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~ Com puter Graphics, Volume 24, Num ber 4, August 199

Particle Transport and Im age SynthesisJames Arvo*

David Kirk*~*Apollo Systems Division of Hewlett-Packard300 Apollo DriveChelmsford, MA 01824

* California Institute of TechnologyComputer Science 256-80Pasadena, CA 91125A b s t r a c tT h e r e n d e r i n g e q u a t i o n i s s i m i l a r t o t h e l in e a r B o l t z m a n ne q u a t i o n w h i c h h a s b e e n w i d e l y s t u d i e d i n p h y s i c s a n d n u -c l e a r e n g i n e e r in g . C o n s e q u e n t l y , m a n y o f t h e p o w e r f u l te c h -n i q u e s w h i c h h a v e b e e n d e v e l o p e d i n t h e s e f i e ld s c a n b ea p p l i e d t o p r o b l e m s in i m a g e s y n t h es i s . I n t h i s p a p e r w ea d a p t s e v e r a l s t a t i s t i c a l te c h n i q u e s c o m m o n l y u s e d i n n e u -t r o n t r a n s p o r t t o s t o c h a s t i c r a y t r a c i n g a n d , m o r e g e n e r a l ly ,t o M o n t e C a r l o s o l u t i o n o f t h e r e n d e r i n g e q u a t i o n . F i r s t , w ed e s c r i b e a t e c h n i q u e k n o w n a s R u s s i a n r o u l e t t e w h i c h c a nb e u s e d t o t e r m i n a t e t h e r e c u r s i v e t r a c i n g o f r a y s w i t h o u ti n t r o d u c i n g s t a t i s t i c a l b i a s . W e a l s o e x a m i n e t h e p r a c t i c eo f c r e a t in g r a y t r e e s i n c l a s s i c a l r a y t r a c i n g i n t h e l i g h t o f aw e l l - k n o w n t e c hn i q ue i n p a r ti c le t r a ns p o rt k n o w n a s spl i t -t ing . W e s h o w t h a t n e i t h e r r a y t r e e s n o r p a t h s a s d e s c r i b e di n [ 1 0] c o n s t i t u t e a n o p t i m a l s a m p l i n g p l a n i n t h e m s e l v e sa n d t h a t a h y b r i d m a y b e m o r e e f fi c ie n t.C R C a t e g o r i e s a n d S u b j e c t D e s c r i p t o r s : 1 . 3 . 7 -[ C o m p u t e r G r a p h i c s ] : T h r e e - D i m e n s i o n a l G r a p h i c s a n dR e a li sm ; 1 . 3 . 3 - - [ C o m p u t e r G r a p h i c s ] : P i c t u r e / I m a g eG e n e r a t i o n ;G e n e r a l T e r m s : A l g o r i t h m s , G r a p h i c sA d d i t i o n a l K e y W o r d s a n d P h r a s e s ~ B o I t z m a n n e q u a -t i o n , M o n t e C a r l o , p a r t i c l e t r a n s p o r t , r a d i o s i t y , r a y t r a c i n g ,r e n d e r i n g e q u a t i o n .1 I n t ro d u c t i o nT h e r e n d e r i n g e q u a t i o n [1 0] p r o v i d e s a f r a m e w o r k i n w h i c haL l c u r r e n t i m a g e s y n t h e s i s t e c h n i q u e s c a n b e b e v i e w e d a sm e t h o d s o f a p p r o x i m a t i o n . B o t h r a d i o s i t y [ 7] a n d r a y t r a c -i n g [1 8] a re e x a m p l e s o f a p p r o x i m a t i o n b e c a u s e t h e y n e g l e c tv a r i o u s o p t i c a l p h e n o m e n a i n o r d e r t o y ie l d a r e a s o n a b l em e t h o d o f s o lu t io n . A n a l t e r n at i v e , in t r o d u c e d b y K a j i y a ,i s t o s o lv e t h e r e n d e r i n g e q u a t i o n d i r e c t l y v i a M o n t e C a r l ot e c h n i q u e s s i m i l a r to t h o s e d e v e l o p e d f o r n e u t r o n t r a n s p o r tp r o b l e m s . S u c h t e c h n i q u e s h a v e a l o n g h i s t o r y a n d h a v e

Pelmission o copy w ithout fee all or part of this material is grantedprovided that the copies are not made or distributed for directcomm ercial advantage, the ACM cop)right notice and the title of thepublication and its date appear, and notice is given that copying is bypermission of the Association for Com puting Machinery. To copyotherwise, or to repu blish, requires a fee and/or specific permission.

be en applied to integral equations of essentially the sa mefo rm as the rendering equati on since the 50's [1].

K a j i y a d e m o n s t r a t e d t h e f e as ib il it y o f th i s a p p r o a c h i nima ge synthesis by successfully solving the rendering equa-tion for scenes including bo th specular an d diffuse eflectorsT h o u g h t h e l e v e l o f re a l is m a t t a i n ab l e i n t hi s w a y i s v e r yh i g h, t h e c o s t c a n b e p r o hi b i ti v e d u e t o s l o w c o n v e r g e n c e o ft h e M o n t e C a r l o m e t h o d . O t h e r m o r e e ff ic ie nt a p p r o a c h e sh a v e b e e n d e v is e d [ 1 6, 1 7] b u t n o n e h a v e c o m p l e t e l y o b v i -ated the ne ed for stochastic appr oxi mat ion with out sacrificing certain mo de s of light transport.

Relat ed statistical pproac hes hav e bee n applied to raytracing. Co ok , et al. [4] described a stochas tic sa mp li ngt e c h n i q u e t e r m e d d i s t r i bu t e d a y t r a c i n g w h i c h p r o v i d e s ame an s of anti-aliasing as w ell as simulating effects such asm o t i o n b l u r, p e n u m b r a e , d e p t h o f f ie ld , a n d f u z z y r ef le ctions. Its central idea is that features in the env iro nme ntw h i c h v a r y i n t i m e a n d s p a c e c a n b e s a m p l e d s t oc h as t ic a llto estimate their contribution to the final image. Bo th oft h es e p a r a d i g m s h a v e a g r ea t d e al i n c o m m o n w i t h M o n t eCarlo techniques applied to particle transport pro ble ms inother fields.

2 Pa r t i c l e T ra n s p o r tT h e c l a ss o f p a r t i c l e t r a n s p o r t p r o b l e m s c o n s i s t s o f t h o s ep r o b l e m s w h i c h s e e k t o c h a r a c t e r i z e t h e d i s t r i b u t i o n o f i d e -a l i z e d p a r t i c l e s t a k i u g a c c o u n t o f t h e i r m o t i o n a n d i n t e r -a c t i o n w i t h a m e d i u m [ 5, 12 ,1 9] . S u c h p r o b l e m s a p p e a r i nn u c l e a r e n g i n e e r i n g as n e u t r o n t r a n s p o r t [1 5], i n h e a t t r a n s -f e r a s p h o t o n t r a ns p o rt [ 13 ], a n d i n s e m i c o n d u c t o r d e v ic esimulation as carrier transport [6]. M a n y of the equationsgoverning these transport processes ultimately derive fromt h e B o l t z m a n n e q u a ti o n w h i c h a r os e f r o m t h e k i ne t ic t h e o ryo f g a s es . I n i t s s i m p l e s t f o r m t h e l i n e a r B o l t z m a n n e q u a t i o nc a n b e w r i t t e n a s

( P ) = S ( P ) + f K ( P ' --~ P ) q,(P') e ' (IJ f lw h e r e P r e p r e s e n t s p a r t i c l e p o s it i o n , d i r e c t i o n , a n d e n e r g yan d ~( P) is the density of radiation at P due to emissionfr om the source S as well as contributions scattered intP f r o m a l l P ' [ 11 ]. T h e f u n ct i o n K i s k n o w n a s t h e s c a tt e e i n g k e r n e l , a n d t h e d o m a i n o f in t eg r at i on , ~ , c o n si s t s o

1 9 9 0 ACM -0-89791-344-2/90/008/0063 $00.75 6

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O SIGGRAP H '90, Da l las, August 6-10, 1990ia l l p o s i t i o n s , d i r e c ti o n s ~ a n d e n e r g i e s . T h i s i s a n o t o r i o u s l yd i ff ic u l t e q u a t i o n t o s o l v e a n a l y t i c a l l y i n a l l b u t t h e m o s tt r i v i a l p r o b l e m i n s t a n c e s [ 2, 5] . T h i s i s t r u e o f t h e r e n d e r i n ge q u a t i o n a s w e l l w h i c h i s e s s e n t i a l l y a v a r i a n t o f t h e l i n e a rB o l t z m a u n e q u a t i o n . T h e p r i n c i p a l d i f fe r en c e i s t h a t t h es c a t t e r i n g k e r n e l i s r e p h r a s e d a s a g e o m e t r y t e r m , g~ w h i c ha c c o u n t s fo r o c c l u s i o n a n d i n v e r s e s q u a r e a t t e n u a t i o n , a n da tr i v a r i a t e s c a t t e r i n g t e r m , p , w h o s e a r g u m e n t s a r e s u r -f a c e p o i n t s ( S e e [1 0 ]) . T h e l a t t e r e n c o d e s t h e d i r e c t i o n s o fi n c i d e n c e a n d r e f l e c t i o n i m p l i c i t l y t h r o u g h t h e p o s i t i o n s o ft h e s o u r c e a n d d e s t i n a t i o n e l e m e n t s r e l at i v e t o t h e p o i n t o fr e fl e c ti o n . T h e s i m i l a r it y o f t h e r e n d e r i n g e q u a t i o n t o t h el in e a r B o l t z m a n n e q u a t i o n s u g g e st s t h a t m a n y o f t h e p o w e r -f u l t e c h n i q u e s w h i c h h a v e b e e n d e v e l o p e d f o r o t h e r p a r t i c l et r a n s p o r t p r o b l e m s m a y b e a p p l ie d t o p r o b l e m s i n i m a g es y n t h e s i s .W e n o t e t h a t t h e r e a r e s e v e r al a s p e c t s i n w h i c h t h e r e n -d e r i n g e q u a t i o n is so m e w h a t m o r e t r a c ta b l e t h a n t h e t r an s -p o r t e q u a t i o n s i n fi el d s s u c h a s n u c l e a r e n g i n e e r i n g .

1 ) T h e p a r t i c l e s ( i. e. p h o t o n s ) d o n o t i n f l u en c e o n e a n o t h e r ,a l t e r t h e e n v i r o n m e n t , c a r r y a c h a r g e , o r r e p l i c a te v i af i ss i o n . T h u s s c a t t e r i n g i s i n d e p e n d e n t o f a s w e l l a se x t e r n a l f o r c e s , m a k i n g t h e e q u a t i o n l i n e a r.2 ) I n t h e a b s e n c e o f p a r t i c i p a t i n g m e d i a , c o l li s io n s o c c u ro n l y a t s u r f a c e s . T h e p a r t i c l e s t h e r e f o r e h a v e a r e l a -t i v e l y l a r g e m e a n f r e e p a t h .3 ) W e s ee k o n l y th e s t e a d y s t a t e s o l u t i o n , n o t t r a n s i e n td i s t r i b u t i o n s o n t h e w a y t o e q u i li b r i u m .

T h e s e p r o p e r t i e s m a n i f e s t t h e m s e l v e s l a r g e ly i n t h e r e l a-t i v e l y s im p l e f o r m o f t h e s c a t t e r i n g k e r n e l w h i c h i s c o m -p r i s e d o f t h e b i d i r e c t i o n a l r e f l e c t a n c e f u n c t i o n s a s s o c i a t e dw i t h t h e s u r f a c e s. A f t e r p r o b a b i l is t i c a ll y d e t e r m i n i n g a n e wp a r t i c l e d i r e c t i o n a t e a c h s c a t t e r i n g e v e n t t h e n e x t c o l ] . i s i o n -s i te a l o n g t h e r a n d o m w a l k is c o m p l e t e l y d e t e r m i n e d , e l im -i n a t i n g s t o c h a s t i c d i s t a n c e c a l c u la t i o n s . H o w e v e r , t h e r e a r et w o r e s p e c t s i n w h i c h t h i s t r a n s p o r t p r o c e s s is m a d e m o r ed i ~ c u l t t h a n t y p i c a l l y e n c o u n t e r e d i n o t h e r d i sc i p li n e s.F i r s t, t h e g e o m e t r y o f t h e s i m u l a t e d e n v i r o n m e n t s c a nb e a r b i t r a r i l y c o m p l e x . W h i l e s im u l a t i o n s o f r e a c t o r c o r e sa n d s e m i c o n d u c t o r d e v i c e s b e n e f i t f r o m f a i rl y c o n s t r a i n e dg e o m e t r i e s a n d e x p l o i t s p e c i a l p r o p e r t i e s o f l a t ti c e s , c y l i n -d e r s , s l a b s , e t c . [1 2 ], t h e t r e n d i n c o m p u t e r g r a p h i c s i s t om o v e t o w a r d g r e a t e r a n d g r e a t e r s c en e c o m p l e x i t y . T h i si s e x e m p l i f ie d b y r e c e n t w o r k i n v o l v i n g b i ll i o n s o f g e o m e t -r i ca l p r i m i t i v e s [ 1 4]. T h i s c a n b e f u r t h e r c o m p l i c a t e d b yt i m e - d e p e n d e n t s c en e g e o m e t r y . S i m u l a t i o n o f t h e r e s u l ti n gm o t i o n b l u r r eq u i r e s t i m e a v e r a g i n g s t e a d y - s t a t e s o l u t i o n sa t i n t e r m e d i a t e s c e n e c o n f i g u r a t io n s .

S e c o n d l y , t h e p r o b l e m o f i n t e r e st i n i m a g e s y n t h e s i s i s t oc o m p u t e t h e i n t e n s i t y o f i l l u m i n a t i o n i m p i n g i n g o n a si n g lep o i n t , t h e " e y e " , t h r o u g h s m a ll a p e r t u r es w h i c h c o r r e s p o n dt o " p i x e l s" . A n a l o g o u s s it u a t i o n s o c c u r i n r e a c t o r s h i e ld -i n g p r o b l e m s w h i c h s i m u l a t e p o i n t r a d i a t i o n d e t e c t o r s [3 ].T h e s e a r e i n h e r e n t l y m o r e d i i B c n lt t o s o lv e t h a n t h e t y p i c a lp r o b l e m s w h i c h i n v o l v e f l u x a v e r a g e s o v e r v o l u m e s .M a n y i m p o r t a n t p r o b l e m s i n p a r ti c le t r a n s p o r t d o n o ta d m i t a n a l y t i c s o l u t i o n s a n d a r e al s o p r o h i b i t i v e l y e x p e n s i v et o s o lv e v i a n u m e r i c a l i n t e g r a t i o n d u e t o t h e h i g h d i m e n s i o no f t h e p h a se sp a ce i n w h i c h t h e y o p e r a t e ( e .g . t h r e e s p a -t i a l d i m e n s i o n s , t w o d i r e c t i o n a l d i m e n s i o n s , a n d a n e n e r g yd i m e n s i o n ) . T h e o n l y r e c o u r s e f o r s o l v i n g t h e s e t y p e s o fp r o b l e m s a p p e a r s t o b e M o n t e C a r l o m e t h o d s w h i c h t r a c k

t h e b e h a v i o r o f l a r g e n u m b e r s o f p a r t i c le s o b e y i n g t h e p r e -s c r i b e d l a w s o f m o t i o n e x p r e s s e d a s s c a t t e r i n g p r o b a b i l i ti e s .E a c h p a r t i c l e u n d e r g o e s a s e q u e n c e o f col l i s ions o r sca t ter-i n g even t s w h i c h p r o b a b i l i s t i c a ll y a l t e r it s t r a j e c t o r y a t e a c hc o l l i s i o n - s i t e a n d c o n t r i b u t e t o t h e h i s to ry o f t h e p a r t i c l e .E a c h p a r t i c le h i s t o r y , o r r a n d o m w a lk , i s u s e d a s a s t a t i s t i c a le s t i m a t o r o f a v e r a g e c a se b e h a v i o r . R a y t r a c i n g i s a m e c h a -n i s m f o r c o m p u t i n g p o i n t s o f c o l li s io n , a n d a s t o c h a s t i c r a yp a t h [1 0] i s t h e r e s u l t in g r a n d o m w a l k o f a p a rt i c le . T h er e n d e r i n g e q u a t i o n p r o v i d e s a l i n k w h i c h a l lo w s u s t o v i e wi m a g e s y n t h e s i s i n t e r m s o f p a r t i c le t r a n s p o r t . T h r o u g h t h i sc o n n e c t i o n w e c a n g a i n u s e f u l in s i g h t i n t o t h e f e a t u r es a n dl i m i t a t i o n s o f i m a g e s y n t h e s i s t e c h n i q u e s .F o r e x a m p l e , c o n s i d e r t h e u s e o f d e c o u p l e d p a s s e s o f ra yt r a c i n g a n d r a d i o s i t y t o m o d e l s p e c u l a r a n d d i ff u se m o d e so f t r a n s p o r t i n d e p e n d e n t ly . I t h a s b e e n o b s e r v e d t h a t s im -p l y c o m b i n i n g t h e r e s u l ts o f t h e s e p a s s e s fa il s to a c c o u n tf o r s o m e i m p o r t a n t p h e n o m e n a o f g e o m e t r i c al o p t i c s [ 16 ].T h e m o s t o b v i o u s e x a m p l e i s a c a u s ti c f o r m e d b y s p e c u l a r l yt r a n s m i t t e d o r r e f le c t e d l i g h t f al l in g o n a d i ff u s e s u r f a c e .B o t h c l a s s i c a l r a y t r a c i n g a n d r a d i o s i t y t o t a l l y n e g l e c t t h i sm o d e o t r a n s p o r t , t h e r e f o r e t h i s d e fi c i en c y c a n n o t b e r e m e -d i ed b y s u m m i n g t h e ir c o n t r i b u t i o n s a posteriori . W a l l a c ed e s c r ib e s a s o l u t i o n f o r th i s p a r t i c u l a r c a s e o f s p e c u l a r - t o -d i ff u se t r a n s p o r t , b u t i t is im p o s s i b l e t o a c c o u n t f o r a ll su c hs e q u e n c e s o f t r a n s p o r t a s s p e c i a l c a se s . T h i s p h e n o m e n o nh a s b e e n o b s e r v e d in o t h e r l i n e a r t r a n s p o r t p r o b l e m s a n di s a t t r i b u t e d t o t h e f ac t t h a t e q u a t i o n 1 , t h o u g h l i n e a r i nt h e s o u r c e t e r m , S , i s n o n l i n e a r w i t h r e s p e c t t o t h e s c a t -t e r i n g k e r n el , K . W h i l e t h e l i n e a r i ty i n S a l lo w s u s t o s u mt h e i n d e p e n d e n t c o n t r i b u ti o n s m a d e b y d i ff er e n t s o u rc e s a n dw a v e l e n g t h s o f l i g h t, t h e a n a l o g o u s d e c o u p l i n g f a il s w h e n t h ek e r n e l is p a r t i t i o n e d i n t o , f o r e x a m p l e , Kspec+Kdiff. A f a i t h -f n l s i m u l a t i o n o f a l l m o d e s o f t r a n s p o r t c a n o n l y b e a c h i e v e db y c o u p l i n g t h e m i n t h e s o l u t i o n p r o c es s .3 R u s s i a n R o u l e t t eT h e albedo o f a s u r f ac e is th e p r o b a b i l i t y t h a t a n i n c i d e n tp a r t i c l e w i l l b e r e - r a d i a t e d a f t e r c o l l is i o n [ 3] . I n M o n t e C a r l os i m u l a t i o n s t h i s p r o b a b i l i t y i s n o r m a l l y u s e d t o a d j u s t a n u -m e r i c a l weig h t a s s o c i a t e d w i t h t h e p a r t i c l e r a t h e r t h a n p r o b -a b i l is t i ca l l y t e r m i n a t i n g t h e h is t o r y . T h i s te c h n i q u e , t e r m e dimp l i c i t ca p tu re [ 12 ], h a s b e t t e r s t a t i s t i c a l p r o p e r t i e s o w i n gt o l o n g e r p a r t i c l e h i s t o r i e s.A p r o p e r t y o f i m p l i c it c a p t u r e i s t h a t p a r t i c l e h i s to r i e sc a n o n l y t e r m i n a t e a t s u r f a c e s o f z e r o a lb e d o o r b y leakage,t h a t i s , b y e s c a p i n g t h e s y s t e m . H o w e v e r , it is n e a r l y a l w a y si m p r a c t i c a l t o c o n t i n u e t r a c i n g a p a t h u n t i l o n e o f t h e s ec o n d i t i o n s i s m e t . E v e n i f w e co u l d g u a r a n t e e t h e e v e n t u a lt e r m i n a t i o n o f e v e r y h is t o ry , w e w o u l d s p e n d a n i n o r d i n a t ea m o u n t o f t i m e c o m p u t i n g c o l li s io n s i n v o l v i n g p a r ti c l e s o fn e g l i g i b l e w e i g h t . O n e s o l u t i o n i s t o p l a c e a l i m i t o n t h en u m b e r o f s c a t t e r i n g e v e n t s i n a p a r t i c le h i s t o r y a n d t o ig -n o r e a ll c o n t r i b u t i o n s b e y o n d th i s p o i n t . A b e t t e r s o l u t i o ni s t o u s e w e i g h t c u t o ~ w h i c h t r u n c a t e s t h e p a r t i c l e 's h i s t o r yo n l y w h e n i t s w e ig h t f a ll s b e l o w s o m e t h r e s h o l d [1 2]. T h ei d e a o f u s i n g w e i g h t c u t o f f t o t e r m i n a t e r a y t r a c i n g r e c u r s i o nw a s i n t r o d u c e d b y H a l l [8 ] a n d t e r m e d a d a p t i ve t ree d ep thcontrol. B o t h o f t h es e t e c h n iq u e s a r e c o m m o n l y e m p l o y e di n r a y t r a c i n g i m p l e m e n t a t i o n s i n o r d e r t o a v o i d e x ce s s iv e l yd e e p r a y t r e es a n d , i n e x t r e m e c a s es , e v e n u n e n d i n g r e c u r -s i o n d u e t o o p p o s i n g m i r r o r s o r t o t a l i n t e r n a l r e fl e ct i o n . T h ed i f fi c u lt y w i t h t h i s t y p e o f p o l i c y is t h a t t r u n c a t i o n i n t r o -

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~ Computer Graphics, Volume 24, Num ber 4, August 1990

i f w e i g h t < T h r e s h t h e nb e g i ns a m p l e s u n i f o r m l y f r o m [0 , 1]i f s < .P t h e n t e r m i n a t e p a t he l s e w e i g h t e - w e i g h t ~ ( 1 - P )e n d

Fig ure 1: The Russian Roulette algorithm which is used to ter-minat e particles with insignificant weights without i ntrodu cingbias. The value P can be any probability less than 1.

duces a systematic bias to the estimator which may becomesignificant if applied to a large nu mbe r of paths.Fortuna tely, this bias can be elim inate d by a simple tech-nique known as R u s s i a n r o u l e t t e [3,12,15]. Acco rding to thistechnique, once the weight of a particle has fallen below ap r e - d e f i n e d t h r e s h o l d w e t e r m i n a t e i ts h i s t or y p r o b ab i l i st i -c al ly , w i t h s o m e g i v e n p r o b a b i l it y , P . I f t h e p a r t i c l e " s u r -v i v e s, " it s w e i g h t i s i n c r e a s e d b y a f a c t o r o f 1 / ( 1 - P ) . L e tw d e n o t e t h e w e i g h t o f a p ar t i c l e b e f o r e p l a y i n g R u s s i a nr o u l et t e a n d l e t t h e r a n d o m v a r i ab l e W d e n o t e i t s s u b s e-q u e n t we i g h t . T h e e x p e c t e d v a l u e o f W , d e n o t e d E ( W ) , i sthen given by

E ( W ) = Prob( t e r m i n a t i o n ) 0 +113Prob( s u r v i v a l ) * 1--~- f (2)

But the probabil i ty of termin at ion is P, and that of survivalis 1 - P, so we haveW

P*0 + (l -P) * 1- P - w (3)which is the t he ori ginal weight of the particle. On average,then, the par ticle will have the appro priate weight. We maytherefore ignore the maj orit y of the insi gnificant particlesby artificially inflating the co ntrib ution s of those which sur-vive. Altho ugh eli minat ing the bias in this way does in factincrease the variance slightly, if applied to particles of suf-ficiently low weight this can be more tha n c ompe nsat ed forby the additi onal samples we can collect for the same overallcost . Perhaps more important ly, el iminat ing the bias guar-antees tha t we will converge to the correct result in the lim itif the sample mea n converges at all. The Rus sian Roulet tealgor ithm is outline d in figure 1.4 S p l i t t in g : P a t h s v s . T r e e sA n o t h e r t e c h n i q u e w h i c h i s c o m m o n l y u s e d t o i m p r o v e t h ee f f i c ie n c y o f p a r t i c l e t r a n s p o r t s i m u l a t i o n s i s s p l i t t i n g . h i l eR u s s i a n r o ul e t t e r e d u c e s t h e n u m b e r o f s c a t t e r i ng e v e n t sa t t h e e x p e n s e o f a s li g h t i n c r ea s e i n v a r i an c e , t h e g o a l o fs p l it t i ng i s t o r e d u c e v a r i a n c e b y i n t r o d u c i n g m o r e s c a tt e r -i n g e v e n t s . I t w o r k s b y p a r t i t i o n i n g a s i n g l e p a r t i c l e i n t oa m u l t i p l i c i t y o f p a r t i cl e s , t r a c k i n g t h e i r d i v e r g i n g h i s t o r i e si n d e p e n d e n t l y , t h e n d o w n w e i g h t i n g t h e i r c o n t r ib u t i o ns a p -p r o p ri a t e ly . I n r e a c to r s i m u l a t i o n s s p l i tt i n g i s u s e d w h e n an e u t r o n e n c o u n t e r s a r e g i o n w h i c h i s p a rt i c u l ar l y i m p o r t a n to r o f h i g h s e n si t iv i t y . T h o u g h t r a c k i n g m a n y l i gh t w e i g h t

p a r t i c l e s i s c o s t l y , i t i s j u s t i f i e d i f t h e v a r i a u c c i s r e d u c es u ff i ci e nt l y. B e c a u s e i t i s u s e d s t r i c t l y a s a v a r i a n c e r e d u ct i o n t e c h n i q u e a n d n o t a s a m e a n s o f s i m u l a t i n g f i ss i on , i t ia p p l ic a b l e t o p h o t o n t r a n s p o r t a s w e l l.

I n t h e c l a ss i c a l a p p r o a c h t o r a y t r a c i n g i n t r o d u c e d b yW h i t t e d [ 1 8] , a s i n g l e r a y c a n r e c u xs i v e ly s p a w n a mu l t i pl i ci t y o f r a y s a t s u r f ac e s w h i c h b o t h r e f le c t a n d t r a n s m i t l i g hs p e c nl a r l y. C o o k , e t a l. [4] g e n e ra l i z e d t h i s a p p r o a c h b yr e p l ac i n g t h e d e t e r m i ni s t i c b r a n c h i n g s t e p s b y p r o b a b i l i st ic o n e s d i s t r i b u t ed o v e r s p a t i a l a n d t e m p o r a l d i m e n s i o n s .T h r o u g h t h e g e n e r al i t y o f M o n t e C a r l o i n t e g r a t i o n, t h i s a ll o w e d a w i d e r v a r i et y o f o p t i ca l ef f e ct s t o b e s i m u l a t e d w i t ht h e s a m e n u m b e r o f s a m p l e s . T h e r e s u l t i n g m e t h o d o f p r o ba b i li s t ic b r a n c h i n g i s e s s e n t i a l l y a n a p p l i c a t i o n o f s pl i t ti n g

A s K a j i y a o b s e r v e d , h o w e v e r , t h i s a p p r o a c h c r e a t es u nn e c e s sa r i l y b u s h y r a y t r e e s a n d e x p e n d s m o s t o f t h e e ff o ra t t h e l e a v es ( h i g h e r g e n e r a t i o n r a y s ) w h i c h m a k e o n l y s m a l l c o n t r i b u t i o n [ 1 0 ]. T h o u g h R u s s i a n r o u l e t t e (S e c. 3c a n h e l p t o l i m i t t h e d e p t h o f t h e s e t r e e s b y t e r m i n a t i n gl o w - w e i g h t b r a n c h e s f ai rl y, i t d o e s n o t i n i ts e lf r e d u c e t hb u s h i n e s s o f t h e t r e e . K a j i y a s u g g e s t s t h a t i t i s m o r e a p -p r o p r i a t e t o t r a c e p a t h s i n s t e a d o f t r e e s. A t e a c h c o ll i si oe v e n t, e x a c t l y o n e r a y i s f o l l o w e d b y p r o b ab i l i st i c a l ly c h o o s -i n g o n e s c a t te r i n g m o d e t o s a m p l e f r o m ; f o r e x a m p l e , e i th et h e r e f l e c t e d o r t h e t r a n s m i t t e d l i gh t .

W e c a n c o m p a r e t h e t w o a p p r o a c h e s u s i n g t h e f i g u r e om e r i t [ 1 2 ] o r e f f i c i e n c y 1 1 ] o f t h e r e s u l t i n g e s t i m a t o r s . T h i sm e a s u r e , w h i c h w e s h a l l d e n o t e b y ~ , i s d e f i n e d b y

1- ( 4 )i f 2 Twhere 0.~ is the va rianc e of the est ima tor an d ~- is the cosa s s o c i a t e d w i t h d r a w i n g a s i n g l e s a m p l e . I n t h i s c a s e a s a m -p l e c o n s is t s o f a c o m p l e t e p a r t ic l e h i s t or y . A t e a c h c o ll i si oe v e n t w e w i s h t o s a m p l e t h e i nc i d e nt i l l u m in a t i on i n s u c h w a y t h a t t h e e n t i re e s t i m a t o r i s a s e f f ic ie n t a s p o s s i bl e . T h ei d e a b e h i n d p a t h t r a c in g i s t o u s e a s i n g l e p a r ti c l e, t h e r e b yr e d u c i n g v , w h i c h i n c l ud e s t h e c o s t o f t r a c i ng e a c h r a y ithe en vkonm ent. This cost can be considerable for compleenvironments. On the other hand, averaging many part iclehistories leads to an estimator with a smaller variance, 0"2A r e t h e r e a n y s i t u at i o n s i n w h i c h t h i s r e d u c t i o n i n v a r i a n co u t w e i g h s t h e c o s t o f t r a c k i n g m u l t i p l e p a r t i c l e h i s t or i e sT h o u g h i t i s d i ff i cu lt i n g e n e r a l t o e s t i m a t e b o t h ~- a n d 0 .f o r a n y g i v e n s t r at e g y , w e c a n n e v e r t h e le s s c o n s t r u c t e x a m -p l e s in w h i c h s p l i t ti n g c o n f er s a c l e a r a d v a n t a g e .

C o n s i d e r a pa r t i c le w h i c h e n c o u n t e r s N i d e a l m i r r o r r eR e c t o r s b e f o r e r e a c h i n g a d i f f u s e r e fl e ct o r. I f w e e s t i m a t et h e i n c i d e n t i l l u m i n a t i o n a t t h e d i f f u s e s u r f a c e b y t r a c i n g s i n g le p a t h , o b t a i n i n g a v a r i a n c e o f o- 1 a t a c o s t v , t h e n t he f f i c i en c y o f t h e e n t i r e e s t i m a t o r i s

1e l - 0 . ~ ( N o ~ + r ) ( 5

w h e r e a i s t h e a v e r a g e c o s t o f t r a c i ng a s i n g l e r a y i n t he n v i r o n m e n t . O n t h e o t h e r h a n d , i f w e a c h i e v e a s l ig h tll o w e r v a r i an c e , 2, ~ , b y s p l i tt i n g i n t o m p a t h s o f t h e s a m ec o s t a f t e r tr a c i ng a s i n g l e p a t h t o t h e d i f f us e r e f l ec t o r t h et h e e f f i c i e n c y o f t h e e n t i r e e s t i m a t o r i s

1~ = 0 . ~ ( N a + m r ) ( 6To see that spl i t t ing can be advantageous in some instancewe need only observe that

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O StGGRAP H '90, Da l las, Augu st 6-10, 1990~ r n _ _ C r 1 2N ~ o e l o ' ~ > 1 (7 )

T h i s s h o w s t h a t f o r a n y g i v e n a a n d v , a f t e r a s u f fi c i e nt l yl a r g e n u m b e r o f m i r r o r r e f l e c t io n s s p l i t t i n g i n t o m u l t i p l ep a t h s i s a m o r e e f f i c i e n t s t r a t e g y t h a n c o n t i n u i n g a s i n g l ep a t h .A n o t h e r i n s t a n c e i n w h i c h s p l i t t i n g i s a d v a n t a g e o u s i sw h e n m u l t i p l e s a m p l e s r e d u c e t h e v a r i a n c e s i g n i f i c a n t l y .M o r e p r e c is e l y , i f a t a n y p o i n t a l o n g a p a t h w e c a n e m p l o y ms a m p l e s o f e q u a l c o s t t o e s t i m a t e t h e i n c i d e n t i l l u m i n a t i o n2 s u c h t h a tn d a c h ie v e a v a r i a n c e O ' m

~ m < - - ( s )m

w h e re o-12 s t h e v a r i a n c e o f a s in g e s a m p le , t h e n e m > e l a n dw e h a v e a n in c re a s e in e l~ ic ie nc y. U n d e r c e r t a in c o n d i t io n s ,s u c h a r e d u c t i o n i n v a r i a n c e c a n b e o b t a i n e d t h r o u g h samplestratif ication, a c o m m o n M o n t e C a r l o t e c hn i q u e i n w h ic ht h e d o m a i n o f i n t e g r a t i o n i s p a r t i t i o n e d i n t o d i s j o i n t re g i o n sw h i c h a r e s a m p l e d i n d e p e n d e n t l y .I f t h e i n c i d e n t i l l u m i n a t i o n a t a s u r f a c e p o i n t c a n b es e p a r a t e d i n t o l o w - v a r ia n c e s t r a t a w h o s e m e a n v a l ue s d if f erg r e a tl y , t h e n s p l i t t i n g i n t o o n e p a t h f o r e a c h s t r a t u m w i llr e s u l t i n a m o r e e f fi c ie n t e s t i m a t o r [ 9 ]. T w o s t r a t a w h i c h w i llo f t e n m e e t t h e s e c r i t e r i a a r e t h e i n t e n s e d i r e c t i l l u m i n a t i o nf r o m l i gh t s o u rc e s a n d t h e a t t e n u a t e d i n d i r e ct i l l u m i n a t io nf r o m t h e r e m a i n d e r o f t h e e n v i r o n m e n t [ 10 ].W h i l e r a y t r e e s g e n e r al l y p la c e t o o m u c h o f t h e c o m p u -t a t i o n a l b u r d e n a t t h e l e a ve s , t h e s e e x a m p l e s i n d i c a t e t h a tt h e r e e x i s t c a s e s i n w h i c h t r e e s l e a d t o g r e a t e r e f f i c i e n c yt h a n a s t r ic t a p p l i c a t i o n o f p a t h t r a c i n g . T h i s s u g g e s ts t h a ta h y b r i d m e t h o d c a n a c h i e v e h i g h e r e f f i ci e n cy t h a n e i t h e rs t r a t e g y a l o n e i f i n e x p e n s i v e h e u r i s t i c s f o r s t r a t e g y s e l e c t i o na r e e m p l o y e d .5 ConclusionA s t h e l e v e l o f r e a l i sm i n c o m p u t e r g e n e r a t e d i m a g e s h a sg r o w n , t h e u n d e r l y i n g i l l m n i n a t i o n m o d e l s h a v e e n c o u n t e r e dm a n y o f t h e c o m p l i c a t i o n s c o m m o n t o o t h e r p a r t i c l e t r a n s -p o r t p r o b l e m s . T h i s i s n o t s u r p r i s i n g w h e n o n e v i e w s t h er e n d e r i n g e q u a t i o n a s a f o r m o f t he l i n e a r B o l t z m a n n e q u a -t i o n, a t r a n s p o r t e q u a t i o n w h i c h h a s a p p l i c a t i o n in m a n y a r -e a s o f s c i en c e a n d e n g i n e e r i n g . W e c a n e x p l o i t t h i s s i m i l a r i t yb y d r a w i n g u p o n t e c h n i q u e s d e v e l o p e d f o r o t h e r t r a n s p o r tp r o b l e m s a n d a p p l y i n g t h e m t o i m a g e s y n th e s i s. F i e ld s s u c ha s n u c l e a r e n g i n e e r i n g a r e r i c h s o u r c e s o f s t a t i s t i c a l t e c h -n i q u e s w h i c h a r e a p p l i c a b l e t o s t o c h a s t i c r a y t r a c i n g a n dt o M o n t e C a r l o s o l u t i o n o f t h e r e n d e r i n g e q u a t i o n . A s e x -a m p l e s , w e h a v e d i s c u s se d t h e u s e s o f R u s s i a n r o u l e t t e a n ds p l i t t i n g i n t h i s c o n t e x t . F i n a l l y , b e c a u s e i m a g e s y n t h e s i sp r e s e n t s a d d i t i o n a l c h a l l e n g e s d u e t o f e a t u r e s s u c h a s c o m -p l e x s c e n e g e o m e t r y , t e c h n i q u e s d e v e l o p e d f o r i m a g e s y n t h e -s i s m a y a l s o b e u s e f u l i n o t h e r d o m a i n s .Re fe r e n c e s[1 ] A l b e r t , G . E . , " A g e n e r a l t h e o r y o f s t o c h a s t i c e s t i m a t e so f t h e N e u m a n n s e ri e s fo r s o l u ti o n o f c e r t a i n F r e d h o l mi n t e g r a l e q u a t i o n s a n d r e l a t e d s e r i e s , " i n S y m p o s i u m o nMonte Carlo Methods, e d i t e d b y M . A . M e y e r , J . W i l e y ,N e w Y o rk 1 9 5 6 , p p . 3 7 -4 6 .

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Computer Graphics, 2 1 (4 ) , J u ly 1 9 8 7 , p p . 3 1 1 -3 2 0 .[ 17 ] W a r d , G r e g o r y J . , F r a n c i s M . R u b i n s t e i n , a n d R o b e r tD . C l e a r , " A R a y T r a c i n g S o l u t i o n f o r D i f f u se I n t e rr e f l e c -t i o n , " Compute r Graph ic s , 2 2 (4 ) , A u g u s t 1 9 8 8 , p p . 8 5 -9 2 .[ 18 ] W h i t t e d , T u r n e r , " A n I m p r o v e d I l l u m i n a t i o n M o d e l f orS h a d e d D i s p l a y , " C o m m u n i c a t i o n s o f t h e A C M , 3 2 (6 ) ,J u n e 1 9 8 0 , p p . 3 4 3 -3 4 9 .[ 1 9 ] W i l l i a m s , M . M . R . , " M a t h e m a t i c a l M e t h o d s i n P a r t i -c l e T r a n s p o r t T h e o r y , " J . W i l e y , N e w Y o r k , 1 97 1 .

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