A radiosity method for non-diffuse environments - UCL Department of

Dallas, August 18-22

Volume 20, Number 4, 1986 I

I

A RADIOSITY METHODFOR NON-DIFFUSE ENVIRONMENTS

David S. Immel, Michael F. Cohen, Donald P. Greenberg Cornell University Ithaca, N.Y. 14853

ABSTRACT A general radiosity method accounting for all interreflections of light between diffuse and nondiffuse surfaces in complex environments is introduced. As contrasted with previous r a d i o s i t y methods, surfaces are no longer required to be perfectly diffuse reflectors and emitters. A complete, viewer independent description of the light leaving each surface in each direction is computed, allowing dynamic sequences of images to be rendered with l i t t l e additional computation per image. Phenomena such as "reflection tracking", reflections following a moving observer across a specular surface are produced. Secondary l i g h t sources, such as the light from a spotlight reflecting off a mirror onto a wail are also accounted for. CR Categories and Subject Descriptors: 1.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism: 1.3.3 [Computer Graphics]: Picture/Image Generation General Terms: Algorithms Additional Keywords and Phrases: Radiosity, intensity, bi-directional reflectance, hiddensurface, depth buffer, non-diffuse reflection. INTRODUCTION The production of r e a l i s t i c images requires the a b i l i t y to simulate the propagation of light in an environment. This requires two steps: modeling the interaction of l i g h t with an individual surface and combining the effects of emission, transmission and reflection between a l l the surfaces in the environment. In the past, computer generated pictures were obtained by simulating how l i g h t is reflected from an individual surface, ignoring the interreflections from surface to surface. A number of reflection models were developed, ranging from a simple Lambertian diffuse

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model in which a l l l i g h t is reflected with equal intensity in all directions, to more complex models including non-uniform specular d i s t r i butions based on geometric and electromagnetic properties of the surface. For any given reflection model, images were created by determining the visible surface at each screen pixel location, and then computing the intensity of l i g h t leaving that surface in the direction of the eye. T h i s intens i t y was t y p i c a l l y found f r o m the geometric relationships between the visible surface, the light sources, and the eye. Ray-tracing was the f i r s t method introduced which attempted to include the effects of shadowing and reflections from neighboring surfaces [1]. In ray-tracing, an additional intensity contribution from secondary reflected and transmitted rays is added from the mirror and refracted directions. The fact that a limited number of rays are traced from the eye into the environment precludes the a b i l i t y to completely account for the global illumination arising from complex interreflections within the environment. Although ray-tracing techniques have been able to produce s t r i k i n g l y r e a l i s t i c images, only radiosity methods can account for the complex d i f fuse interreflections within an environment. The radiosity method, borrowed from thermal engineering techniques [2] [3], was introduced in the context of computer graphics by Goral in 1984 [4] and was extended for use in complex environments by Cohen [5] [6] and Nishita [7]. T h i s method is based on the fundamental Law of Conservation of Energy within closed systems. I t entails performing a simultaneous global solution for the intensity of l i g h t leaving each surface by constructing and solving a set of linear equations describing the transfer of diffuse l i g h t energy between all surfaces. Calculations are performed independently for any number of discrete wavelength bands. The advantages of t h i s method are the increased accuracy in simulating the global i l l u mination effects and the fact that a l l of the intensity calculations are independent of viewing parameters, thus allowing e f f i c i e n t rendering of dynamic sequences. An underlying assumption in a l l previous radiosity methods has been that a l l surfaces exhibit perfect Lambertian diffuse reflections. Environments c o u l d not contain surfaces with specular properties. T h i s paper demonstrates a method which removes this restriction by generalizing the radiosity formulation to include

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S I G G R A P H

directional reflection properties, while s t i l l maintaining the advantages of accurate global illumination and view independence. To correctly determine the illumination of an environment, one must simulate the spatial and spectral intensity d i s t r i b u t i o n of l i g h t leaving each d i f f e r e n t i a l surface area (or point) within the environment. There are two requirements to predict this intensity d i s t r i b u t i o n . F i r s t , the spatial and spectral d i s t r i b u t i o n of the l i g h t arriving at the point must be specified. This " i n c i d e n t " l i g h t i s a f u n c t i o n of the p o i n t ' s geometric relationship to a l l other surfaces in the environment and the l i g h t leaving those surfaces in the direction of the point. Second, a reflection model must be specified which describes how l i g h t of any spectral content, which arrives from any arbitrary direction, is scattered from the surface. This may include transmission as well as reflection and is usually referred to as the bidirectional reflectance of a surface. Since the l i g h t arriving at any surface is a function of the l i g h t leaving a l l other surfaces, i t is apparent that a simultaneous solution for the global illumination must be performed. The solution would contain a l l of the i n f o r mation necessary to render an image of any part of the environment from any viewpoint. As with standard diffuse radiosity solutions, the complex interreflections within an environment are correctly modeled, yielding "color bleeding" effects, which occur when one surface acts to illuminate another surface, as well as s o f t shadows and penumbrae, which are associated with area l i g h t sources. The new procedure for non-diffuse environments inherently contains additional directional phenomena such as specular reflections, which follow the viewer as the eye position changes. This "reflection tracking" can be seen by moving your head from side to side while looking at a shiny surface. Secondary l i g h t sources, such as the l i g h t from a spot l i g h t bouncing o f f a mirror onto the f l o o r , are also accounted f o r . This source of secondary i l l u m i nation has been ignored by a l l previous image synthesis methods. THEORETICAL CONCEPTS The intensity of a point in a certain direction, or "directional i n t e n s i t y " , is the sum of the total reflected intensity from the point in that direction and, i f the surface is an emitter, the l i g h t intensity emitted from the point in that direction:

/,(~o) =/,(~o) +/,(~o)

(i)

Figure 1 shows the hemisphere of directions surrounding the normal to a point, or d i f f e r e n t i a l area, dAl , through which and f r o m which i t receives and reflects l i g h t . The intensity emitted d i r e c t l y from ~ l i n outgoing direction o is simply:

[,('~Foj) = el(~IJo.1)

(2)

The subscript 1 has been added to indicate that the emittance and reflected i n t e n s i t y terms emanate from surface 1.

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'86

Table 1 D eflnition o f T e r m s W represents a vector direction to or from a surface. ~Fi.1 = incoming direction to surface 1 ~i.2 = incoming direction to surface 2 ~Po.l = incoming direction from surface 1 "Fo.2 = incoming direction from surface 2 8t = angle away from the n o r m a l to surface 1 El = directional emission from surface 1 p'~ = bidirectional refletance o f surface 1 dz01 = differential solid angle a r o u n d surface 1 Ei.l = energy incident to surface 1 I , j = intensity reflected from surface 1 I,.~ = intensity e m i t t e d from surface 1 I,.1 = t o t a l directional intensity leaving surface 1 I,.2 = t o t a l directional intensity leaving surface 2 flea = t o t a l solid angle encompassing surface 2 when viewed from surface 1

•

I energy

J Hemisphere

of Directions

to

a Differential

Area

FIGURE 1 The spectral and s p a t i a l d i s t r i b u t i o n of the r e f l e c t e d l i g h t is a f u n c t i o n of the material prop e r t i e s of the surface from which i t r e f l e c t s . The l i g h t i n t e n s i t y r e f l e c t e d from a p o i n t can be found given a f u l l d e s c r i p t i o n of the l i g h t arriving at that point and the surface's bidirectional reflectance. The bidirectional reflectance of a surface defines the relationship between the l i g h t energy arriving at some point and the intens i t y of l i g h t leaving that point:

#':

I,

(3)

The t o t a l incoming energy i s a f u n c t i o n of the incoming i n t e n s i t y l i , the s o l i d angle ~ i through which i t a r r i v e s , and the projected d i f f e r e n t i a l area. The incoming energy per u n i t area from direction ~IJi is thus:

Ei(~'IJl) = li(~IJi)cos~dcoi

(4)

The r e f l e c t e d i n t e n s i t y in d i r e c t i o n To from energy incident from direction '~I~iis therefore:

1,(%, ~ ) = p"(~o, ~DLcos~,~.oi

(5)

Dallas, August 18-22

Volume 20, Number 4, 1986

Consider an environment consisting of only two surfaces, A i and Az (Figure 2). Intensity radiates from d i f f e r e n t i a l area ~L~l in direction o due to both emission and r e f l e c t i o n . The reflected i n t e n s i t y in the outgoing direction results from energy a r r i v i n g a t ~ 1 fromA2. Let Ii.i be the incident i n t e n s i t y through solid angle ~ . By integrating Equation 5 over the solid angle encompassing area A2, the total i n t e n s i t y reflected in direction o due to the l i g h t energy a r r i v i n g from A2 i s :

[r(~'iJ°'l) = fnl., ~Od~(~'~°''~'Tffi'l)[i'l(~'ili'l)c°sOi'ld~'Oi'l l

-I Outgoing intensity from dL~z,

][o, l 1

(6 )

In Equation 6, the incoming i n t e n s i t i e s , Ii.l , result from and are equal to the outgoing i n t e n s i t i e s , ItS , fromA2:

i~ i(~:) =/,~(%,=)

z'(%a)

= / a i.=Pl" (~iJo.l,~'iJi,t)l t,2(~ff o.2)c°s~,ldcoi,1

(8)

Therefore, the d i r e c t i o n a l i n t e n s i t y from surface dAl in d i r e c t i o n o i s the sum of the emitted i n t e n s i t y (Equation 2) and the reflected i n t e n s i t y (Equation 8):

Similar equations define i n t e n s i t i e s leaving ~ t for a l l other directions within the hemisphere in the simple two-surface environment considered above. Each outgoing directional i n t e n s i t y from ~ is a function of the directional intensities leaving area A 2 . In an equivalent manner, the directional i n t e n s i t i e s leaving each d i f f e r e n t i a l area of Az are a function of the directional i n tensities leaving area A~. Thus, a l l of the outgoing directional i n t e n s i t i e s from both surfaces must be solved simultaneously. In a d d i t i o n to the d i r e c t i o n a l v a r i a t i o n , intensities may vary continuously according to spatial position across a surface, or according to wavelength across the v i s i b l e spectrum, since the spectral d i s t r i b u t i o n s and bidirectional reflectances are wavelength dependent. Thus, there are t h e o r e t i c a l l y an i n f i n i t e number of unknown intens i t y values to be simultaneously solved. The problem is made tractable by discretizing the surfaces, the v i s i b l e spectrum, and the hemisphere of incoming and outgoing directions. As with previous radiosity approaches, surfaces are divided into small discrete areas, or "patches", with the assumption that for each direction, the intensities arriving and leaving the patch are constant over the whole patch. The v i s i b l e spectrum is likewise divided into a small number of wavelength bands. Each wavelength band is assumed to be independent f r o m each other, allowing a separate solution for the l i g h t i n t e r ation for each band. The hemisphere above the patch is also discretized into a f i n i t e number of directions, or solid angles. Thus, for a particular patch, the

o.2

A2

/

/

Outgoing intensities from

i,i

intensities to dA

A simple e n ~ r o ~ m e n t w i t h its o w n

r

of two surfaces, each

set of directions.

FIGURE 2 d i r e c t i o n a l i n t e n s i t y in d i r e c t i o n f then becomes i t s emission in d i r e c t i o n f plus the sum of the reflected i n t e n s i t i e s from discrete directions d, which encompasses the hemisphere above the patch (Equation 10). d- I represents the outgoing directions from patch 2 which point towards patch I , and are thus in the opposite d i r e c t i o n from d. D

/, t(f) = etV') + ~ # i ' ( f , m / , ~ ( a - 1 ) c ~ e ~ ClO)

I,,1(~'lJo, l) = ~(~'lao,t) + / P~(~lJo.t, ~'IJi,l)ILl(~'lJi,l)cos~,Idg.ot,l

(~)

/

A~

(7)

Substituting the outgoing i n t e n s i t i e s from A2 f o r the incoming i n t e n s i t i e s to ~ I , Equation 6 becomes:

~/

/

d: 1

Figure 3 shows the results of two incoming directions with several outgoing directions from a patch. A set of simultaneous l i n e a r equations can now be formed, defining the unknown d i r e c t i o n a l i n t e n s i t i e s f o r a l l d i r e c t i o n s , given the b i d i r e c t i o n a l reflectivities and emissions of each patch. So f a r , the method has been i l l u s t r a t e d by an environment consisting of only two surfaces. The form of the relationships remains the same for more complex environments except that light incoming

dlstrlbuti~ l

ipt=~=ity

/ The outgoing distributions sumzl~ed, resulting i n an

from all incoming energies aggregate distribution.

are

FIGURE 3

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S I G G R A P H

arrives at a patch from many surfaces. An additional problem arises from the possible occlusion of one patch from another due to intervening surfaces. Thus, the directional i n t e n s i t i e s of each patch depend on the directional i n t e n s i t i e s of a l l other patches and the v i s i b i l i t y between them. Let N be the number of patches in an environment and D be the number of directions on the hemisphere. The outgoing i n t e n s i t y of a patch i is given by: N

'86

m:id -ModelLng Mesb.i.nti

(Interactive)

•

lobnl

Cube Preprocessi

(Non-interactive)

D

z,, ~f) = ~i(f) + ~ ~ ' ( f , j=l~=l

a)1,j (a-')cosOd~dZID(i d, d)

E

(11)

GlobalSolution (Non-interactive)

where

HID(i,j,d)

=

I Patch i sccs patch j in direction d 0 Otherwise (12)

New Eye | RenderizLg Position L (Non-interactive)

for environments with hidden surfaces. At f i r s t glance, Equation 11 appears immensely more complicated than previous radiosity solutions, since i n t e n s i t i e s must be solved for each patch for each direction. However, certain properties of the matrix and the environment can be e f f e c t i v e l y exploited to obtain tractable solutions. For example, i f in an environment, there are 1000 patches, and the hemisphere of directions is divided i n t o 1000 small s o l i d angles, then there are 1,000,000 unknowns and 1,000,000 squared or I0 x2 matrix c o e f f i c i e n t s . Fortunately, t h i s matrix is VERY sparse. This can easily be seen since each patch receives energy from only 1000 directions. T h u s , each row of the coefficient matrix has only 1000 non-zero HID terms, and the matrix is 99.9% sparse. In addition, i f some surfaces are d i f f u s e , each of t h e i r patches w i l l e x h i b i t a single constant directional i n t e n s i t y over a l l d i r e c t i o n s , and the 1000 d i r e c t i o n a l equations can be reduced to a single linear equation. These facts are taken advantage of in the implementation, which is described in the following sections. PROGRAM OVERVIEW ~'i~e directional radiosity analysis is comprised of a series of separate steps analagous to previous diffuse radiosity methods (Figure 4 ) . F i r s t , the environment geometry and material properties such as the bidirectional reflectance must be specified. An additional i n i t i a l requirement for radiosity analyses is that each surface must be subdivided into discrete surface patches. The second step is to determine the geometric and v i s i b i l i t y relationship between every pair of patches in the environment. Given the set of equations, a simultaneous solution for the directional i n t e n s i t i e s of each patch is then performed. This global patch solution is then used along with the vertex to patch v i s i b i l i t y relationships to determine the directional i n t e n s i t y d i s t r i b u t i o n at each vertex grid point on every surface. The resulting f u l l description of the i n t e n s i t y leaving each surface in a l l directions provides the means to render an image of the environment from any given viewpoint. Given an eye position and viewing direction, the i n t e n s i t i e s pointing back to the eye from each vertex are used to render an image. Succesive images of the environment can be rendered from d i f f e r e n t viewpoints.

136

Program

Flow FIGURE 4

2oj2 :;b;;:(Ztod / J

Everyotherpatch inthe environment is projected onto the

cube.

\

\ \

FIGURE 5

THE GLOBAL CUBE The l i g h t l e a v i n g a patch depends on the l i g h t a r r i v i n g at i t and the direction(s) from which i t arrives. This impinging l i g h t is the result of l i g h t leaving the other patches in the environment, and thus a determination must be made of which patches are v i s i b l e from each patch. Each patch has a view of i t s environment across the hemisphere of directions surrounding i t s normal. This continuum of directions is approximated by finding the v i s i b l e patches in a f i n i t e number of discrete solid angles which cumulatively are equivalent to the hemisphere. The hemisphere can be replaced by the upper half of a cube, or "hemi-cube" (Figure 5), creating f i v e separate 90 degree viewing frustums through which to project the environment [ 5 ] .

Dallas, August 18-22

Volume 20, Number 4, 1986

Each surface of the cube becomes an imaginary screen which can be d i v i d e d i n t o small square cells. Each of these cells defines a small solid angle of view from the patch at the center of the cube. Depth buffer algorithms are performed for each frustum to determine the patch "seen" through each c e l l . To obtain an accurate view of the environment from a patch, a series of sample points within the patch from which to view the environment are selected. A grid is established on the patch defining a series of sub-patches or "elements". A view is taken from each grid vertex and may vary across the grid. The f i n a l view of the environment from a patch is taken as a weighted average of the views from the individual grid vertices. This is analogous to the "substructuring" described in [ 6 ] . The hemi-cube can be oriented so t h a t the patch normal always coincides with the center of the hemi-cube's top face. This allows for the same set of small solid angles to be located in the same directions r e l a t i v e to each patch. However, since the orientations are always in the

patch's local coordinate system, there is no single directional relationship between hemi-cubes (Figure 6). In previous diffuse radiosity methods, since each patch was defined to have a s i n g l e i n t e n s i t y equal in a l l d i r e c t i o n s , the directional relationship was unimportant. When the i n t e n s i t i e s are allowed to vary d i r e c t i o n a l l y , i t is important to equate the i n coming directions of the receiving patch with the reciprocal outgoing directions from the sending patches. Rather than reorienting a hemi-cube for each patch, a f u l l cube surrounding each vertex is maintained, oriented along the global axes (Figure 7). The advantage l i e s in the fact that i f patch i sees patch j through the c e l l oriented in the X,Y,Z d i r e c t i o n , then the reciprocal cell from patch j is simply in the -X,-Y,-Z d i r e c t i o n . The cube is stored as a vector of v i s i b l e patch ident i f i c a t i o n numbers. I f the faces of the cube are oriented as in Figure 8a and the vector is ordered as in Figure 8b, then the reciprocal global cube locations may be t r i v i a l l y calculated. For a global cube with D discrete directions, the cell with the reciprocal direction from a cell with index c tO ~

9 Patch4 Patch 7 Patch 9

latch4matrix .... FI If arch7trix ....ixl

atc° h 'trXt°r°s]

Incorrect T o d e c r e a s e the n u m b e r of iterations, solve for t h e p a t c h e s r o u g h l y the s a m e order that light is p r o p o g a t e d t h r o u g h t h e environment.

FIGURE 10

in

RESULTS The following images were rendered from two global solutions of the same environment. Each set of f i g u r e s contains a series of views (a) i l l u s t r a t i n g the view independence of the solution. Notice the reflections in the f l o o r tracking the eye as the viewer position is changed. The two larger images (b and c) in each set are enlargements from the series above them. The primary difference between the two sets of images is the bidirectional reflectance models. The algorithms outlined in t h i s paper are independent of any p a r t i c u l a r reflectance model. The s o l u t i o n f o r the images in Figure 12 used the Phong-like reflectance model. For Figure 13, the solution used the simpler mirror-like bidirectional reflectance. The environmental modeling and meshing for the images was performed using the testbed image synthesis system at Cornell's Program of Computer Graphics. The system was written in C under a VMS operating system and ran on a VAX 11/780. An Evans and Sutherland picture system was used for the modeling and meshing, a Floating Point Systems FPS264 attached to a VAX 11/750 was used for the solution process, and the pictures were displayed using a Rastertek 1280x1024x24 bits frame buffer.

139

,~ S I G G R A P H '86

(a)

(b)

(c) PBONG-LIKE REFLECTION

Number Number Number Number Number Number

of of of of of of

Specular Patches : Diffuse Patches : Specular Vertices: Diffuse Vertices : Cube Directions : Vavelengths :

PROCESS

64 93 1089 1400 24576 4

.

.

.

.

.

.

.

.

.

CPUTIME (Hrs) .

.

.

.

.

.

.

.

.

.

Visible Surfaces Solution

Render

.

.

.

.

.

.

.

.

.

.

.

.

8 192 1

FIGURE 12

As can be seen from the s t a t i s t i c s l i s t e d below each set of images, the computation time, p a r t i c u l a r l y for the solution process, is nontrivia]. The time required f o r the s o l u t i o n process is a f u n c t i o n of the number of cube directions and the number of patches and vertices, more s p e c i f i c a l l y , the number of specular patches and vertices. The amount of time is s i g n i f i c a n t l y reduced by using the simpler m i r r o r - l i k e reflection model in Figure 13 since incoming energy is not scattered in a l l directions. An a r t i f a c t due to the nature of the algorithm becomes v i s i b l e when the reflection becomes more mirror-like. Recall that the rendering is based on the vertex solution which is in turn based on the patch solution. Thus, the discrete patches become v i s i b l e in the r e f l e c t i o n . T h i s resulted in the increased number of d i f f u s e patches in Figure 13.

140

CONCLUSION A general radiosity method which accounts for a l l i n t e r r e f l e c t i o n s from both diffuse and nondiffuse surfaces has been introduced. The procedure computes an intensity d i s t r i b u t i o n for every surface in an environment, consisting of directional intensities for a number of discrete directions. The rendering process then becomes one of only looking up the intensities that point back to the eye and displaying them. This method makes i t possible to precalculate the energy leaving specular surfaces regardless of the viewing direction. Since this global solution is view independent, successive views of the same environment can be calculated using the same directional intensity information, and the phenomenon of reflection tracking can be observed. The a d d i t i o n of specular surfaces to the radiosity method is s i g n i f i c a n t , since radiosity methods need no longer be limited to environments

Dallas, August 18-22

Volume 20, Number 4, 1986 IIII

I

I

(a)

(c)

(b) MIRROR-LIKE REFLECTION Number Number Number Number Number Number

of of of of of of

Specular Patches : Diffuse Patches : Specular Vertices: Diffuse Vertices : Cube Direetlons : Vavelengths :

PROCESS

64 237 1089 1400 15000 4

Visible Surfaces Solution Render

CPU TIHE ( R r s ) 7 25 1

FIGURE 13 containing only purely d i f f u s e surfaces. The methodology is independent of a particular reflection model, t h u s any reflection model which relates incoming energy from one direction to outgoing intensity in another direction can be used. The only restriction placed on the model is that i t must conform to the physical Taws of reciprocity and the conservation of energy as discussed in the paper. A few problems exist that must be dealt with in the future. Errors in approximation occur because environments have been discret~zed. Discret i z i n g the number of directions on the global cube results in misrepresenting the incoming energy because of aliasing on the faces of the global cube. Reflections from a highly reflective surface may appear blurred due to d i s c r e t i z i n g the surface, as well as interpolating between outgoing intensities in rendering.

Future work should include the addition of non-planar and textured surfaces, as well as transparent and translucent surfaces. The framework of the method is general enough to accomodate these additions by simply using different reflection functions. Although this method currently requires very large computational resources for complex environments due to the volume of information that must be processed, this research was undertaken to provide a s c i e n t i f i c b a s i s for modeling complex interreflections within an environment. In the future, as storage and computational power increases, global solutions of more complex environments w i l l be within reach.

141

~. S I G G R A P H '86

ACKNOWLEDGMENTS The authors would l i k e to thank a l l those who helped in the preparation of the article. Thanks go to Professor Kenneth Torrance, Holly Rushmeier, and Kevin Koestner for their help in laying the theoretical groundwork. Thanks also go to Dan Baum, Janet Brown-Aist, and Emil Ghinger for their help in coding and preparation of the article. This research was conducted at the Cornel] University Program of Computer Graphics, under a grant from the National Science Foundation #DCR-8203979. The VAX computers used were made possible by a large grant from the Digital Equipment Corporation and use of the Floating Point Systems Advanced Processor through Cornell University's Center for Theory and Simulation in Science and Engineering. REFERENCES [I]

Whitted, Turner, "An Improved Illumination Model for Shaded Display," Communications of the ACM, Vol. 23, No. 6, June 1980, pp. 3 4 - ~ 9 .

[2]

Siegel, Robert and John R. Howell, Thermal Radiation Heat Transfer, Hemisphere Publishing Corp., Washington DC., 1981.

[3]

Sparrow, E. M. and R. D. Cess, Radiation Heat Transfer, Hemisphere Publishing Corp., Washington DC., 1978.

[4]

Goral, Cindy M., Kenneth E. Torrance, Donald P. Greenberg, Bennet Battaile, "Modeling the Interaction of Light Between Diffuse Surfaces," C ACoMm p u t e r Graphics (Proceedings 1984), pp. 213-222.

[5]

Cohen, Michael F. and Donald P. Greenberg, "A Radiosity Solution for Complex Environments," ACM Computer Graphics (Proceedings 1985), pp. 31-40.

[6]

Cohen, Michael F., Donald P. Greenberg, David S. Immel, Philip J. Brock, "An E f f i c i e n t Radiosity Approach for Realistic Image Synthesis," IEEE Computer Graphics and Applications, March 1986.

[7]

Nishita, Tomoyuki, and Eihachiro Nakamae, "Continuous Tone Representation of Three-Dimensional Objects Taking Account of Shadows and Interreflection," ACM Computer Graphics (Proceedings 1985), pp. 22-30.

[8]

Phong, Bui Tuong, Illumination for Computer Generated Images, P h . D . Dissertation, University of Utah, 1973.

[9]

Hornbeck, Robert W., Numerical Methods, Quantum Publishers, New York, NY, 1974, pp. 101-106.

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