MONTE Carlo ray tracing is the method of choice for

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1 IEEE TRASACTIOS O VISUALIZATIO AD COMPUTER GRAPHICS, VOL. 11, O. 5, SEPTEMBER/OCTOBER 25 1 Radiance Caching for Efficient Goba Iuination Coputation Jarosav Kriváne, Pasca Gautron, Suanta Pattanai, Meber, IEEE, and Kadi Bouatouch, Meber, IEEE Abstract In this paper, we present a ray tracing-based ethod for acceerated goba iuination coputation in scenes with owfrequency gossy BRDFs. The ethod is based on sparse saping, caching, and interpoating radiance on gossy surfaces. In particuar, we extend the irradiance caching schee proposed by Ward et a. [1] to cache and interpoate directiona incoing radiance instead of irradiance. The incoing radiance at a point is represented by a vector of coefficients with respect to a heispherica or spherica basis. The surfaces suitabe for interpoation are seected autoaticay according to the roughness of their BRDF. We aso propose a nove ethod for coputing transationa radiance gradient at a point. Index Ters Goba iuination, ray tracing, heispherica haronics, spherica haronics, directiona distribution. æ 1 ITRODUCTIO MOTE Caro ray tracing is the ethod of choice for coputing iages of copex environents with goba iuination [2]. Even for the radiosity ethod, high quaity iages are created by fina gathering, often using Monte Caro ray tracing [3]. Monte Caro ray tracing is, however, expensive when it coes to coputing indirect iuination on surfaces with ow-frequency BRDFs (bidirectiona refectance distribution functions). Too any rays have to be traced to get a reasonaby precise estiate of the outgoing radiance at a point. Fortunatey, a high degree of coherence in the outgoing radiance fied on those surfaces [1], [4], [5], [6] can be expoited by interpoation [1], [7] to obtain a significant perforance gain. Our goa is to acceerate Monte Caro ray tracing-based goba iuination coputation in the presence of surfaces with ow-frequency gossy BRDFs. We achieve it through sparse saping, caching, and interpoating radiance on those surfaces. In particuar, we extend Ward et a. s irradiance caching [1], [8] to gossy surfaces. Irradiance caching is based on the observation that refected radiance on diffuse surfaces due to indirect iuination changes very sowy with position. However, this appies to surfaces with arbitrary ow-frequency BRDFs. Motivated by this observation, we extend Ward et a. s wor to cache and interpoate the directiona incident radiance instead of the irradiance. This aows us to acceerate indirect ighting coputation on surfaces with gossy BRDFs. We ca the new ethod radiance caching.. J. Kriváne, P. Gautron, and K. Bouatouch are with IRISA, Capus de Beauieu, 3542 Rennes, France. E-ai: {jrivane, pgautron, adi}@irisa.fr.. S. Pattanai is with the Schoo of Coputer Science, Coputer Science Buiding, University of Centra Forida, Orando, FL E-ai: suant@cs.ucf.edu. Manuscript received 26 Juy 24; revised 22 Dec. 24; accepted 17 Jan. 25; pubished onine 12 Juy 25. For inforation on obtaining reprints of this artice, pease send e-ai to: tvcg@coputer.org, and reference IEEECS Log uber TVCG The incoing radiance at a point is represented by a vector of coefficients with respect to spherica or heispherica haronics [9]. Due to the basis orthogonaity, the iuination integra evauation (1) reduces to a dot product of the interpoated incoing radiance coefficients and the BRDF coefficients. Radiance interpoation is carried out by interpoating the coefficients. We enhance the interpoation quaity by the use of transationa gradients. We propose nove ethods for coputing gradients that are ore genera than the ethod of Ward and Hecbert [8]. Radiance caching shares a the advantages of Ward et a. s wor. Coputation is concentrated on visibe parts of the scene; no restrictions are iposed on the scene geoetry; ipeentation and integration with a ray tracer is easy. Our approach can be directy used with any BRDF represented by (hei)spherica haronics, incuding easured BRDFs. This paper extends the initia description of radiance caching given in [9]. The ain contributions of this paper are the extension of irradiance caching to gossy surfaces, an autoatic ethod for seecting BRDFs suitabe for radiance caching, new ethods for coputing transationa radiance gradient, and integration of radiance caching in a ray tracer. The rest of the paper is organized as foows: Section 2 suarizes the reated wor. Section 3 gives an overview of how radiance caching wors and how it is integrated in a ray tracer. Section 4 detais different aspects of radiance caching. Section 5 presents the resuts. Section 6 discusses various topics not covered in the agorith description. Section 7 concudes the paper and suarizes our ideas for future wor. 2 RELATED WORK 2.1 Interpoation Interpoation can be used in goba iuination whenever there is a certain eve of soothness in the quantity being coputed. The radiosity ethod uses interpoation in the for of surface discretization, e.g., [1], [11], [12]. In the context of Monte Caro ray tracing, approaches have been /5/$2. ß 25 IEEE Pubished by the IEEE Coputer Society

2 2 IEEE TRASACTIOS O VISUALIZATIO AD COMPUTER GRAPHICS, VOL. 11, O. 5, SEPTEMBER/OCTOBER 25 proposed for screen space interpoation [5], [13], [14], [15]. The goa of these ethods is to dispay an approxiate soution quicy. However, they do not acceerate the coputation of the fina high quaity soution, which is the objective of our wor. Object space interpoation has aso been used for the purpose of fast previewing [16], [17]. Sparse saping and interpoation for high quaity rendering was used in [7], [1]. The approach of Baa et a. [7] is suitabe ony for deterinistic ray tracing. Ward et a. [1] use interpoation ony for diffuse surfaces. Our approach extends this wor to support caching and interpoation of the directiona incoing radiance on gossy surfaces. 2.2 Caching Directiona Distributions Caching directiona distributions has been used to extend the radiosity ethod to support gossy surfaces, e.g., [18], [19], [2], [21], [22], [23], [24], [25]. It has aso been used in Monte Caro ray tracing on diffuse surfaces [6], [26], [27]. Susae et a. [6] and Kato [26] use reprojection of radiance sapes. Tawara et a. [27] seectivey update a radiance sape ist in tie to expoit tepora coherence. Storing ight partices in the scene can aso be thought of as caching a directiona distribution [28], [29]. 2.3 Spherica Function Representation A representation of functions on a (hei)sphere is necessary for incoing radiance caching. Piecewise constant representation [6], [24], [26], [27] is sipe but prone to aiasing and eory deanding. Uness higher order waveets are used, even waveet representation [21], [22], [23], [25], [3] does not reove the aiasing probes. evertheess, waveets are a viabe aternative to the use of spherica or heispherica haronics, especiay for higher frequency BRDFs. Spherica haronics [18], [2], [31], [32], [33], [34], [35], [36], [37] reove the aiasing probe and are efficient for representing ow-frequency functions. However, representation of sharp functions requires any coefficients and ringing ight appear. Heispherica haronics [9] are better suited for representing functions on a heisphere. Basis functions siiar to spherica haronics are Zernie poynoias [38], [39] and the heispherica haronics of Mahotin [4]. Unie for spherica haronics, the rotation procedure is not avaiabe for these basis functions. We choose heispherica and spherica haronics because they are the ony basis for which an efficient rotation procedure is avaiabe. They aso have good antiaiasing properties, ow storage cost, and are easy to use. 2.4 Iuination Gradient Coputation Arvo [41] coputes the irradiance Jacobian due to partiay occuded poygona eitters of constant radiosity. Hozschuch and Siion [42] hande poygona eitters with arbitrary radiosity. Ward and Hecbert [8] copute irradiance gradient using the inforation fro heisphere saping. Our gradient coputation is aso based on heisphere saping. We use gradients to iprove the soothness of the radiance interpoation. One of the agoriths we use for gradient coputation was independenty deveoped by Annen et a. [43]. 2.5 Irradiance Caching Ward et a. [1] propose irradiance caching as a eans of coputing indirect diffuse interrefections in a ray tracer [2]. They sape the irradiance sparsey over surfaces, cache the resuts, and interpoate the. For each ray hitting a surface, the irradiance cache is queried. If irradiance records are avaiabe, the irradiance is interpoated. Otherwise, a new irradiance record is coputed by saping the heisphere and added to the cache. In [8], the interpoation quaity is iproved by the use of irradiance gradients. We retain the basic structure of the origina agorith, but each record stores the incoing radiance function over the heisphere. This aows us to appy the interpoation to gossy surfaces. 3 ALGORITHM OVERVIEW Radiance caching is a part of a ray tracing approach to goba iuination. At every ray-surface intersection, the outgoing radiance is evauated with the iuination integra: Lð o ; o Þ¼ Z 2 Z =2 L i ð i ; i Þfð i ; i ; o ; o Þ cos i sin i d i d i ; where L is the outgoing radiance, L i is the incoing radiance, and f is the BRDF. The integra is spit into parts and each of the is soved by a different technique:. Direct iuination uses a deterinistic ethod for point ight sources and area saping for area ight sources [44].. Perfect specuar refections/refractions are soved by tracing a singe deterinistic secondary ray.. Ward s irradiance caching coputes the indirect diffuse ter for purey diffuse surfaces.. Two different techniques ay be used for gossy surfaces. - Low-frequency BRDF. Our radiance caching coputes the indirect gossy and diffuse ters. - High-frequency BRDF. Monte Caro iportance saping coputes the indirect gossy ter and irradiance caching coputes the indirect diffuse ter. Radiance caching is not used for high-frequency BRDFs since any coefficients woud be needed. Moreover, highfrequency BRDFs are we-ocaized and iportance saping provides good accuracy with a few secondary rays. The distinction between ow and high-frequency BRDFs is done autoaticay, as described in Section 4.1. The steps of the rendering agorith reated to radiance caching are shown in Fig. 1. The ith radiance cache record contains:. position p i,. oca coordinate frae ðu i ; v i ; n i Þ,. heispherica haronics coefficient vector i representing the incoing radiance,. two @x representing the transationa gradient, and. haronic ean distance R i of objects visibe fro p i. denotes a coefficient vector and denotes a coefficient, that is, ¼f g. Each record stores the incident radiance function in a view-independent anner so that it can be ð1þ

3 K RIV AEK ET AL.: RADIACE CACHIG FOR EFFICIET GLOBAL ILLUMIATIO COMPUTATIO 3 Fig. 2. Adaptive BRDF representation for (a) Phong BRDF with exponent h ¼ 15 and (b) anisotropic Ward BRDF [46] with d ¼, s ¼ 1, x ¼ :6, y ¼ :25. The order of the heispherica haronics representation adapts to the BRDF without ever exceeding the specified axiu representation error (here, 5 percent). The coor diss represent BRDF representation error for different outgoing directions ð o ; o Þ. Directions are apped on the dis with the paraboic paraetrization. (One can iagine the diss as ooing at the heisphere fro the top.) The graphs represent one scanine fro the dis iages (i.e., fixed y and varying x coponent of the outgoing direction). In the case of the Ward BRDF, radiance caching is not used for soe directions since the representation error woud be too high. Fig. 1. Outine of the radiance caching agorith. (HSH stands for heispherica haronics.) reused for different viewpoints. The records are stored in an octree, as described by Ward et a. [1]. 4 RADIACE CACHIG DETAILS 4.1 BRDF Representation We represent the BRDFs using the ethod of Kautz et a. [35] which we briefy describe here. We discretize the heisphere of outgoing directions. For each discrete outgoing direction ð o ; o Þ, we use heispherica haronics to represent the BRDF over the heisphere of incoing directions. The nth order representation of a cosine weighted 1 BRDF f ðo; Þ for an outgoing direction ð o ; o Þ is where H f ðo; oþð i ; i Þ Xn 1 c ð o ; o Þ¼ Z 2 Z =2 ¼ X ¼ c ð o ; o ÞH ð i ; i Þ; fð o ; o ; i ; i ÞH ð i ; i Þ sin i d i d i : are the heispherica haronics basis functions [9]. We sape the outgoing heisphere for ð o ; o Þ using the paraboic paraetrization [45]. 1. A BRDFs are utipied by the cosine ter cos i before coputing the haronics representation. ð2þ ð3þ Adaptive BRDF Representation. Heispherica haronics representation for a scene BRDFs is coputed before the rendering starts. For each outgoing direction, the adaptive representation of f ðo ; o Þ uses the iniu order n sufficient to avoid exceeding the user specified axiu error, easured as described in [33] (see Fig. 2). If no order n<n ax is sufficient for the specified error, the heispherica haronics representation is discarded: Radiance caching wi not be used for that BRDF and that outgoing direction. After appying this procedure, ony ow-frequency BRDFs are represented using the haronics and radiance caching is used for the. This constitutes an autoatic criterion for discerning ow and high-frequency BRDFs in our rendering fraewor. n ax is user specified; we use n ax ¼ 1 for our exapes. The ai is to have n ax such that a BRDF is cassified as ow-frequency if and ony if radiance caching is ore efficient than Monte Caro iportance saping. Whie n ax ¼ 1 was a good vaue for our scenes, it woud not have to be so in other ones. Higher n ax aows using radiance caching for higher frequency BRDFs. The higher the frequency of the BRDFs, the ore rays wi be needed to sape the heisphere for a new record and the ess reuse wi be possibe for that record. 4.2 Incoing Radiance Coputation Whenever interpoation is not possibe at a point p, a new radiance record is coputed and stored in the cache. We represent the incoing radiance L i by a vector of heispherica haronics coefficients ¼f g as L i ð; Þ P n 1 P ¼ ¼ H ð; Þ, where n is the representation order. The coefficients for an anaytica L i woud be coputed with the integra ¼ Z 2 Z =2 L i ð; ÞH ð; Þ sin dd:

4 4 IEEE TRASACTIOS O VISUALIZATIO AD COMPUTER GRAPHICS, VOL. 11, O. 5, SEPTEMBER/OCTOBER 25 Our nowedge of L i is based ony on saping (ray casting). Hence, we copute by a Monte Caro quadrature with unifor saping: ¼ 2 X L i ð ; ÞH ð ; Þ; ð4þ ¼1 where L i ð ; Þ is the incoing radiance coing fro the saped direction ð ; Þ and is the nuber of saped directions. We use a fixed, but adaptive heisphere saping [47], [48] is desirabe. The order n for the incoing radiance representation is equa to the order of the BRDF representation at p. This cuts off high frequencies fro the incoing radiance. The approach is justified by a ow-frequency BRDF acting as a ow-pass fiter on the incoing radiance [34]. If incoing radiance vectors with different nuber of coefficients are interpoated, the shorter vectors are padded with zeros. 4.3 Transationa Gradient Coputation We want to copute the transationa gradient r for each coputed with (4). The gradient is used to iprove the interpoation soothness. We did not succeed in extending the gradient coputation of Ward and Hecbert [8] to hande our case since their ethod tighty coupes the cosine probabiity density and the cosine weighting used for irradiance coputation. Instead, we have deveoped two new ethods for coputing transationa gradient r. The first, nuerica, dispaces the center of the heisphere. The second, anaytica, is based on differentiating the ters h of (4). i In both cases, we copute the gradient @y ; by coputing the partia =@x =@y. The gradient is defined in the oca coordinate frae at the point p. The derivative with respect to z is not coputed since typica dispaceents aong z are very sa and using it does not visuay iprove the interpoation quaity. We copute the gradients siutaneousy with the coputation of the coefficients during the heisphere saping uerica Gradient Coputation To copute the =@x nuericay, we dispace the point p, aong the oca x-axis, by x to p (Fig. 3). For each Monte Caro sape L i ð ; Þ, we: 1. Copute the new direction ð ; Þ at p as ð ; Þ¼q p r. Here, q is the point hit by the ray fro p in direction ð ; Þ and r ¼q p.we wi aso denote r ¼q p. See Fig. 3 for the various ters used here. 2. Copute the soid ange associated with the new direction ð ; Þ. The soid ange associated with each direction in (4) is unifor and equa to 2=. With the dispaceent of the point p, the soid anges no onger reain unifor. The change in soid ange is due to the change in distance r ¼ q p and orientation of the surface at q, as seen Fig. 3. Gradient coputation by dispacing the heisphere center fro p to p ((a) before and (b) after the dispaceent). The quantities changing with the dispaceent are (shown in red): sape ray direction ð ; Þ, the soid ange associated with this sape, and the ange between the sape direction and the surface nora at the hit point q. either the hit point q nor the area A visibe through change with the dispaceent. fro the heisphere center p or p. The soid ange cos before the dispaceent is ¼ A ¼ 2 r 2, where is the ange between the surface nora at q and the vector fro q to p. The area A ¼ r 2 cos 2 is the part of the environent visibe through. It does not change with the dispaceent because we assue that the environent visibe fro p and p is the sae. After the dispaceent, the soid ange subtended by A becoes ¼ A cos r 2 We now estiate the coefficient ¼ 2 X r 2 r 2 ¼1 ¼ 2 r 2 r 2 at p as cos L i ð ; ÞH cos cos : cos ; ; and, finay, we =@x ¼ð Þ=x. The coputation =@y proceeds in a siiar way. This copetes the nuerica estiation of the transationa gradient r at the point of interest Anaytica Gradient Coputation We rewrite (4) as ¼ X ¼1 L i ð ; ÞH ð ; Þ; with ¼ 2 for unifor heisphere saping. We have seen that does not reain constant with dispaceent of p and, therefore, it has to be incuded in the su and differentiated. The partia =@x is coputed by differentiating the ters of the su in (5): ð5þ

5 K RIV AEK ET AL.: RADIACE CACHIG FOR EFFICIET GLOBAL ILLUMIATIO COMPUTATIO 5 Pugging (11) and (7) into (6), we get the copete forua =@x. The foruas =@y are siiar; ony (8) ust be repaced by (9). A siiar gradient cacuation was aso proposed in [29]. This ethod disregards the change of and, hence, does not provide good resuts. The anaytica ethod was aso independenty deveoped by Annen et a. [43]. A code fragent evauating one ter of the su in (6) is given in the accopanying ateria [5]. Fig. 4. Quantities in the ¼ L i ð ; ÞH ð ; Þ ¼1 X L i ð @x H ð ; Þ ð ; Þþ ¼1 The derivative of the basis function ð ; with @H ð =@x ¼ cos cos =r =@x ¼ sin =ðr sin Þ: Those derivatives with respect to y woud be ð ; Þ @ =@y ¼ cos sin =r ; =@y ¼ cos =ðr sin Þ: =@ =@ are given in Appendix A. The derivative of the soid @x A cos r 2 ¼ r 2 : The area A ¼ 2 r 2 cos is the part of the environent visibe through. It does not change with the dispaceent. The change of cos =r 2 with the dispaceent of p is opposite to its change with the dispaceent of q ¼ðq x ;q y ;q z Þ, r 2 x r 2 : The x can be coputed with the assuption that p ies at the origin because ony the reative r 2 position of p and q atters qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (see Fig. 4). Since cos ¼ n q r and r ¼q ¼ cos. r 2 q 2 þ q2 y þ q2 z, we x r 2 n x q x þ n y q y þ n z q x ðqx 2 þ q2 y þ q2 z Þ3=2 ¼ r n x þ 3q x cos r 4 : ð8þ ð1þ Here, n ¼ðn x ;n y ;n z Þ is the surface nora at q. Cobining this resut with A ¼ 2 r 2 cos, ¼ 2 r n x þ 3q x cos r 2 cos : ð11þ Discussion For the derivation of both nuerica and anaytica ethods, we assued:. The radiance L i ð ; Þ fro the point q incident at p does not change with the dispaceent of p.. The visibiity of A, the sa area around q, does not change with the dispaceent of p. Though none of these assuptions is necessariy vaid in a scenes, they are reasonabe for sa dispaceents. The nuerica and anaytica ethods are equivaent, their resuts are indistinguishabe. The nuerica ethod is easier to ipeent since we do not need to evauate the basis function derivatives. The anaytica ethod is nuericay ore stabe near edges and corners and aso sighty faster to evauate Irradiance Gradient Coputation ote that both ethods we propose can sti be used if H is repaced by any other heispherica function. We aso do not rey on unifor heisphere saping. Any probabiity density pð; Þ can be used for saping. The ony change is 1 that the soid ange becoes ¼ pð ; Þ instead of ¼ 2, used for the unifor saping. As an exape, we copute the irradiance gradient re with a cosine-weighted heisphere saping. H ð; Þ is repaced by cos, the probabiity density of saping in direction ð; Þ is pð; Þ ¼ cos and, therefore, ¼ cos. The resuting forua for the anaytica ethod ¼ X ¼1 L i ð cos ¼ r n x þ 3q x cos cos r 2 cos : We ipeented this irradiance gradient coputation ethod and that of Ward and Hecbert [8] and we copared the on a sape scene (Fig. 5). The resuts were siiar for both ethods. Ward and Hecbert s ethod gives better resuts when soe surfaces are seen at very sharp grazing anges fro the saping point p. Otherwise, our ethod gives sighty soother resuts. Even though the quaity of Ward and Hecbert s ethod is subty superior to ours, our ethod provides any advantages. It wors with any saping distribution and with any function used to weight the radiance sapes. The contribution of radiance sapes to the gradient is independent of each other and, therefore, our ethod is ore easiy aenabe for paraeization or hardware ipeentation. We do not assue any stratification of

6 6 IEEE TRASACTIOS O VISUALIZATIO AD COMPUTER GRAPHICS, VOL. 11, O. 5, SEPTEMBER/OCTOBER 25 Fig. 6. Rotation R i aigns the coordinate frae at p i with that at p to ae interpoation possibe. Fig. 5. Coparison of irradiance gradient coputation. The scene is a diffuse Corne box; ony first bounce indirect iuination is coputed. The coor-coded iages show the difference between the gradientbased interpoation and the reference soution (1, sapes per heisphere at each pixe). RMS error of the iages is.125 for Ward s ethod and.131 for our ethod. The graph shows the reative error of the interpoation aong a singe scanine as copared to the reference soution. Ward s ethod gives better resuts when there are surfaces seen at very sharp grazing anges fro the saping point. Otherwise, our ethod gives a sighty ower error. the heisphere. This aows us to use our gradient coputation with different saping strategies, e.g., quasi Monte Caro saping. 4.4 Radiance Interpoation If a query to the radiance cache succeeds, the incoing radiance is interpoated as described in this section. We use a weighted interpoation schee siiar to the one proposed in [8] for interpoating the coefficient vectors i at any required surface point p. The difference is that we repace the use of the rotationa gradient by truy rotating the incident radiance function. This aigns the coordinate frae at the position p i of the cache record and the frae at p (see Fig. 6). The weight w i ðpþ of record i with respect to p is w i ðpþ ¼ p p i p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 n n i R ; i where n is the surface nora at p, n i is the surface nora at p i, and R i is the haronic ean distance to objects visibe fro p i. The coefficient vector of the interpoated radiance is coputed as a weighted average: P S i i þ d i þ w i ðpþ ðpþ ¼ P S w ; ð12þ iðpþ where the set S of radiance records used for interpoation at p is defined as S ¼fijw i ðpþ > 1=ag and a is a user-defined desired accuracy. The definition of the set S effectivey represents the criterion used to decide which radiance cache records can be used for interpoation. If S is nonepty, interpoation (or extrapoation) is possibe. d x and d y are the dispaceents of p p i aong the x and y axes of record i s oca coordinate frae. Dispaceent aong z is not taen into account for the gradient-enhanced interpoation since it is typicay very sa. R i is the heispherica haronics rotation atrix [9] that aigns the coordinate frae at p i with the frae at p. The interpoation schee is borrowed fro Ward et a. [1]. They derived it fro the spit sphere ode that estiates an upper bound on the agnitude of the change of irradiance. Athough this ode does not appy for radiance caching, the resuts are satisfying. We observe that a ower a has to be used for higher frequency BRDFs and interpoation errors are ore apparent when surfaces are viewed fro grazing anges. In future wor, we want to devise an interpoation schee suitabe for radiance caching based on these observations. 4.5 Outgoing Radiance Coputation The incoing radiance obtained by interpoation or heisphere saping is integrated against the BRDF to copute the outgoing radiance. With an orthonora basis, the integra evauation reduces to the dot product [51]: Lð o ; o Þ¼ Xn 1 X ¼ ¼ c ð o ; o Þ: ð13þ is an interpoated incoing radiance coefficient and c ð o ; o Þ is a BRDF coefficient at p for the outgoing direction ð o ; o Þ.

7 IVA EK ET AL.: RADIACE CACHIG FOR EFFICIET GLOBAL ILLUMIATIO COMPUTATIO KR 7 Fig. 7. Tiing breadown for the test scenes. RC fiing is the tie spent on coputing and adding radiance cache records. RC interpoation is the tie spent on ooing up the existing radiance cache records and interpoating the radiance. Tota is the tota rendering tie. The difference between the tota tie and the tie spent by radiance caching consists of priary ray casting, direct ighting, and irradiance caching (in Disney Ha and Faingo). A ties given in seconds. 5 RESULTS Fig. 7 gives the breadown of rendering ties for the the three scenes we used to test radiance caching (Corne Box, Wat Disney Ha, Faingo). The tiings were easured on a 2.2GHz Pentiu 4 with 1 GB RAM running Windows XP. The resuting renderings are shown in Figs. 8, 9, and 1 and in the accopanying video [5]. The axiu heispherica haronics order for radiance caching was set to n ¼ 1, which corresponds to approx. 3:6 B sized radiance cache records. We copared the soutions obtained by radiance caching with those obtained by Monte Caro iportance saping. Iportance saping uses the surface BRDF as the iportance function. The two rendering ethods exhibit artifacts with very different characteristics: high-frequency noise for iportance saping ( specs in iages) and ow-frequency error in radiance caching (uneven iuination gradients). It is therefore difficut to copare rendering ties needed to attain the sae visua quaity. Instead, we Fig. 8. Two views of a Corne Box with gossy bac wa rendered using radiance caching (top) and Monte Caro iportance saping (botto). have chosen to fix the rendering tie and copare the iage quaity deivered by the two ethods. 5.1 Corne Box Fig. 8 shows renderings of a Corne Box with a gossy bac wa (Phong BRDF, exponent 22), taen fro two viewpoints at resoution 1; 28 1; 28. Except for the bac wa, a objects are Labertian. Ony direct ighting and first bounce indirect gossy ighting for the bac wa were coputed. Fig. 9. Rendering of a sipe ode of Wat Disney Ha in Los Angees. The top and the idde iages, copared with radiance caching, show the buiding fro two different viewpoints. The botto iage was coputed with Monte Caro iportance saping.

8 8 IEEE TRASACTIOS O VISUALIZATIO AD COMPUTER GRAPHICS, VOL. 11, O. 5, SEPTEMBER/OCTOBER 25 Fig. 1. Fraes fro the Faingo aniation rendered using radiance caching (top row) and Monte Caro iportance saping (botto row). Iages in the top row were coputed using radiance caching with the caching accuracy set to a ¼ :15 and the nuber of rays cast to sape each heisphere set to ¼ 6;. The indirect gossy ter too 33:3 seconds to copute for the eft iage; tota rendering tie was 82:9 sec (direct iuination uses eight sapes per pixe to sape the area source). The nuber of radiance cache records was 6. The tie spent on the indirect gossy ter coputation in the right iage was ony 13:8 sec since the records fro the eft rendering were retained and ony 164 additiona records were required. The indirect gossy ter for each of the two botto iages was coputed in 35 seconds using Monte Caro iportance saping with 12 refected rays per pixe on a gossy surface. Those rendering ethods exhibit a high noise eve whose perception is even apified in the tepora doain, as shown in the video. The average tie spent on radiance caching for a 18 fraes ong aniation with the caera oving between the position in the eft and right iages was just 4:9 sec per frae. Most of this tie is spent on interpoation since ony a few records are needed for additiona fraes. The average frae tie with Monte Caro iportance saping is 35 seconds since this ethod does not reuse any inforation fro the previous fraes. Even though this tie is seven ties onger than for radiance caching, the quaity is uch ower. 5.2 Wat Disney Ha Fig. 9 shows renderings of a sipe ode of Wat Disney Ha in Los Angees. The curved was of the rea buiding are covered with brushed eta ties, whose BRDF we approxiate by a three-obe Lafortune ode [52]. The iuination is due to the sun (odeed as a directiona ight), the sy (odeed as a constant bue ight), and the surrounding urban environent (odeed as a constant brownish ight). Siiary to the rea buiding (see [5]), was refect the sy or the surroundings depending on their noras and the viewpoint (copare the top and the idde iage). A fu goba iuination soution with one ray per pixe was coputed at resoution 1; and then scaed down. Indirect iuination and the iuination coing fro the sy and fro the surroundings was coputed in the sae way: with irradiance caching on diffuse surfaces, with radiance caching on gossy surfaces in the top and the idde iage, and with Monte Caro saping on gossy surfaces in the botto iage. To capture the high indirect iuination variations on the eta was, the caching accuracy was set to a ¼ :1, eading to as any as 38, radiance cache records. We used 8 rays for heisphere saping. The rendering tie was s and s for the top and the idde iages, respectivey. Monte Caro iportance saping in the botto iage used 15 refected rays on each visibe gossy pixe (first bounce) and one ray for other bounces. The rendering tie was siiar to that for radiance caching, but the quaity of radiance caching resuts is higher. The BRDF we used for the etaic was [53] was reativey sharp; the Lafortune obes had exponents of 16, 88, and 186. It is ipossibe to represent such a BRDF accuratey using heispherica haronics of order 1 the average representation error was over 2 percent. onetheess, radiance caching renderings show no serious distortion of the ateria appearance copared to Monte Caro saping. 5.3 Gossy Faingo Fig. 1 shows three fraes fro the Faingo aniation. The bird was assigned the Phong BRDF (exponent 7) and a other surfaces are purey diffuse. The rendering resoution was 1; 28 1; 28 pixes. Fu goba iuination up to four bounces was coputed. Irradiance caching was used to copute indirect ighting on diffuse surfaces. First bounce indirect ighting on gossy surfaces for the iages in the top row was coputed using radiance caching; path tracing was used for deeper recursion eves. The caching accuracy was set to a ¼ :15 and the nuber of rays for heisphere saping was ¼ 1;. Fig. 7 gives the rendering ties for the three iages when rendered

9 K RIV AEK ET AL.: RADIACE CACHIG FOR EFFICIET GLOBAL ILLUMIATIO COMPUTATIO 9 Fig. 12. Inforation oss when radiances are interpoated on a curved surface. Fig. 11. Tie spent on radiance caching for the Faingo aniation (see the accopanying video). Cached records are shared between the fraes. independenty, without retaining radiance cache records fro previous renderings. Fig. 11 shows the tie spent on radiance caching in a 28 fraes ong aniation with caera oving between the shown iages (see the accopanying video). The iages in the botto row use Monte Caro iportance saping instead of radiance caching. In order to eep the rendering tie the sae as for radiance caching, the nuber of refected rays per pixe on a gossy surface was set to 12, 4, 6, respectivey (fro eft to right). This scene is particuary chaenging for radiance caching since the gossy surface is curved. On such a surface, radiance cache records cannot be used for interpoation at as any pixes as on a fat surface. Moreover, costy aignent is required before each interpoation. In the eft iage, the faingo occupies ony a sa part of the screen and, therefore, one does not tae advantage of radiance caching s independence on iage resoution. The quaity advantage of radiance caching over Monte Caro iportance saping can be seen ony by a very cose inspection (see the iages in the accopanying ateria [5]). However, in the other two iages, the noise introduced by Monte Caro saping is ore obvious. otice aso that, for the aniation rendering, the average tie for radiance caching is 15 seconds per frae. Rendering the sae aniation using Monte Caro iportance saping with ony two refected rays on a gossy pixe eads to 27 seconds per frae spent on indirect gossy ighting coputation. otice, in the accopanying video, that the quaity obtained by radiance caching is consideraby better. 6 DISCUSSIO 6.1 Rotation Loss There is a oss of inforation when radiances are interpoated on a curved surface (Fig. 12). A part of the radiance incident at p i shoud disappear under the surface (ared a in Fig. 12) and shoud not contribute to the interpoated radiance at p.a part of the radiance actuay incident at p is not represented by the radiance record at p i (ared b in Fig. 12) and is issing in the interpoated radiance. This probe is not due to using a heispherica basis for representing the incoing radiance, but due to the fact that the incident radiance at a surface point is a heispherica function. Using spherica, instead of heispherica, haronics woud not sove this probe. In practice, the error introduced by this probe is very sa because the difference between the nora at p and the nora at any record used for interpoation at p is sa. ote aso that Ward et a. s irradiance caching suffers fro the sae probe. 6.2 Goba versus Loca Coordinates Incoing radiance at a point can be represented either in the oca frae at that point or in the goba frae. This infuences how the interpoation at p is perfored:. Incoing radiance in the goba frae for (each avaiabe record i) do Update the interpoation sus in (12). end for Aign the resut with the oca frae at p. Copute the dot product.. Incoing radiance in the oca frae for (each avaiabe record i) do Aign the frae at p i with the frae at p. Update the interpoation sus in (12). end for Copute the dot product. The fina dot product (13) is aways carried out in the oca frae at p. On curved surfaces, fewer rotations are needed if the incoing radiance is represented in the goba frae. On the other hand, if the incoing radiance is represented in the oca frae, no aignent (even with the BRDF) is needed on fat surfaces. The owest nuber of rotations is obtained if the incoing radiance is represented in the oca frae on fat surfaces and in the goba frae on curved surfaces. ote that fu spherica function representation (e.g., using spherica haronics) is needed to represent the incoing radiance in the goba frae. 6.3 Suitabiity of Heispherica Haronics We use (hei)spherica haronics since they are sipe, coputationay efficient (anipuations of vectors of foats), avoid aiasing, and rotation is avaiabe for the. The essentia disadvantage is the ac of directiona ocaization. When we create a new radiance cache record, the fu heisphere ust be saped, whatever the incoing ray direction is. The ore directiona the BRDF is, the ore this approach becoes wastefu. With a basis that ocaizes in directions, ony the required part of the heisphere woud need to be saped. For this purpose, one can use piecewise constant representation [6], [26], [27],

10 1 IEEE TRASACTIOS O VISUALIZATIO AD COMPUTER GRAPHICS, VOL. 11, O. 5, SEPTEMBER/OCTOBER 25 but it woud presuaby introduce aiasing. The use of spherica waveets [3] is probaby a good choice, and is eft for further investigation. 6.4 Genera Liitations The BRDF frequencies used in the exape scenes are near the iit of what our technique can currenty hande. The restriction to ow-frequency BRDFs is the ain iitation of our technique because we use heispherica haronics. We beieve that higher frequency BRDFs can be handed within the presented fraewor by using a ocaized basis such as waveets. Additionay, the radiance caching, as a gathering technique, cannot sove certain types of ight transfer phenoena, such as caustics. Those have to be soved by a ore adapted technique. where x!! is the doube factoria (product of a odd integers ess than or equa to x). The partia derivatives for heispherica haronics ð; Þ 8 p 2 ffiffiffi 2K e cosðþ sinðþ dp ð2 cos 1Þ if > >< p 2 ffiffiffi 2K e dp sinð Þ sinðþ ð2 cos 1Þ if < >: 2 ek sinðþ dp ð2 cos 1Þ if ¼ if ¼ ð; Þ H ð; Þ otherwise; p where ek ¼ ffiffiffi 2 K. 7 COCLUSIO AD FUTURE WORK We have presented radiance caching, a ethod for acceerating the coputation of the indirect iuination on surfaces with ow-frequency gossy BRDFs. Radiance caching is based on sparse saping, caching, and interpoating incoing radiance on those surfaces. Radiance is represented by heispherica or spherica haronics in our approach. The interpoation quaity is enhanced by the use of transationa gradients that can be coputed with two nove ethods we have presented in this paper. We have aso proposed an autoatic criterion to decide for which BRDFs radiance caching is suitabe. We have shown in severa exapes that radiance caching is ore efficient than pure Monte Caro saping at every surface point and deivers iages of superior quaity. In future wor, we woud ie to use adaptive heisphere saping to copute the incoing radiance coefficients [29], [48], [54], use a different representation for the incoing radiance that ocaizes better in directions, and devise interpoation criteria better suited for gossy surfaces. In the ong ter, we wish to investigate the reationship between the frequency content of a BRDF and the suitabiity of interpoating radiance on surfaces with that BRDF. APPEDIX DERIVATIVES OF SPHERICAL AD HEMISPHERICAL HARMOICS Partia derivatives for spherica haronics ð; Þ 8 p ffiffi 2 K cosðþ sinðþ dp ðcos Þ if > >< p ffiffi 2 K dp sinð Þ sinðþ ðcos Þ if < >: K sinðþ dp ðcos Þ if ¼ if ¼ ð; Þ Y ð; Þ otherwise; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where K ð2þ1þð jjþ! ¼ 4ðþjjÞ!.The derivative of the associated Legendre poynoias can be found fro the recurrence forua: ( dp 1 ðxþ ¼ x 2 1 xp ðxþ ðþþp 1 ðxþ if < ð 1Þ xð2 1Þ!!ð1 x 2 Þ 2 1 if ¼ ; ACKOWLEDGMETS The authors than Vastii Havran for etting the use his ray tracer and Marina Casiraghi and Eric Risser for the Wat Disney Ha ode. This wor was partiay supported by ATI Research, the I-4 Matching fund, and the US Office of ava Research. REFERECES [1] G.J. Ward, F.M. Rubinstein, and R.D. Cear, A Ray Tracing Soution for Diffuse Interrefection, Proc. SIGGRAPH 88, pp , [2] G.J. Ward, The Radiance Lighting Siuation and Rendering Syste, Proc. SIGGRAPH 94, pp , [3] X. Granier and G. 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Jarosav Kriváne received the Master s degree in coputer science fro the Czech Technica University. He is a PhD student at IRISA/IRIA Rennes and the Czech Technica University in Prague and a visiting research associate at the University of Centra Forida. His research focuses on rea-tie and offine ighting siuation and using visua perception for efficient ighting coputation. Pasca Gautron received the Master s degree in coputer science fro the University of Poitiers. He is a PhD student at IRISA/IRIA Rennes, France. He wors in coaboration with the University of Centra Forida and the University of Poitiers. His ain research interest is high quaity rea-tie rendering using graphics hardware. Suanta Pattanai is an associate professor in the Coputer Science Departent at the University of Centra Forida. Fro 1995 to 21, he was a research associate in the Progra of Coputer Graphics at Corne University. Before that, he was a postdoctora researcher at IRISA/ IRIA ( ), France. His ain fieds of research are reaistic iage synthesis and digita iaging. His current research focuses on reatie reaistic rendering and the appication of visua perception for efficient ighting coputation and accurate dispay. He is a eber of ACM SIGGRAPH, Eurographics, and the IEEE. Kadi Bouatouch received the PhD degree in 1977 and a higher doctorate in coputer science in the fied of coputer graphics in He is an eectronics and autoatic systes engineer (ESEM 1974). His research interests are goba iuination, ighting siuation, and reote rendering for copex environents, rea-tie high fideity rendering, parae radiosity, virtua and augented reaity, and coputer vision. He is currenty a professor at the University of Rennes 1, France, and researcher at IRISA. He is a eber of Eurographics, ACM, and the IEEE. He is and was a eber of the progra coittees of severa conferences and worshops and a referee for severa coputer graphics journas