurface being modelled. Large values yield dullsurfaces and small values yield shiny surfaces.We are interested in the number of facets pointingin the direction of H so the angle a here iscos ’ (N-H).

Since the intensity is proportional to thenumber of facets pointing in the H direction, wemust take into account the observer sees more ofthe surface area when the surface is tilted. Theincrease in area is inversely proportional to thecosine of the angle of tilt. The tilt angle isis the angle between the average surface normal,“0; ;d the eye, E. This explains the division by

l .

Counteracting this effect is the fact thatsome of the facets shadow each other. The degreeto which this shadowing occurs is called the“geometrical attenuation factor”, G. It is avalue from 0 to.1 representing the proportionateamount of light remaining after the masking orshadowing has taken place. Calculation of Gassumes that the micro facets exist in the formof V shaped grooves with the sides at equal butopposite angles to the average surface normal. Weare interested only in grooves where one of thesides points in the specular direction H. Fordiffering positions of the light source and eyeoosition we can have one of three cases illustratedin Figure 1.

Case aNo interference

Case b Y$J!f>Some of the reflected

light is intercepted

Figure 1

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Note that the vectors L and E do not necessarilylie in the plane of the figure (i. e. the planecontaining N and H). We can see this by consid-ering a top view as in Figure 2.

The value of G for case a of Figure 1 is 1.0,signifying no attenuation.

To compute G for case b we need to compute theratio 1-(m/l) which is the proportionate amount ofthe facet contributing to the reflected light.See Figure 3.

We can reduce the problem to two dimensions if weproject E onto the plane containing N and H (theplane of the diagram). Calling this projection Eand labeling relevant angles we have Figure 4. P

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In our case, the angle of incidence is d =

cos-'(L-H) = cos-'(E-H). The interesting thingabout this function is that it has a substantiallydifferent form for' metallic vs. nonmetallic sub-stances. For metals, corresponding to large valuesof n, F(O,n) is nearly constant at 1. For non-metals, corresponding to small values of n, it hasa more exponential appearance, starting out nearzero for 0=0 and going to 1 at C=T/2.

For a particular situation, the effective value ofG will be the minimum of Ga, Gb and G .

The final factor in the specular reflectionis the Fresnel reflection. This gives the fractionof the light incident on a facet which is actuallyreflected as opposed to being absorbed. This is afunction of the angle of incidence on the microfacet and the index of refraction on the substance.It is given by

Examining the diagram for Gc we see that it is thesame as that for Gb but with the roles of L and Eexchanged. Thus

Since Ep is the projection of B onto the N,H planethen N-E =N-E and H-E =H-E so that

p -p

Plugging these into the expression for sinf

Due to the symmetry of the groove and the comple-mentarity of d and a

Since the angles of the triangle must sum to 21we have

Then we note that

FACET DISTRIBUTION FUNCTIONS

One thing in the above model can be improved

upon. This is the facet distribution function.This function takes an angle, a, and a measure of

the shininess of the surface and computes theproportionate area of facets pointing in that

direction. The angle a is the angle between H

and N; we can evaluate its cosine as (N-H).

The Phong model effectively uses the dis-

tribution function of the cosine raised to a

power.

The Torrance Sparrow model uses the standard

Gaussian distribution already mentioned.

A third function has been proposed by

Trowbridge and Reitz [8]. They showed that a

very general class of surface properties could be

generated by modelling the microfacets as

ellipsoids of revolution. This leads to thedistribution function

Where c3 is the eccentricity of the ellipsoids and

is 0 for very shiny surfaces and 1 for very dif-

fuse surfaces.

Each of these functions has a peak value of

1 at a=0 (for facets pointing along the averagesurface normal) and falls off as a increases or

decreases. The rate of fall off is controlled by

the values cl, c2 and c3. In comparing the

functions it is necessary to specify this rate in

a uniform unit. A convenient such unit is the

angle at which the distribution falls to one

half. In terms of this angle, B, the threecoefficients are:

If these three functions are plotted with equal

values of 8 it can be seen that they are verysimilar in shape. However, since there is some

experimental as well as theoretical justification

for D3 and since it is the easiest to compute,it is the one we shall choose.

COMPUTATIONAL CONSIDERATIONS

There are several observations which can be

made to speed up the computation of the hilight

function.

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CnMPARISON WITH PHONG SHADING

Now that we have derived this hilightfunction we should compare it with the Phongfunction to see where and by how much they differ.

Figure 5 shows a plot of the amount of light re-flected from a surface as a result of an incident

ray at 30 degrees from the surface normal. The

distance of the surface in a particular direction

from the center represents the amount of light

reflected in that direction. The incoming ray is

from the right. A vector pointing to the left at

the specular direction is shown for reference.The hemispherical portion of the function is the.

diffuse reflection; equal amounts in each direc-

tion. The bump is the specular reflection.. For

this angle of incidence the functions.are almost

identical. Figure 6 shows the same function for

an incident ray at 70 degrees. Note that the

specular bump is much larger for the Torrance

The Fresnel reflection is a function only of

the index of refraction and the dot product (E-H).

If E is assumed constant at (0 0 -1) then this

calculation needs to be made only once per change

in light source direction. In addition, by some

trigonometric identities it can be shown that the

Fresnel formula can be calculated by:

A simplification which is often made is to

assume that the light source is at infinity. Thusthe vector L is a constant for each point of the

picture. We may also model the eye as being faraway from the object so that E = (0 0 -1). Thisallows the calculation of the direction of H to

be done once per change in light direction.

It is possible to avoid a potential division

by zero when computing G by combining it with the

term 1/(N-E) and finding the minimum of Ga, Gb and

Gc before doing the divisions:

If 8 does not change within a frame thefunction D3 can be calculated using the intermed-iate values (calculated once per frame):

Phong Model

Figure 5

Torrance-Sparrow Model

Comparison of Phong and Torrance-Sparrow reflectiondistributions for incident light at 300 from normal

Sparrow function and not in quite the same direct-ion. This indicates that the new function will bematerially different only for shallow angles ofincident light and that the specular reflectionwill be much higher there. This may be verifiedby the simple experiment of holding a mattesheet of paper edge on to a light and noting thatit looks quite shiny.

Figure 7 shows images of an object madeusing the two hilight functions with both anedge-on lighting direction and a front-on direc-tion. Figure 7a simulates an aluminum metallicsurface using the experimentally measuredparameters :

p, = .4pd= .6n = 200cg = .J‘5

Figure 7b simulates a Magnesium Oxide ceramic (astandard diffuse reflector) using the experimentalparameters:

p, = .667pd = .333n = 1.8

cg = .35

Note that the ceramic looks quite diffuse forlight hitting it almost perpendicularly and veryspecular (even more. so than. the aluminum) forlight hitting it almost tangentially.

VARYING SURFACE SHININESS

In [ 1 ] and [ 3 ] a technique for mappingtexture patterns onto. bicubic surfaces wasdescribed. The object-was defined as a bipara-meteric surface and the parameter values were

Phong Model Torrance-Sparrow Model

Figure 6Comparison of Phong and Torrance-Sparrow reflectiondistributions for incident light at 70” from normal

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Phong Torrance-Sparrow Both Models Essentially SameEdge Lit Front Lit

Figure ?aSimulation of Aluminum Surface

Phong Torrance-Sparrow Both Models Essentially SameEdge Lit Front Lit

Figure 7bSimulation of Magnesium Oxide Surface

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Figure 8Surface shininess varying as a function

of two different texture patterns

used as input to a texture function which scaledthe diffuse component of the reflection. Thisform of mapping is good for simulating patternspainted on the surface but attempts to simulatebumpy surfaces were disappointing. This effectcan, however, be better approximated by using thesame texture mapping approach applied to thelocal surface roughness cg.

If c3 is going to change from place to placeon the surface we must worry about nonnallizationof the D, function. In its original derivation

t in [8] Da differed from that shown here by afactor of c3'. This additional factor wasincluded here as a normallizing constant to makeD3(0)=1. Since, now, c3 is varying across thesurface, we wish to use a constant normallizingfactor based on its minimum value over thesurface. The texture modulated distributionfunction should then be:

c3 = Cmin + (l-c min) t (",v)

I CD3 = min " '

cos2a(cs2-l)+l Iwhere t(u,v)=texture value

Figure 8 shows some images made with varioustexturing functions.

CONCLUSIONS

The Torrance-Sparrow reflection model differsfrom the Phong model in the inclusion of the G, Fand l/(N*E) terms. This has a noticable effectprimarily for non-metallic and edge lit objects.The use of the D3 micro facet distributionfunction provides a better match to experimentaldata and is, happily, easier to compute than D1or D2. This savings effectively offsets theextra computation time for G and F yielding ahilight generation function having a high degreeof realism for no increase in computation time.

REFERENCES

1. Blinn, J. F. and Newell, M. E. Texture andreflection incomputer generated images.Comm ACM 19, 10(Oct 1976), 542-547

2. Bui-Tuong Phong. Illumination for computergenerated images. Comm ACM 18, 6(June 1975)311-317

3. Catmull, E. A. Computer display of curvedsurfaces. Proc. Conf. on Comptr. Graphics.May 1975 (IEEE Cat. No. 75CH0981-1C) 11-17

4. Gilpin, F. H. Effect of the variation of theincident angle on the coefficient of diffusedreflection. Trans. Illum. Eng. Soc. Vol 5, 1910854-873

5. Middleton, W. E. K. and Mungall, A. G. Theluminous directional reflectance of snow.J. Opt. Soc. Am. 42, 8(Aug 1952) 572-579

6. Torrance, K. E. and Sparrow, E. M. Polarization,directional distribution, and off-specular peakphenomena in light reflected from roughenedsurfaces. J. Opt. Soc. Am. 56, 7(Jul 1966)916-925

7. Torrance, K. E. and Sparrow, E. M. Theory foroff-specular reflection from roughened surfaces.J. Opt. Soc. Am.. 57, 9(Sep 1967) 1105-1114

8. Trowbridge, T. S. and Reitz, K. P. Averageirregularity representation of a roughenedsurface for ray reflection. J. Opt. Soc. Am.65, 5(May 1975) 531-536

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