Fast, Arbitrary BRDF Shading for Low-Frequency Lighting Using Spherical Harmonics

Transcription

1 Thrteenth Eurographcs Workshop on Renderng (2002) P. Debevec and S. Gbson (Edtors) Fast, Arbtrary BRDF Shadng for Low-Frequency Lghtng Usng Sphercal Harmoncs Jan Kautz 1, Peter-Pke Sloan 2 and John Snyder 2 1 Max-Planck-Insttut für Informatk, Saarbrücken, Germany 2 Mcrosoft Research, Redmond, WA, USA Abstract Real-tme shadng usng general (e.g., ansotropc) BRDFs has so far been lmted to a few pont or drectonal lght sources. We extend such shadng to smooth, area lghtng usng a low-order sphercal harmonc bass for the lghtng envronment. We represent the 4D product functon of BRDF tmes the cosne factor (dot product of the ncdent lghtng and surface normal vectors) as a 2D table of sphercal harmonc coeffcents. Each table entry represents, for a sngle vew drecton, the ntegral of ths product functon tmes lghtng on the hemsphere expressed n sphercal harmoncs. Ths reduces the shadng ntegral to a smple dot product of 25 component vectors, easly evaluatable on PC graphcs hardware. Non-trval BRDF models requre rotatng the lghtng coeffcents to a local frame at each pont on an object, currently formng the computatonal bottleneck. Real-tme results can be acheved by fxng the vew to allow dynamc lghtng or vce versa. We also generalze a prevous method for precomputed radance transfer to handle general BRDF shadng. Ths provdes shadows and nterreflectons that respond n real-tme to lghtng changes on a preprocessed object of arbtrary materal (BRDF) type. Categores and Subject Descrptors: I.3.7 [Computer Graphcs]: Color, shadng, shadowng, and texture. 1. Introducton Fully general BRDFs are necessary to descrbe many nterestng knds of materals, ncludng brushed metals and fabrcs. Interactve shadng wth such BRDFs has taken two approaches. The frst smply evaluates the BRDF at a number of ponts 9,11,14. Whle effcent for a few pont lghts, ths approach s slow for large lghts snce t requres lght ntegraton over many drectons. The second approach precomputes and tabulates the lghtng ntegral by pre-convolvng the ncdent lghtng. Although ths table reduces to two dmensons for mrror-lke surfaces 8, hghdmensonal tables are requred for general BRDFs 13. Phong-lke BRDFs requre 3D tables where the ncdent lghtng s convolved wth kernels of varable radus; ansotropc BRDFs requre as much as 5D tables. Pre-convolved tables are costly to compute and store and thus dffcult to use wth dynamc lghtng. Pre-convoluton also gnores shadowng and nterreflectons 19. Our approach dffers by usng a sphercal harmonc (SH) bass for lghtng and BRDF functons. Essentally, the BRDF s tabulated not n terms of response to drectonal lghts, but to large lghts coverng the sphere va the orthonormal SH bass. The lghtng envronment also s (a) pont lght (b) glossy (c) ansotropc (d) shadowed Fgure 1: Prevous real-tme shadng methods are lmted to pont lghts (a) or allow smoother lghtng envronments but restrcted to Phong-lke glossy BRDF models (b). We generalze to area lghtng of arbtrary BRDF models, lke ansotropc brushed metal (c) and nclude shadows (d) at lttle extra cost. Our method renders ths model at 2Hz; 10/50Hz after fxng ether the vew/lght. projected nto the SH bass usng a fast, on-the-fly calculaton 19. For low-frequency lghtng, very few coeffcents are requred to accurately compute the lghtng ntegral (25 components). The calculaton s smple and fast even for arbtrary BRDFs, requrng no hgh-dmensonal tables of pre-convolved lghtng. Our approach allows spatally varyng materals by tabulatng the BRDF wth varyng parameters. Furthermore, our method can also be easly combned wth precomputed transfer 19 to obtan real-tme shadows and nterreflectons over preprocessed objects. Our method s benefts are hghlghted n Fgure Prevous Work Preconvolved Envronment Maps. Blnn and Newell 4 proposed envronment maps to approxmate specular reflectons. Greene 8 observed that a pre-convolved envronment map could be used to smulate dffuse reflectons.

2 Kautz et al / Fast, Arbtrary BRDF Shadng Snce then, several approaches have been proposed to smulate glossy reflectons based on pre-fltered envronment maps 6,8,9,12,13. These algorthms assume a smple, fxed BRDF model 8,9,13,16 (e.g, Phong model, Banks model 2 ) or generalze but only to sotropc, radally symmetrc BRDFs 12. More precsely, not the BRDF tself but the BRDF product functon,.e. the BRDF multpled by the cosne factor (Secton 3), must be an sotropc, radally symmetrc functon of the lght drecton; n other words, a Phong-lke model. Such pre-flterng methods thus neglect the cosne factor unless the parameterzaton s vewdependent, as proposed by Cabral et al. 6. But ther method s restrcted to sotropc BRDFs about a central reflecton drecton. None of these methods handle spatally varyng BRDFs well snce they requre even hgher-dmensonal tables to account for the varyng parameters. Generally, these methods cannot change the lghtng envronment on-the-fly snce an expensve convoluton must be appled to the envronment map. Graphcs hardware has been used to accelerate ths convoluton, agan for the restrcted class of radally symmetrc BRDF product functons 13. For the specal case of a dffuse BRDF, Ramamoorth and Hanrahan 16 propose a fast method based on sphercal harmoncs for computng the convoluton and performng real-tme renderng. Sphercal Harmoncs for Shadng. Cabral et al. 5 use sphercal harmoncs to derve sotropc BRDFs and make the observaton that ther use reduces the lghtng ntegral to a dot product. We apply ths nsght to real-tme renderng. Our BRDF product coeffcents are derved from analytc models 1,15 or measured materals 20 rather than heght feld geometry. Ther method assumes a constant vew drecton per object; we generalze to arbtrary vews by usng hardware-supported 2D textures that parameterze BRDF product coeffcents n terms of the vew drecton. Ther method s also lmted to sotropc BRDFs whle we generalze to arbtrary BRDFs. Fnally, we combne ansotropc BRDF shadng wth precomputed transfer to obtan realtme self-shadows and self-nterreflectons. Sllon et al. 18 use sphercal harmoncs to represent ext radance at many ponts n an off-lne, progressve radosty smulaton. In contrast, our goal s real-tme renderng. Ther method represents the vew-dependence of ext radance n the SH-bass, mplctly assumng ths dependence s smooth and so requres few coeffcents. As does Cabral, we make the converse assumpton that ncdent lghtng s smooth and represent ncdent lghtng and the BRDF product functon s lght-dependence n the SH-bass n order to accelerate the lghtng ntegral whch forms the bottleneck n real-tme renderng. We tabulate the BRDF product functon s vew-dependence usng hgh-resoluton 2D textures of SH coeffcents, provdng hgh-frequency vewdependence whch can be sgnfcant even n low-frequency lghtng. Westn et al. 21 use sphercal harmoncs for off-lne BRDF nference from geometrc models. Both vew and lght dependence of the BRDF s represented usng the SH bass va a large matrx. Encodng vew dependence usng the SH bass requres on-the-fly evaluaton of hgh-order bass functons whch we avod by usng a table. Precomputed Radance Transfer. Sloan et al. 19 use the SH bass to represent how an object casts shadows and nterreflectons onto tself, called precomputed transfer. Only radally symmetrc BRDF models are consdered, precludng ansotropc materals and Fresnel effects. Such restrcted BRDFs permt accelerated shadng by explotng the smplcty of convoluton n the SH bass 3,16,19. We generalze precomputed transfer to arbtrary BRDFs, for whch the convoluton property of the SH bass cannot be used. Nevertheless, lttle extra cost s ncurred snce t s possble to fold the necessary rotaton of lghtng coeffcents per pont on the preprocessed object nto the glossy transfer matrx already needed. We just need to do an addtonal BRDF coeffcent vector lookup ndexed by the vew drecton at each pont. Alternatvely, the methods of ths paper can be used wthout precomputed transfer, so that no storage or preprocessng s needed for transfer vectors or matrces over an object. In ths case, the shadng result wll lack shadows as n the case of pre-flterng methods, but works wth arbtrary BRDFs and dynamc lghtng. Ansotropc lghtng models. Analytc models for ansotropc BRDFs have been proposed by Kajya 10, Pouln and Fourner 15, Banks 2, and recently by Ashkmn and Shrley 1. Our approach allows any BRDF model as well as measured data 20 n the presence of arbtrary low-frequency lghtng. 3. Acceleratng Shadng wth the SH bass To shade a pont p, we need to compute the lghtng ntegral, gven by Ú Ú Q v L s f s v s N ds L s f s v ds * p() = () (,)max(0, p) = () (,) where Q p (v) s the outgong radance for vewng drecton v, L s the ncdent llumnaton functon, f s the BRDF, f* s the BRDF product functon, and s s the ncdent llumnaton drecton representng the varable of ntegraton. Both s and v are on S 2. We frst parameterze f* by local vew drecton to get sphercal functons, whch we represent n the SH bass va n * fv () s = Â c() v y() s = 1 where y (s) are the SH bass functons. (Formulas for y (s) appear n 7,18.) We then tabulate the c n terms of the vew drecton v. For each v, the result s a vector of 25 coeffcents representng a sphercal functon (Fgure 2). We use a parabolc parameterzaton 9 of the hemsphere of drectons v to get a 2D texture mage where each texel has 25 components, obtanng good results wth textures. By representng the lghtng functon n the SH bass n = 1 Ls () = Â L y() s, the lghtng ntegral then reduces to the smple dot product n Q () v = Â L c() v. (1) p = 1 Inconvenently, f s represented usng a local coordnate frame whch vares over the object whle the ncdent lghtng typcally uses a global coordnate system shared across the whole object. Thus to perform the ntegraton, we frst need to rotate the lghtng nto ths local frame at

3 Kautz et al / Fast, Arbtrary BRDF Shadng constant band lnear band quadratc band cubc band quartc band Fgure 2: BRDF product textures for an example Ashkhmn-Shrley model 1. The top rows of smaller mages show component textures for each frequency band, where band has 2+1 components (red=postve, blue=negatve). The bottom row shows squared energy summed over all the band s components. The parabolc parameterzaton of a hemsphere s used to map vew drectons to samples n a square. Note the smoothness of the textures and sgnal attenuaton at hgher bands. each p. Fortunately, ths can be performed wth a smple computaton (see Appendx). Calculatng BRDF Coeffcents. Our approach allows any BRDF model (sotropc or ansotropc) or tabulated data (whch we nterpolate usng radal bass functons to create a smooth BRDF). Because the SH bass s orthonormal, the coeffcents g of the least-squares best approxmaton of a functon g(s) by a fnte seres s gven by the sphercal ntegral of g tmes the bass functon y (s). Thus, as a preprocess we compute wth numercal ntegraton c() v = Ú y() s f(,)max(0, s v sz) ds where s z denotes the z coordnate of the drecton s, assumng the local coordnate frame maps the normal to the z axs. Computng these coeffcents va the above ntegral s called SH-projectng the functon, n ths case f*. To vary the BRDF s parameters across the object, we just need a hgher-dmensonal c (v,q) table ndexed by whchever parameters q are to be spatally vared. Calculatng Lghtng Coeffcents. SH-projecton s also appled to the ncdent lghtng L(s) to obtan ts coeffcents L. Ths can be done ether as a precomputaton f the lghtng s fxed, or on-the-fly for dynamc lghtng. Hardware renderngs can be used to quckly sample ncdent lghtng at many ponts 19. We have also mplemented arbtrary rotatons of fxed envronments va on-the-fly SH rotaton (Appendx), and crcular lght sources of parameterzed sold angle and drecton converted to SH coeffcents va a smple formula. 4. Renderng Algorthm The renderng algorthm s farly smple; we only need to evaluate equaton (1). In the followng, we use a prmed symbol (e.g., v p ) for quanttes expressed n a global coordnate system for a whole object and an unprmed (e.g., v p ) for the correspondng quantty transformed nto the local coordnate frame. At each pont p to be shaded, havng vew vector v p and local coordnate frame R p mappng p s normal N p to the postve z axs and two chosen tangent vectors to the x and y axes, we: 1. rotate lghtng coeffcents L p, to frame R p to obtan rotated coeffcents L p, (see Appendx) 2. rotate vew vector v p to local frame to obtan v p 3. lookup BRDF coeffcents at v p yeldng c (v p ) 4. compute dot product of L p, wth c (v p ) In the smplest case, the global (untransformed) ncdent lghtng coeffcents L p, can be constant for all ponts p over an object. Ths s analogous to usng an envronment map representng lghtng at nfnty. For fnte lghtng (that vares over ponts p on the object), we can use an ncdent lght feld whch nterpolates lghtng sampled at varous ponts around the object 19. Rotaton of lghtng coeffcents (step 1) s too complcated for current graphcs hardware, so we currently perform ths operaton on the host and upload the rotated lghtng coeffcents to the GPU. The remanng operatons (steps 2-4) are performed on the GPU. Vew vector rotaton s done n a vertex shader, the BRDF lookup s just a texture access, and the fnal dot product s computed n a pxel shader. Current hardware (we use an ATI Radeon 8500) computes the dot product for only a sngle color channel n one pass due to an nstructon count lmt n the pxel shader, so three renderng passes are needed. By fxng ether lght or vew, ths renderng algorthm can be accelerated by pre-rotatng the SH coeffcents. In the case of fxed lghtng, we frst rotate the lghtng coeffcents nto the local coordnate system at every vertex p and store the resultng L p,. We can then reuse these coeffcents for a new vew;.e., we only need to perform steps 2-4 of the renderng algorthm. The fxed vew case s smlar. For a gven vew, we rotate the vew vector v p nto the local coordnate frame at every vertex p and do a lookup nto the BRDF texture wth the local vew vector v p to retreve BRDF coeffcents c (v p ). These coeffcents are then rotated nto the global coordnate system and stored per vertex. Ths s done so that we do not need to rotate the lghtng coeffcents, whch are then the same for every vertex n the case of ncdent lghtng at nfnty. After a lghtng change, we only need to compute the dot-product between the stored, pre-rotated

4 Kautz et al / Fast, Arbtrary BRDF Shadng BRDF coeffcents and the lghtng coeffcents, performed by a vertex shader. An added advantage s that all 3 color channels can be done n one pass. Spatally varyng BRDFs are handled wth the same renderng algorthm, where the BRDF lookup now depends on addtonal parameters that are vared spatally across the object as well as the vew drecton. 5. Combnng wth Precomputed Transfer Sloan et al. 19 use a lnear transformaton to represent how an object casts shadows and nterreflectons onto tself. Ths lnear transformaton, called a transfer matrx, s stored on many ponts p over the object and appled to the SH coeffcents of the ncdent lghtng. The result of ths transformaton s a new set of SH coeffcents that represent the ncdent lght at the pont p but account for selfshadowng and nterreflectons due to the object, called transferred radance. Transfer matrces are derved usng an offlne global llumnaton smulaton but can be appled at run-tme to arbtrary low-frequency lghtng. We can easly combne precomputed radance transfer wth our technque to obtan self-shadowng and selfscatterng effects on ansotropc materals. Instead of usng the ncdent lghtng vector L p, drectly, we apply the transfer matrx to t and substtute the resultng transferred radance at each p nto the renderng algorthm. Steps 1-4 are then performed as before; the BRDF s also tabulated n exactly the same way. As an optmzaton, we can fold the requred rotaton of transferred radance from the global to local frame nto the transfer matrx tself as a preprocess, thus elmnatng step 1 but stll requrng a lnear transformaton of the ncdent lghtng at each pont. In general, the transfer transformaton s less sparse than the rotaton transformaton. To nclude nterreflectons, the desred BRDF must be fxed and accounted for n the global llumnaton smulaton 19 snce nterreflectons depend on t. If only shadows are consdered, then the BRDF can be changed on the fly wthout addtonal smulaton. 6. Results and Dscusson We acheve nteractve frame rates for all models; for the fxed vew/fxed lght modes we acheve real-tme frame rates (Table 1). Performance does not depend on the BRDF model used and s ncreased only slghtly when precomputed radance transfer s ncluded. Renderng qualty can be judged from Fgures 1-8, all of whch were read back from a PC. Results show monochromatc BRDFs (colors are due solely to colored lghtng) but colored BRDFs ncur no performance penalty, snce we currently compute all color channels separately anyway. All mages except Fgure 6 were computed usng 25 SH coeffcents (ffth order projecton) for lghtng and BRDF table outputs. Fgure 6 vares the SH projecton order, showng that a lmted number of coeffcents support only a lmted frequency range n reflectons on hghly specular surfaces. If both lghtng and materal BRDF contan hgher frequences, they wll be cut off, resultng n blurred reflectons. However, even for sharp lghtng, accurate results are obtaned f the BRDF s not too specular, snce more dffuse BRDFs effectvely low-pass flter ncdent lghtng 17. Rngng s also a problem 5,19 whch can be mtgated by #fps Model #vertces no transfer #fps transfer #fps #fps fx lght fx vew Teapot Head Buddha Brd Tyra Table 1: Tmng results. Tmngs were done on a 2.2Ghz Intel P4 PC wth ATI Radeon 8500 card (720x760 mage). wndowng (.e., smoothng the lghtng) and s masked on bumpy, complex models (compare rows of Fgure 8). Fgure 3 shows the ansotropc AS model appled to a statue head, placed n varous lghtng envronments. Renderng performance s not hampered by dynamc lghtng and does not depend on the number or sze of lght sources; n fact, accuracy ncreases as lghts get bgger. Fgure 1d and 5b llustrate our approach combned wth precomputed transfer, whch provdes self-shadowng. These examples use the Ashkhmn-Shrley (AS) BRDF model 1. Fgure 7 also shows precomputed transfer, focusng on the effect of our generalzaton to ansotropc BRDFs. All AS model mages use a smple ansotropc or brushng drecton derved from projectng a constant vector to the tangent space at each pont. Fgure 4 shows a teapot model wth a spatally varyng BRDF. We vared a sngle parameter of the AS model accordng to a nose functon, requrng a 3D texture. The fgure s left column shows spatal varaton of the glossy power (sotropc model); the rght vares the ansotropy. We observe neglgble performance decrease when usng BRDF spatal varaton. In a dynamc lghtng envronment, such spatal varaton would be dffcult for pre-flterng methods snce a hgh-dmensonal table would have to be bult each tme the lght changed. Fgure 8 compares varous BRDF models ncludng ansotropc AS, ansotropc Pouln-Fourner (PF) 15, and measured alumnum fol and vnyl 20. Note the dfference between columns (a) and (b), especally for the teapot (mddle row), due to the ncreased specularty at grazng angles produced by the AS model. Precomputaton tmes for creatng the BRDF table depend on the model used. Analytc models were projected n a few mnutes (usng 1000 samples for each ntegraton); measured models took up to 90 mnutes. 7. Conclusons We have presented an nteractve technque to render objects wth general, spatally-varyng BRDFs llumnated by dynamc, low-frequency lghtng. Combnng t wth precomputed transfer methods ncorporates self-shadowng and self-nterreflecton effects. Our technque yelds fast, very realstc renderngs of any materal lt by lghtng envronments contanng large lght sources, somethng mpossble to acheve wth prevous technques. Our mplementaton s not yet real-tme for the general case where the vew and the lghtng change smultaneously, snce per-vertex rotaton of lghtng cannot be mplemented on current GPUs. We expect next generaton hardware to be flexble enough to perform ths operaton, allowng fully general, real-tme performance.

5 Kautz et al / Fast, Arbtrary BRDF Shadng Fgure 3: Brushed metal head n varous lghtng envronments. A straghtforward extenson of ths approach would be to decompose lghtng nto hgh and low-frequency terms, usng our approach for the low-frequency part and exstng pont lght methods 9,11 for the hgh frequences. Acknowledgments We would lke to thank Greg Ward for BRDF data, Paul Debevec for lghtng envronments, and Hans-Peter Sedel for support. References 1. ASHIKHMIN, M, AND SHIRLEY, P, An Ansotropc Phong BRDF Model, Journal of Graphcs Tools (2000), 5(2): BANKS, D, Illumnaton n Dverse Codmensons, SIGGRAPH 94, BASRI, R, AND JACOBS, D, Lambertan Reflectance and Lnear Subspaces, In Internatonal Conference on Computer Vson, BLINN, J, AND NEWELL, M, Texture and Reflecton n Computer Generated Images. Communcatons of the ACM 19 (1976), CABRAL, B, MAX, N, AND SPRINGMEYER, R, Bdrectonal Reflecton Functons from Surface Bump Maps, SIGGRAPH 87, CABRAL, B, OLANO, M, AND NEMEC, P, Reflecton Space Image Based Renderng, SIGGRAPH 99, EDMONDS, A, Angular Momentum n Quantum Mechancs, Prnceton Unversty, Prnceton, GREENE, N, Envronment Mappng and Other Applcatons of World Projectons, IEEE CG&A, 6(11):21-29, HEIDRICH, W, SEIDEL H, Realstc, Hardware-Accelerated Shadng and Lghtng, SIGGRAPH 99, KAJIYA, J, Ansotropc Reflecton Models, SIGGRAPH 85, KAUTZ, J, AND MCCOOL, M, Interactve Renderng wth Arbtrary BRDFs usng Separable Approxmatons, Eurographcs Workshop on Renderng 1999, KAUTZ, J, AND MCCOOL, M, Approxmaton of Glossy Reflecton wth Prefltered Envronment Maps, Proceedngs Graphcs Interface (May 2000), KAUTZ, J, VAZQUEZ, P, HEIDRICH, W, AND SEIDEL, H, A Unfed Approach to Pre-fltered Envronment Maps, Eurographcs Workshop on Renderng 2000, MCCOOL, M, ANG, J, AND AHMAD, A, Homomorphc Factorzaton of BRDFs for Hgh-Performance Renderng, SIGGRAPH 01, POULIN, P, AND A. FOURNIER, A Model for Ansotropc Reflecton, SIGGRAPH 90, RAMAMOORTHI, R, AND HANRAHAN, P, An Effcent Representaton for Irradance Envronment Maps, SIGGRAPH 01, RAMAMOORTHI, R, AND HANRAHAN, P, A Sgnal-Processng Framework for Inverse Renderng, SIGGRAPH 01, SILLION, F, ARVO, J, WESTIN, S, AND GREENBERG, D, A Global Illumnaton Soluton for General Reflectance Dstrbutons, SIGGRAPH 91, SLOAN, P, KAUTZ, J, AND SNYDER, J, Precomputed Radance Transfer for Real-Tme Renderng n Dynamc, Low-Frequency Lghtng Envronments, SIGGRAPH 02, to appear. 20. WARD, G, Measurng and Modelng Ansotropc Reflecton, SIGGRAPH 92, WESTIN, S, ARVO, J, TORRANCE, K, Predctng Reflectance Functons from Complex Surfaces, SIGGRAPH 92, Appendx: SH Rotaton A sphercal functon represented by a set of SH coeffcents can be exactly rotated va a lnear transformaton of those coeffcents 7. Ths lnear transformaton has some addtonal propertes: t s tself a hgher-dmensonal rotaton matrx, and ts result for each SH band only depends on coeffcents from that band (thus mplyng the transformaton s sparse). SH bands have successvely 1, 3, 5,, 2m- 1 coeffcents where the total number of coeffcents for an order m projecton s n=m 2. We have mplemented two methods for computng rotatons. The frst, useful for very low-order projectons (up to order 2 or 3), s smply to use symbolc ntegraton to fnd the lnear transformaton explctly as a functon of matrx components of the desred rotaton R, va Mj = Ú yj ( Rs) y ( s) ds where the rotated coeffcents c are computed from the unrotated c va the lnear transformaton c = Â M j j cj. The second method becomes more effcent as the projecton order ncreases. It frst converts the rotaton matrx R nto ts zyz Euler angle decomposton. Rotaton around z can be computed usng a smple formula that performs a 1D rotaton between pars of coeffcents 5,18. To perform the rotaton around y, we further decompose t to a rotaton around x by 90, a general rotaton around z, and then a rotaton around x by -90. The two fxed rotatons around x are computed usng tabulated coeffcents derved from numercal ntegraton, and can be represented by very sparse matrces.

6 Kautz et al / Fast, Arbtrary BRDF Shadng (a) varyng exponent (b) varyng ansotropy Fgure 4: Spatally-Varyng BRDFs. (a) unshadowed (b) shadowed Fgure 5: Combnng wth Precomputed Transfer. orgnal lghtng envronment projected lghtng envronment shaded sphere AS (12,4) shaded sphere AS (24,8) shaded sphere AS (48,16) n=9 n=25* n=49 n=225,unwndowed n=225, wndowed Fgure 6: Comparson of SH order and glossness. Columns represent SH projecton order for both the lghtng envronment and BRDF product functon: from low (9 coeffcents) at left, to hgh (225 coeffcents) at rght. Other fgures n ths paper were computed usng 25 coeffcents, shown here as the starred column. The last two columns use the same number of SH components, but wth and wthout wndowng whch reduces rngng artfacts but blurs the lghtng. The top row shows the lghtng envronment; the next three rows then show a shaded sphere lt by that envronment usng the Ashkmn-Shrley (AS) BRDF model. Varous ansotropc glossness exponents are used: from more dffuse (12,4) at top, to more specular (48,16) at bottom. Low-order projecton suffces for less specular BRDFs, even though the lghtng envronment loses much of ts detal.

7 Kautz et al / Fast, Arbtrary BRDF Shadng (b) ansotropc (c) ans., dfferent brushed drecton (a) sotropc19 Fgure 7: Generalzng Precomputed Transfer to Ansotropc BRDF Models. (a) AS (analytc) (b) PF (analytc) (c) vnyl (measured) (d) alum. fol (measured) Fgure 8: Comparson of BRDF models appled to varous geometrc models and lghtng envronments.