Stochastic Rendering of Density Fields. Jos Stam. Department of Computer Science. University of Toronto.

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1 Stochastc Renderng of Densty Felds Jos Stam Department of Computer Scence Unversty of Toronto Abstract Stochastc models are often economcal to generate but problematc to render. Most prevous algorthms rst generate a realzaton of the stochastc model and then render t. These algorthms become expensve when the realzaton of the stochastc model s complex, because a large number of prmtves have to be rendered. In stochastc renderng we also model the ntensty as a random eld, and the statstcs of the ntensty eld are related to the statstcs of the stochastc model through an llumnaton model. Stochastc renderng algorthms then generate a realzaton of the ntensty eld drectly from these statstcs. In other words, a random component s shfted from the modellng to the renderng. Ths paradgm s not entrely new n computer graphcs, so related work wll be dscussed. The man contrbuton of ths paper s a stochastc renderng algorthm of gaseous phenomena modelled as random densty elds such as clouds, smoke and re. A smpled verson of the scatterng equaton s used to derve the statstcs of the llumnaton eld. Our algorthm s therefore an mprovement over smlar algorthms both n terms of computatonal speed and generalty. Resume Le rendu de modeles stochastques est souvent problematque. La plupart des algorthmes exstants generent d'abord une realzaton du modele stochastque avant le rendu. Ces algorthmes devennent couteux dans le cas ou le modele est complque a cause du grand nombre de prmtves qu dovent ^etre rendues. Dans la methode du rendu stochastque nous modelsons l'ntenste comme un champs aleatore, les statstques du champs d'ntenste sont calculees a partr des statstques du modele stochastque en utlsant un modele d'llumnaton. La methode du rendu stochastque genere une realzaton du champs d'ntenste drectement a partr de ces statstques. En d'autres termes la composante stochastque est translatee du modellage au rendu. La contrbuton majeure de ce paper consste en un algorthme pour le rendu de phenomenes gazeux tels que les nuages, la fumee et le feu. Keywords: stochastc modellng, smulaton of gaseous phenomena, scatterng equaton, sold textures, ray tracng. 1 Introducton The use of stochastc models to capture the complexty of natural phenomena s well establshed n computer graphcs. Phenomena such as terran, oceans and clouds have all been successfully modelled usng ths approach. Most research n ths area n the past has focused on developng algorthms that generate a realzaton of the stochastc model. For example, a realzaton of a fractal terran s gven by a set of connected trangles [4]. In general, the realzaton s a set of smple geometrc prmtves whch can be rendered usng a standard renderng algorthm. The cost of the renderng s therefore drectly related to the complexty of the realzaton. For complex phenomena ths means that the renderng can become prohbtvely expensve. To avod ths computatonal exploson we propose a derent approach n renderng stochastc models, based on the observaton that the ntensty eld resultng from the nteracton of lght sources wth the stochastc model s random. Therefore, ths random ntensty eld can also be descrbed by a stochastc model. The statstcs of the ntensty eld are related to the statstcs of the phenomenon through an llumnaton equaton. Thus, an mage of the phenomenon can be computed by generatng a realzaton of the ntensty eld drectly from ts statstcs. We call ths approach stochastc renderng. In other words, nstead of perturbng the model, we perturb the ntensty n a way whch s consstent wth both the model and the llumnaton equaton. We shall apply ths paradgm to the ecent renderng of gaseous phenomena modelled as random densty elds: functons that assgn a random densty to each spatal locaton. Clouds, for example, are modelled as a densty of water vapour. Prevous algorthms generate realzatons of the densty eld by usng ether procedural models [14] or spectral methods [2]. The realzaton s then sampled on a three-dmensonal lattce and rendered usng a volume-tracer. Ecent sngle-scatterng volume tracers have been developed by Ebert [3] and Sakas [16]. The eects of multple-scatterng can be calculated pror to volume tracng at the expense of an ncrease n computaton tme. Rushmeer et al. descrbe a radosty-based algorthm for densty elds havng constant scatterng propertes [15]. Kajya et al. descrbe a more general multple-scatterng algorthm based on a decomposton of the ntensty eld nto sphercal harmoncs [1], however very few detals are gven about the mplementaton. For scenes contanng complex gaseous

2 phenomena such as a typcal sky on a partly cloudy day, these algorthms become very expensve n terms of both computaton and storage. Consequently, more ecent algorthms are needed. In ths work we descrbe one soluton, a stochastc renderng algorthm for random densty elds. To get a tractable algorthm we separate the ntensty eld nto both a drect and ndrect component. The statstcs of the drect ntensty eld are calculated usng the scatterng equaton from radatve transfer. We use the fact that ndrect ntensty usually has low varance and can therefore be calculated usng only average propertes of the densty eld. Under these condtons an ecent renderng algorthm can be derved. Ths paper wll be dvded nto eght sectons. In Secton 2 we wll revew the basc theory of random elds n order to precsely dene the statement \statstcal descrpton". Secton 3 descrbes the stochastc renderng paradgm n more detal and refers to related work. In Secton 4 we wll delneate the statstcal model used to model random densty elds. In Secton 5 we wll brey state the scatterng equaton and specfy our smplfyng assumptons. Secton 6 explcates our new algorthm. In Secton 7 we wll summarze results obtaned. Conclusons and further work wll be dscussed n Secton 8. 2 Random Felds In ths secton we precsely explan the notons of random eld and statstcal descrpton. A random eld R(x) s a functon that assgns a random varable for each spatal poston x = (x; y; z). Partcular nstances of the random varables dene a (real) functon known as a realzaton of the random eld. A random eld s entrely dened f the sequence of dstrbuton probabltes F n (n 1) of ts realzatons s speced: F n(r 1; x 1;... ; r n; x n) = Prob(R(x ) r ; = 1;... ; n): In practce, however, there s nether a way to nd ths sequence, even for nte n, nor a need for ths snce a random eld can be descrbed by ts statstcal moments. These moments are obtaned by averagng over many realzatons R (1) ;... ; R (N ) : hr(x) = lm N!1 1 N NX n=1 R (n) (x): The most mportant statstcal moments of a random eld are ts mean R and ts covarance K R gven by R(x) = hr(x) and K R(x ; x ) = hr(x )R(x )? R(x )R(x ): The mean s the average value at each pont and the covarance s the amount of correlaton between two ponts. The spread around the mean s gven by the varance R(x) 2 = K R(x; x). In practce, the mean and the covarance are enough to fully characterze a random eld. It s customary n computer graphcs [13] to assume that hgher-order moments are rrelevant for vsual models. In ths paper a statstcal descrpton refers to the speccaton of these two moments. In case the covarance depends only on the derence between two ponts, the random eld s homogeneous. Homogeneous random elds are useful because they have a representaton n the frequency doman. Ths s desrable snce ecent algorthms based on the fast fourer transform exst to generate such random elds [2]. The drawback, however, s that homogeneous random elds have a constant mean and varance. Many phenomena that exhbt large scale varatons cannot therefore be modelled by homogeneous random elds. In ths work we consder nonhomogeneous random elds that are smple transformatons of homogeneous random elds, speccally random elds of the form R(x) = f(x) + g(x)q(x); (1) where Q s a homogeneous random eld wth zero mean and covarance K Q(x ; x ) = K Q(x? x ; ). The determnstc functons f and g are used to model a desred mean and varance. A smple calculaton shows that the mean and covarance of the transformed eld are equal to R(x)=f(x) and K R(x ; x )=K Q(x?x )g(x )g(x ): (2) In partcular the varance s gven by 2 R(x) = g 2 (x). A random eld wth an arbtrary mean and varance can therefore be modelled by choosng the functons f and g accordngly. We have chosen ths partcular transformaton as t gves us total control over both the mean and varance. We wll now dscuss the use of random elds n renderng. The temporal behavour of the random eld can be modelled by allowng the functons f and g to vary over tme and by specfyng the temporal statstcs of the homogeneous perturbaton Q. 3 Stochastc Renderng In general, the ntensty of lght at a gven pont s a functon of the geometry of the scene, the ntal lghtng condtons (lght sources), the reectve propertes of surfaces and the scatterng propertes of a partcpatng medum [9]. If any of these parts s random, then the resultng ntensty eld wll also be random. The statstcal descrpton of the ntensty eld s a functon of the statstcal descrptons of the parts (va the llumnaton model used). In cases when ths statstcal descrpton can be establshed, a realzaton of the random ntensty eld can be computed. In many nstances ths wll result n an ecent renderng algorthm. To our knowledge no work has been devoted n computer graphcs to calculatng these statstcs, except for the dervaton of reectance models for surfaces. Followng s a bref dscusson of these models. Many reectance models for surfaces are derved usng a statstcal descrpton of the surface. Usually only the mean value of the ntensty reected from the surface s calculated. Torrance and Sparrow model the surface by a random dstrbuton of normals [19]. Ther model has been used n many local llumnaton models. The uctuatons around the mean are usually accounted for by addng an arbtrary value through a texture map or by perturbng the mean normal of the surface [1]. Recently He et al. generalzed the work of Torrance and Sparrow to a wave-physcs descrpton of lght [7]. The only work n computer graphcs whch addresses the problem of calculatng the covarance of the reected ntensty eld to

3 our knowledge s the work of Krueger [12]. Hs calculatons become tractable for certan dealzed stuatons (e.g., planar surfaces). The ncluson of the covarance n a reectance model remans, however, an unexplored area of research n computer graphcs. Ths s due manly to the mathematcal complexty of the task. In the case of volume renderng, attempts to calculate the statstcal descrpton of the ntensty eld have not been publshed. However, Gardner's algorthm to render clouds has the semblance of a stochastc renderng algorthm n that he perturbs the transparency rather than the densty of the clouds [5]. Therefore randomness s added n the renderng nstead of n the modellng as n our algorthm. Hs model s heurstc and s not based on any well establshed equaton that descrbes the transfer of lght n densty elds. Instead Gardner's model s a modcaton of a standard llumnaton model for surfaces. Because hs algorthm s surface based t has some shortcomngs. For example, objects cannot dsappear smoothly through hs clouds. Recently Kaneda et al. have used Gardner's model n conjuncton wth ther atmospherc llumnaton model [11]. They have rendered the most realstc pctures of clouds to date. In ths paper we explore only a small set of the possble applcatons of stochastc renderng. For example, algorthms could be devsed to deal wth random lght sources. 4 Statstcal Descrpton of the Densty Feld Complex gaseous phenomena are convenently modelled by a random densty eld, a functon whch assgns a random densty varable (x) to each spatal locaton x. In order to dene the random densty eld, both ts mean and covarance must be speced. The mean descrbes the large scale varatons (\global shape") of the densty eld. A natural model for the mean s therefore a smooth functon. Many choces are possble, from splnes to nterpolaton schemes. In ths work we choose to model the mean as a smooth blendng of scattered data values, so that the model can be used both n desgn and vsualzaton. In desgn the data s provded by a user, thus allowng control over the global features of the densty [17]. In vsualzaton the data s sampled usng a measurng devce. More formally, gven values for the masses m at n ponts x, the mean s gven by: (x) = nx =1 m W (x? x ) = nx =1 (x); (3) Rwhere W s a smooth blendng functon satsfyng W (x) dx = 1. A smlar mean s obtaned f we model the densty as a sum of n random elds wth each havng a mean. In other words, the random eld can be seen as a weghted sum of \random blobs". For our purposes ths nterpretaton s very convenent. Consequently, we descrbe the covarance functons of each blob nstead of the covarance of the entre densty eld. We want the varance of each blob to be proportonal to ts mean. A natural choce then, s to make the varance drectly proportonal to the mean: ; 2 =. Wth ths Fgure 1: 2 ; =, 2 ; = and 2 ; = F ( ) choce, varatons tend to become too large near the centre of the blob whch s unrealstc. We expect the varance to drop o wth the mean but to not grow to exceedngly large values. We can avod the latter by \clampng" the values of the mean: ; 2 = F ( ). The clampng functon s dened as: F (t) = t f t < otherwse: Fgure 1 shows the eect of the clampng functon on the resultng random blob. The cuto parameter and the magntude are ether provded by a user or are estmated from data. We assume that the random blobs are transformatons of a sngle homogeneous random eld Q (see Eq. 1): = + ;Q. The covarance of each blob therefore has the followng form: K ;(x ; x ) = K Q(x? x ) ;(x ) ;(x ): (4) We assume that the covarance of the homogeneous random eld Q s a gaussan wth standard devaton : 1 1 K Q(x) = exp? kxk2 : (5) (2) Realzatons of homogeneous random elds wth a gaussan covarance are easy to generate, because the fourer transform of a gaussan s well dened [2]. 5 Illumnaton Model In ths secton we descrbe the llumnaton model that we use to derve the llumnaton eld statstcs. We begn wth a bref statement of the scatterng equaton. 5.1 The Scatterng Equaton A densty eld alters the ntensty of lght usng three eects: absorpton, scatterng and emsson. To descrbe how these eects alter the ntensty of lght along a ray, consder a ray x u = x? us wth orgn x and drecton?s. The lght at pont x shnng n drecton s s the sum of two contrbutons [8]: 2 I(x ; s) = (; b)i(x b; s) + (; u)(x u) tj(x u; s) du: (6) The rst term s the proporton of lght emtted at pont x b that s not absorbed or scattered n other drectons 1 Ths assumpton s sgncant n Secton 5.2 of ths paper. 2 All functons n ths secton are understood to be wavelength dependent.

4 by the densty eld. The attenuaton factor s called the transparency and s gven by Z u2 (u 1; u 2) = exp? t (x u) du u 1 ; (7) where t s quantty characterzng the amount of lght that s absorbed or scattered per unt length. By denton, the transparency has values between zero (total opacty) and one (total transparency). The second term accounts for the ntensty of lght due to the densty eld. The functon J, the source functon, descrbes the ncrease n ntensty at each pont. Ths s due to both the emsson and scatterng of lght from other drectons nto the drecton of the ray: J(x; s) = (1? )E(x) + 4 Z 4 p(s; s )I(x; s ) d! : The functon E denotes the emsson of ntensty by the densty, such as n the case of a re. Ths s usually proportonal to the densty. The second term on the rght hand sde s an ntegraton of all ncomng ntensty from all possble sold angles. The functon p s called the phase functon and models the sphercal dstrbuton of scattered lght. The factor 2 [; 1] known as the albedo gves the fracton of lght that s scattered. We can rewrte Eq. 6 n terms of the transparency only by denng the average source functon along the ray by: J(x ; s) = = R b R b (; u)(xu)tj(xu; s) du R b (; u)(xu)t du (; u)(xu)tj(xu; s) du : (8) 1? (; b) Wth ths denton the ntensty of lght reachng pont x along the ray can be expressed as a lnear combnaton of the background llumnaton I(x b; s) and the average source functon along the ray: I(x ; s) = (; b)i(x b; s) + (1? (; b))j(x ; s): (9) Followng we descrbe the statstcs of ths equaton. 5.2 Statstcal Equatons Because the densty eld s random, the transparency and the average source functon n Eq. 9 are also random. To derve the statstcs of the transparency, we make the assumpton that all rays orgnate from the xy plane and that ther drectons are n the postve z drecton: x u = (x ; y ; u). In other words, we assume an orthographc projecton. The ntegral n the equaton of the transparency (Eq. 7) then becomes a functon of the locaton (x ; y ) n the xy plane: T (x ; y ) = = (x ; y ; u) du = nx =1 T (x ; y ): nx =1 (x ; y ; u) du Each ntegral T s a random eld whch has a well dened mean and covarance [6]: K T;(x ; y ; x 1; y 1)= T (x ; y ) = (x ; y ; u) du; K ;(x ;y ;u; x 1;y 1;u )dudu : (1) In other words, the mean and covarance of T are drectly related to the mean and covarance of the densty through an ntegral. Usng the fact that the covarance of the homogeneous random eld Q s gaussan, the covarance of the ntegral T becomes (see Eq. 4): 3 K T;(x ; y ; x 1; y 1) = K Q(x 1?x ; y 1?y ; ) K Q(; ; u?u) ;(x ; y ; u) ;(x 1; y 1; u ) dudu K Q(x 1?x ; y 1?y ; ) ;(x ;y ; u) ;(x 1;y 1;u)du: (11) Eq. 11 depends on the fact that the support of the covarance K Q s much smaller than the support of the varance 2 ;: K Q(; ; u?u) ;(x 1; y 1; u ) du ;(x 1; y 1; u): In the lmtng case where the covarance s a delta functon the above relaton s exact. Equaton 11 demonstrates that the spatal structure of each ntegral T s essentally the same as the spatal structure of a \slce" of the densty eld. They der n that they have a dfferent mean and varance. In partcular the varance s gven by: 2 T;(x ; y ) = K Q(; ; ) F ( (x ; y ; u)) du: (12) In case both the mean and the covarance are computable, then a realzaton T () for each of the ntegrals can be generated: T () (x ; y ) = T (x ; y ) + T;(x ; y )Q () (x ; y ; ); (13) where Q () s a realzaton of the homogeneous random eld. A realzaton P for the transparency s then equal to n exp(? t T () =1 (x ; y )). In the next secton we wll descrbe an ecent computaton of the mean and the varance. The statstcs of the average source functon are tremendously more complcated to calculate. In ths work we wll assume that the average source functon s only a functon of the average of the densty eld. There are good justcatons for ths assumpton. Frst, the average source functon s n fact a convoluton of the source functon by the \weghtng" functon (; u)(x u) t along the ray. Second, the source functon tself nvolves an ntegral accountng for scatterng 3 We use the multplcatve property of a gaussan: K Q(x; y; z) = K Q(x; ; )K Q(; y; )K Q(; ; z).

5 Ray u u? u d u? u u + Fgure 3: Clampng of the Blob Fgure 2: Geometry of Ray and Blob from derent drectons. Ths scatterng acts to smooth the ntensty eld. In fact t s sometmes modelled as a duson process [8]. The same smoothng phenomenon due to ndrect llumnaton s also observed n radosty envronments [2]. 6 Implementaton In ths secton we gve an ecent algorthm to calculate both the mean and the varance of the ntegrals n the transparency. We also gve an algorthm to calculate the average source functon from the mean densty eld. 6.1 Calculaton of the Transparency We assume that the smoothng kernel that denes the mean densty (see Eq. 3) depends only on the magntude of ts argument W (x) = W (kxk). In ths case the ntegral of the mean of each blob only depends on the dstance d of the ray to the centre of the blob. Refer to Fgure 2 for the geometry of the stuaton. Let u be the parameter value of the pont on the ray closest to the centre of the blob. The ntegral s then gven by: T = m W pd 2 + (u? u)2 du = m (Table[d ; b? u ]? Table[d ;? u ]) : The table s computed for dscrete values pror to renderng: Table[d; x] = Z x W pd 2 + u 2 du: The ntegral can then be evaluated ecently by nterpolatng entres n ths table. In order to calculate the varance, a clamped verson of the mean densty must be ntegrated (see Eq. 12). Let u? and u + be the two parameter values on the ray between whch the mean densty of the blob s clamped,.e., for whch m W (kx u? x k) = (see Fgure 3). These values may not be dened n case the maxmum of the mean s smaller than the clampng value, n whch case we set u? = u + = u. The ntegral can be computed by ntegratng over each nterval separately: F ( (x u)) du = Z u? (x u) du + Z u + u? du + (x u) du = m? Table[d; u?? u ]? Table[d ;? u ] + (u +? u? )+ Table[d ; b? u ]? Table[d ; u +? u ] : u + varance dstance Fgure 4: Varance 2 T; versus the dstance d for derent values of the clampng value In Appendx A we gve complete equatons for the case where the smoothng kernel W s gaussan. Realzatons of the ntegrals T can therefore be generated ecently usng Eq. 13: T () = T + T;Q () (x u ). The choce of x u (the pont closest to the centre of the blob) guarantees that the perturbaton s sampled on a plane for each frame and that the samples are chosen coherently from frame to frame. The latter s mportant n anmated sequences n whch the vewng condtons change. In Fgure 4 we show plots of the varance T; 2 versus the dstance to the centre of the blob d for derent values of the clampng value. As expected, the growth of the varance becomes smaller for small values of the dstance,.e., when the ray s near the centre of the blob. The plan lne ndcates no clampng and s equal to the mean T. It s nterestng to note that Gardner arbtrarly vared the threshold of hs transparency functon n a smlar way. To derve the statstcal descrpton of the ntegrals T, we have assumed that the vewng rays are parallel. Ths s of course not the case n most practcal stuatons. However, f the sze of each blob s small wth respect to the mage, then the rays emanatng from a pont are approxmately parallel across each blob. 6.2 Calculaton of the Average Source Functon As we stated n Secton 5.2 we assume that the average source functon J(x ; s) (see Eq. 8) depends only on the mean densty. We can therefore drectly use the renderng algorthm descrbed n [18] whch renders densty dstrbutons modelled as a weghted sum of blobs of the type gven n Eq. 3. The renderng algorthm assumes sngle-scatterng. We refer the reader to reference [18] for a descrpton of the algorthm.

6 6.3 Overvew of the Algorthm Followng we gve an overvew of our stochastc renderng algorthm. Ths algorthm can be mplemented drectly nto a standard ray tracer. For each ray, the ntensty of lght I(x b; s) n absence of a densty eld s calculated usng a standard ray-tracer. Ths llumnaton s then attenuated and the lght created wthn the densty eld s added to t as follows: T = I b = I(x b; s) for each blob ntersectng the ray do Calculate av T = T and sgma T = T; Sample Q at mdpont: Q = Q () (x u ) Update ntegral: T = T + av T + sgma T*Q end for Calculate transparency: tau = exp(- t*t) Calculate av J = J(x ; s) from mean densty eld usng the algorthm descrbed n [18] Combne: I = tau*i b + (1-tau)*av J In cases when a shadow ray s computed, only the transparency tau must be calculated. The complexty of ths algorthm s O(n), where n s the number of blobs. Unlke volume renderng algorthms, our algorthm s therefore ndependent of the resoluton of the random perturbaton of the densty eld. On average our algorthm can be mproved by an order of magntude by groupng the blobs together nto a herarchcal data structure of boundng volumes [18]. In the next secton we descrbe results obtaned usng our algorthm. 7 Results We mplemented the stochastc renderng algorthm nto a standard ray-tracer. Pror to renderng, we precompute the tables for the transparency and we generate a realzaton of the homogeneous random eld Q. The latter s generated usng an nverse spectral method [2]. Ths method has the advantage that the resultng realzaton can be dened and s perodc on a regular threedmensonal lattce. Therefore t can be evaluated ecently usng trlnear nterpolaton and s dened for all ponts n space. The artfacts caused by the perodcty of the eld are not vsble n our mages because the eld s used only to perturb the llumnaton. We used a table of sze n all of our results. A user has control over the shape of the densty eld by specfyng the poston, centre and mass of each blob of the mean densty (see Secton 4). The perturbaton s controlled by provdng the magntude of the varance, the clampng value and the covarance functon of the homogeneous eld Q (standard devaton, see Eq. 5). The llumnaton propertes of the densty eld are dened by the followng parameters: the extncton coecent t, the albedo and the emsson functon E (see Secton 5.1). In all the results we have assumed a constant phase functon p,.e., scatterng s constant n all drectons. Next we specfy some of our results obtaned usng the stochastc renderng algorthm. Clouds. We model clouds n a way smlar to Gardner. An ndvdual cloud s generated by randomly placng blobs n an ellpsod provded by a user. The sze and mass of each blob are made nversely proportonal to ther dstance from the centre of the ellpsod to ensure that the densty s maxmum n the centre of the cloud. Clusters of ndvdual clouds can be generated by randomly generatng such ellpsods. In Fgure 5 we show four pctures of the same cloud wth derent values for the magntude and the clampng value. Clouds n the left column have a lower perturbaton than the ones on the rght; and the ones on the bottom have a lower threshold value than the ones on the top. Fgure 6 shows one of the clouds wth self shadowng, two derent clusterngs of such clouds and an areal vew of downtown on a foggy day. Clouds are characterzed by a hgh amount of scatterng, therefore we have set the albedo close to one and have used a non zero value for the emsson functon, n an attempt to model multple-scatterng. Fgure 7 shows four frames from an anmated sequence of a cloud nteractng wth Toronto's CN Tower. Ths demonstrates that our model handles the nteracton of clouds wth sold objects. Ths s an mprovement over Gardner's llumnaton model. Fre. We have used our algorthm to smulate re. The ame n Fgure 8 was modelled usng 14 blobs wth dfferent emsson parameters. The emsson values drop o as a functon of the dstance of the ame. We have also changed the colour of the emsson for hgher realsm. The smoke on the top of the ame has zero emsson and s therefore a perfect absorber. The albedo was set to zero. The llumnaton of the ame onto the walls and the oor was modelled by placng 5 pont lght sources wthn the ame. Note the random patterns generated by the sources. The renderng tmes are of the same order of magntude as the computaton of the average source functon [18]: one to ten mnutes for a typcal frame (64 48) on an Irs Indgo. 8 Conclusons and Further Work In ths paper we have presented a new algorthm to render random densty elds. Our algorthm s based on a new paradgm called stochastc renderng. Instead of generatng the random densty eld and then usng volume renderng, we derve the statstcs of the ntensty eld from the statstcs of the densty eld. From ths statstcal descrpton we then generate a realzaton of the ntensty eld drectly. Ths has the advantage that only a few random samples have to be evaluated, versus the many samples needed n the case of volume renderng. Our results demonstrate that many nterestng densty dstrbutons can be modelled and rendered ecently usng our algorthm. Our algorthm s also an mprovement over smlar algorthms snce t does not have the artfacts and drawbacks of Gardner's surface-based model. However, we needed to make certan smplfyng assumptons n order to get a tractable algorthm. Our algorthm therefore s not as general as the more sophstcated volume renderers. Improvements of our algorthm wll be the focus of future research. Ths ncludes: modellng of multple-scatterng, dervng the statstcs for a perspectve projecton and calculatng the statstcal descrpton of the average source functon. Generally speakng, the calculaton of the statstcs of an ntensty eld s a dcult task. The lack of such descrptons for

7 surface reectance models demonstrates ths. There are stll many other applcatons of the stochastc renderng paradgm whch have not yet been explored. For example, the llumnaton from ckerng lght sources has not been studed to our knowledge. 9 Acknowledgments Thanks to Eugene Fume for supervsng ths work. Thanks to Pamela Jackson for correctng and mprovng the wrtng of ths paper. Thanks to the Natural Scences and Engneerng Councl of Canada, the Ontaro Informaton Technology Research Centre, and the Unversty of Toronto for ther nancal support. A Integrals for a Gaussan Smoothng Kernel Let W be a gaussan wth standard devaton h: W h(x) = 1 (2) 3 2 h 3 exp? kxk2 2h 2 The mean ntegral T for each blob s gven by: m T = exp? d2 + (u? u ) 2 (2) 3 2 h 3 2h 2 m = exp (2) 3? d2 b?u 2 h 2 2h 2 h : du??u h In ths case the table s one-dmensonal because of the multplcatve propertes of the gaussan. The functon s dened as: (u) = Z u exp? t 2 2 dt: The values of u? and u + for whch the gaussan s equal to the clampng factor are gven by u = u u, where: vu u u = t 2 log References m (2) 3 2 h 3! exp? d2 : 2h 2 [1] J. F. Blnn. \Smulaton of Wrnkled Surfaces". ACM Computer Graphcs (SIGGRAPH '78), 12(3):286{292, [2] S. E. Chen, H. E. Rushmeer, G. Mller, and D. Turner. \A Progressve Mult-Pass Method for Global Illumnaton". ACM Computer Graphcs (SIGGRAPH '91), 25(4):165{174, July [3] D. S. Ebert and R. E. Parent. \Renderng and Anmaton of Gaseous Phenomena by Combnng Fast Volume and Scanlne A-buer Technques". ACM Computer Graphcs (SIGGRAPH '9), 24(4):357{ 366, August 199. [4] A. F. Fourner, D. Fussell, and L. Carpenter. \Computer Renderng of Stochastc Models". Communcatons of the ACM, 25(6):371{384, June : [5] G. Y. Gardner. \Vsual Smulaton of Clouds". ACM Computer Graphcs (SIGGRAPH '85), 19(3):297{ 384, July [6] W. A. Gardner. Introducton to Random Processes. MacMllan Publshng Company, New York, [7] X. D. He, K. E. Torrance, F. X. Sllon, and D. P. Greenberg. \A Comprehensve Physcal Model for Lght Reecton". ACM Computer Graphcs (SIG- GRAPH '91), 25(4):175{186, July [8] A. Ishmaru. VOLUME 1. Wave Propagaton and Scatterng n Random Meda. Sngle Scatterng and Transport Theory. Academc Press, New York, [9] J. T. Kajya. \The Renderng Equaton". ACM Computer Graphcs (SIGGRAPH '86), 2(4):143{ 15, August [1] J. T. Kajya and B. P. von Herzen. \Ray Tracng Volume Denstes". ACM Computer Graphcs (SIG- GRAPH '84), 18(3):165{174, July [11] K. Kaneda, T. Okamoto, E. Nakamae, and T. Nshta. \Photorealstc Image Synthess for Outdoor Scenery under Varous Atmospherc Condtons". The Vsual Computer, 7:247{258, [12] W. Krueger. \Intensty Fluctuatons and Natural Texturng". ACM Computer Graphcs (SIGGRAPH '88), 22(4):213{22, August [13] J. P. Lews. \Generalzed Stochastc Subdvson". ACM Transacton on Graphcs, 6(3):167{19, July [14] K. Perln. \An Image Syntheszer". ACM Computer Graphcs (SIGGRAPH '85), 19(3):287{296, July [15] H. E. Rushmeer and K. E. Torrance. \The Zonal Method for Calculatng Lght Intenstes n the Presence of a Partcpatng Medum". ACM Computer Graphcs (SIGGRAPH '87), 21(4):293{32, July [16] G. Sakas. \Fast Renderng of Arbtrary Dstrbuted Volume Denstes". In F. H. Post and W. Barth, edtors, Proceedngs of EUROGRAPHICS '9, pages 519{53. Elsever Scence Publshers B.V. (North- Holland), September 199. [17] J. Stam and E. Fume. \ A Multple-Scale Stochastc Modellng Prmtve". In Proceedngs of Graphcs Interface `91, pages 24{31, June [18] J. Stam and E. Fume. \Turbulent Wnd Felds for Gaseous Phenomena". In Proceedngs of SIG- GRAPH '93, pages 369{376. Addson-Wesley Publshng Company, August [19] K. E. Torrance and E. M. Sparrow. \Theory for O-Specular Reecton From Roughened Surfaces". Journal of the Optcal Socety of Amerca, 57(9):115{1114, September [2] R. P. Voss. \Fractal Forgeres". In R. A. Earnshaw, edtor, Fundamental Algorthms for Computer Graphcs. Sprnger-Verlag, 1985.

8 Fgure 5: Fgure 7: Fgure 6: Fgure 8: