Diffuse and specular interreflections with classical, deterministic ray tracing

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1 Dffuse and specular nterreflectons wth classcal, determnstc ray tracng Gergely Vass Dept. of Control Engneerng and Informaton Technology Techncal Unversty of Budapest Budapest, Hungary Abstract Today almost all the free as well as the commercal 3d graphcs applcatons - whch clam to be photorealstc - are capable to render mages usng the ray tracng algorthm. Ths method handles only one knd of lght transport between surfaces, namely the total, mrror-lke reflecton. In ths paper a smple practcal method s gong to be presented to extend the capabltes of the classcal ray tracng algorthm. The proposed workflow makes dffuse lght transport possble and the use of area lghts nstead of the common pont, spot and drectonal lghts. The possble dffcultes applyng the proposed procedure n dfferent software envronments s gong to be dscussed as well. Keywords: renderng, global llumnaton, ray tracng, reflecton 1. Introducton The great mprovement n the computng power of standard desktop computers made possble to use complex renderng softwares to generate photorealstc mages [5]. There are many commercal 3d graphcs applcatons on the market, whch smulate the behavor of lght so convncngly, that the resultng mage looks real. These programs use state of the art technology and are hard to get acces to. The wdely used and the free 3d applcatons generate mages n much less tme, but use smpler renderng methods. These algorthms - lke ray tracng - make lot of compromse n order to avod long render tmes keepng the mage as real as possble. In the next part these smple and commonly used renderng methods wll be dscussed focusng on ther strengths and weaknesses. 1.1 The Common Renderng Algorthms Today's graphcs systems are pxel based. Ths makes the basc step of mage synthess the calculaton for each pxel's color. Ths color value s proportonal to the ncomng radance to the camera from a gven drecton. Ths physcal value s a very hard and resource ntensve problem to solve for, even f the wave propertes of lght or the presence of lquds and gases are gnored. The problem can be expressed n the form of the renderng equaton [6]:

2 e L( ω ) = L ( + L( ω ), ω ) f ( ω, cosθ dω Ω r (1) Ths equaton shows how the radance ( L( x, ) can be calculated from a gven surface element: the surface's lght emsson ( L e ( ) and the fracton of the ncomng lght that reflects to the camera should be added together. In order to express the second term the lght from all possble ncomng drecton should be taken nto account weghted wth the bdrectonal reflectance dstrbuton functon (BRDF) and the cosne of the ncomng angle. Let us ntroduce the τ ntegral operator, whch descrbes the lght-surface nteracton: ( τl)( = L( ω ), ω ) f ω x ω θ dω r (,, ) cos Ω (2) Usng the operator above the renderng equaton can be wrtten n a compact form: e L = L +τ L (3) The unknown radance L s both dfgd nsde and outsde of the operator, whch makes the problem very complex. Because the operator cannot be nverted t os not possble to elmnate the couplng. Fortunately the problem can be solved usng a smple trck: the whole rght sde should be substtuted n the unknown functon of the same sde. Ths transforms the ntegral equaton nto an nfnte seres: = L = 0 e e e 2 e 3 e τ L = L + τ L + τ L + τ L +... (4) The elements represent the drect emsson, the sngle reflecton, the secondary reflecton and so on. The elements of the seres are ntegrals wth ncreasng dmensons. The smple renderng algorthms do not even try to solve the equaton n ths form. The local llumnaton method tres to

3 solve for only the frst two elements of the seres, ray tracng elmnates the ntegral from every element makng the seres nto a much smpler sum. Local Illumnaton Almost all of the commonly used 3d graphcs programs make use of the local llumnaton model. Ths s a drastc smplfcaton of the renderng equaton. It s assumed that the look of real world surfaces can be determned by calculatng drect emsson and sngle reflecton only. The rest of the nfnte seres s expressed n one, user defned ambent term. Usng ths assumpton only one ntegral of one dmenson should be calculated. Ths ntegral s supposed to sum the ncomng radance from all drectons. Because ths calculaton s stll hard and tme consumng to do the local llumnaton model smplfes the equaton even more. It s known that ntegratng any combnaton of Drac delta functons s very easy, snce the ntegral equals the sum of few values. In the case of the renderng equaton the ncomng radance s a combnaton of Drac delta functons only f the lght sources are nfntesmally small. The local llumnaton model uses only ambent, drectonal, pont and spotlghts - and gnores objects wth self llumnaton - when calculatng the ncomng radance n order to elmnate the hard ntegral. Ths model would be useless f the generated pctures were far from real but fortunately that s not the case. In real lfe and especally n case of artfcally lt envronments the domnant lght sources determne the look of objects, and the ntersurface lght reflectons have only secondary mportance. In cases when reflected lght acts as mportant lght source n the local llumnaton envronment the user have to defne more abstract lghts to acheve the desred effect. Ths method s physcally ncorrect and t mght be uncomfortable as well, however n producton envronment the method has proven t's usablty [1]. The renderng equaton usng the local llumnaton model turns out to be very smple due to the Drac delta type lght sources: numberoflghts e L( = L ( + LIGHT ( ω ) = 1 f ( ω, ω ) cosθ + AMBIENT r (5) Where the functon LIGHT ( represents the the ncomng radance from the -th lghtsource to the pont x from the drecton ω. In ths equaton the renderng equaton (1) s reduced to the sum of few, easly computable terms. Recursve Ray tracng The ray tracng algorthm [4] s a very common extenson of the local llumnaton method. Usng ray tracng the total reflecton and the refracton of lght can be smulated correctly. Ths effect s very mportant snce the reflecton seen on shny surfaces - lke chrome, glass or car pant - determne the overall look of these objects. The mplementaton of ray tracng s based on the local llumnaton model. Knowng the normal vector of the rendered surface element the reflecton and refracton drectons can be calculated. Callng the already mplemented local llumnaton algorthm n these drectons the value of the ncomng lght reflected or refracted toward the camera can be easly calculated. Ths process s able to work recursvely and follow a ray" from the camera along the path of the ncomng lght rays.

4 numberoflghts e L( = L ( + LIGHT ( ω ) f r ( ω, cosθ + = 1 reflect reflect refract refract + L( ω ), ω ) RL + L( ω ), ω ) RR + AMBIENT (6) Ths formula s the extenson of the equaton for local llumnaton (5). The two new terms are the reflected and refracted lght rays whch can be calculated recursvely. If no abstract lghts are used for the local llumnaton the only thng that effects the fnal renderng s the self llumnaton and ray tracng: e reflect reflect refract L( = L ( + L( ω ), ω ) RL + L( ω ), ω refract ) RR Ths equaton s practcally equaton (6) wthout the terms of equaton (5). If only non-transparent surfaces are used the thrd term can be gnored. Consderng the frst two terms t s surprsngly smlar to the renderng equaton n the form of equaton (3). The recursve ray tracng algorthm doesn't neglect any term of the nfnte seres, the only smplfcaton comparng to the orgnal s the BRDF whch conssts of Drac mpulses. Ths lkelness makes the use of the ray tracng method possble to calculate the renderng equaton, or n other words: to solve global llumnaton problems. Global Illumnaton There are numerous methods to solve the renderng equaton [6] wth no - or not much - smplfcaton. These algorthms - lke Monte Carlo ntegraton [8] - often handle not only mrror lke surfaces but glossy and dffuse objects as well and are called global llumnaton renderers. The goal s to acheve the same realstc result wth the commonly used ray tracng softwares wthout the need of modfyng the program. (7) 1.2 Fake Radosty" There are some trcks used n the 3d graphcs communty ntended to smulate the effect of global llumnaton. These trcks - often called fake radosty" methods - can be classfed nto two major groups: Non-automatc process: the user places secondary pont lghts near brght areas smulatng the effect of dffuse lght reflecton. Ths can be very tedous and slow process. Automatc process [3]: ths trck s smlar to the proposed method n ths paper, but works for only one object at a tme. The user has to flter or blur manually a very bad qualty - nosy - result and usually composte t back to the rendered mage calculated wth local llumnaton. The proposed method works for all objects n the scene and s far more physcally accurate than the avalable fake radosty" trcks.

5 Fgure 1. Images generated usng local llumnaton, ray tracng and global llumnaton methods. 2. Extendng Ray tracng The man goal s to prepare the ray tracng algorthm to handle glossy and dffuse reflecon. The local llumnaton capablty of the softwares s not used at all, snce t doesn't calculates any secondary lght reflecton. However, local llumnaton usng models lke Phong and Lambert could be used to calculate the frst shoot or the drect llumnaton of lghts but ths opton was thrown away because the followng reason. It requred to have pont lghts defned nstead of lght emttng objects thus the characterstc shadows would be not soft at all. In fact the local llumnaton method s not able to generate realstc soft shadows, whch s overcomed usng ray tracng for the frst shoot too. 2.1 Creatng Dffuse Surfaces It has to be nvestgated how dffuse materals work n the real world to acheve the same result on the vrtual surfaces. The perfectly dffuse behavor s descrbed by the Lambert model. Ths model supposes that the ncomng lght s reflected n all drectons wth the same ntensty. In realty dffuse objects have such mcroscopc structure that the ncomng rays are reflected to all drecton ndependently from the ncomng drecton. The algorthms whch use the mcrofacet model [2,9] smulate ths effect wth lttle, randomly orented mcromrrors called facets. Ths dea s very useful to create dffuse or glossy materals whle renderng wth ray tracng. The 3D programs offer usually 3 alternatve ways do defne the geometry or the structure of the surface [11]. One way should be chosen to create these random dsplacements n order to create dffuse surfaces.

6 2.2 Defnng Structure Mcrostructure: Usng the local llumnaton model an emprcal BRDF - lke Phong or Lambert model s assgned to every surface. Ths means that the lght reflecton propertes of some materal due to the mcroscopc and atomc structure are smulated by abstract mathematcal formulas. Durng ray tracng these llumnaton models are unmportant n determnng the reflecton and refracton drectons, thus these formulas wll not be used. Macrostructure: The structure of dffuse and glossy surfaces can be modeled usng standard and wdespread surface modelng methods. Ether NURBS, polygon or subdvson surfaces could be used to descrbe the geometry of the surface. However, ths approach requres lot of memory - way more than an average user has - to store even a very smple and rough estmaton of some dffuse surface. Mesostructure: To model the vsble roughness of a surface bump maps are used. These requre very lttle memory - especally when usng procedural textures - and are able to change the reflecton and refracton drectons durng ray tracng. The man dea of bump maps s that usng the partal dervatves of a standard texture the normal vector of the surface can be perturbed. Ths creates the mpresson n the vewer as f there were bumps on the surface. It s the deal for us to choose ths approach snce t s supported n all 3d graphcs packages, t s fast and requres almost no memory. 2.3 Settng Up the Materal Attrbutes The only thng that affects the lght reflecton propertes of the materals s the bump map. The goal s to create surfaces that can reflect lght n every possble drecton. To make a dffuse materal based on the Lambert rule t s necessary that the probablty densty of the drecton of the reflected ray to be constant. Ths means the bump map rotates the normal vector n any drecton wth the same probablty. If the bump s smoother more rays wll be reflected towards the deal reflecton drecton and the Fgure 2: Mcrostructure Fgure 3: Macrostructure Fgure 4: Mesostructure

7 materal wll look glossy. To acheve ths effect a very dense, hgh frequency bump map should be used snce the user does not want to see the actual mcroscopc structure of the materal. However, there are lots of materals n real lfe whch do not work ths way. Satn, velvet and dense har have such mcroscopc structure that the majorty of the reflected rays pont not toward the deal reflecton. These non-sotropc structures [2] can be modeled wth such bump maps that reproduce the effect of mcroscopc features shadowng certan drectons or reflectng lght to another specfc drecton. The frst attempt to defne the bump map was to generate textures that would represent a user gven normal vector dstrbuton. Ths approach has had seruos dsadvantages: - The repeatng texture was notcable on the fnal renderng. - If the bump map was scaled very small to avod the artfact above, the bump smply dsappeared because of the renderng software s ant-alasng method [10]. The next approach was to perturb the normal vector wth a custom scrpt. Ths would not work, however, because even the most advanced 3D applcatons are not prepared to gve ths opton to the user. The way of generatng bump maps n the research turned out to be usng bult n textures. These textures generate nosy dsplacements wth no vsble repettons. Modfyng the gan of the texture the depth of the bumps could be changed, thus the lght reflecton propertes of the materal also changed. By modfyng the texture parameters and renderng test pctures the dffuse or glossy behavor could be acheved. Fgure 5: Usng the same texture wth dfferent gan to generate the bump map.

8 3. Renderng Usng the ray tracng renderng method to defne the pcture, the program shoots rays from each pxel to ht a surface. If an object s ht - and no local llumnaton s used the color of the current surface element s calculated usng equaton (7). Snce ths process s a recursve one the rays from the camera do not stop at the frst ntersecton but follow certan paths along the reflectons. These paths wll be used certanly more for each pxel to calculate the color of the pxel defned by the renderng equaton (1). 3.1 Gatherng Walks As stated above the rays shot from the camera follow a path or gatherng walk snce they are reflected at each surface collectng the ncomng radance. Every tme one of these rays hts a surface the collected lght s the sum of the self llumnaton and the lght from the reflecton. There mght be two knd of dangerous stuatons whle renderng: - The path may not reach any object emttng lght. Ths s because ray tracng s stopped after certan number of reflecton. Ths threshold number s user defned. To overcome ths stuaton very large objects were used as lght sources. - If the lght emttng objects are also reflectve the ray tracng process does not stop untl reachng the threshold number. Because the reflecton seen on lght sources are usually not vsble non-reflectve objects were used to emt lght. 3.2 Samplng Apparently one ray shot from each pxel s not enough. To understand why many gatherng walks should be started from each pxel the common ntegral formula should be nvestgated. Ths s useful snce the goal s to calculate the ntegral of equaton (1). V f ( z) dz 1 n n = 1 f ( z ) s( z ) (8) where n s the number of samples, s() s the weght functon and z are the samplng ponts. In the actual case the radance value gven by the renderng equaton should be estmated for each surface element ht wth a ray. Ths means the samples wll be the values of reflected lght rays n dfferent drectons. The more rays are shot from the pxels the more accurate the result wll be. If a bump map defnng glossy reflecton s appled more rays samples wll be reflected toward the drecton of the deal reflecton. If the bump defnes a dffuse materal all samples wll have probably dfferent drecton. No gatherng walk s more mportant than the other, thus no weght functon s used: n e L( L ( + =1 L( ω ), ω ) R n (9) where n s the number of samples for each pxel and ths equaton the refracton s gnored. ω s the reflecton drecton of the -th ray. In

9 The man queston s: what n should be used to acheve good qualty? There s no global answer snce the amount of nose seen on the pcture and renderng tme s n drect proporton to the number of samples. 3.3 Fgure 6: The same scene rendered usng 1,3,10,32 and 320 samples for each pxel

10 3.3 Reducng Nose The 3D applcaton used [7,10] for the tests let the number of samples to be 32 at maxmum whch turned out to be not enough. To overcome ths lmtaton more mages should be rendered from the same scene. If the renderer uses stochastc samplng and the random generator uses dfferent seeds for each frame - the startng ponts of the gatherng walks wll be at slghtly dfferent poston. Averagng these pctures the result wll be as f more samples were used for each pxel. Apparently the user should not rely on the renderer ths much snce the same result can be acheved usng only one sample for each pxel even f the samplng postons are statc. The trck s to anmate the poston of the bump texture. Dong ths the reflecton drectons wll obvously change snce bump map changed whle the probablty densty of the reflecton drecton wll fortunately not be affected. The mages rendered ths way can be averaged together producng the same mage as f t was rendered wth a very hgh number of samples. 4. Conclusons Usng the proposed method global llumnaton problems can be solved wthout any major smplfcaton usng standard 3D packages wth ray tracng capabltes. The user can defne lght emttng objects nstead of abstract pont lght thus generatng realstc soft shadows. The specular and dffuse nterreflectons can be smulated as well not lke whle usng the renderer the conventonal way. The applcaton of the method does not requre any specal tranng or programmng knowledge, any standard even freeware renderers can be used. Unfortunately the generaton takes long and the fnal pctures contan sgnfcant nose. To reduce ths artfact the number of samples should be extremely ncreased causng the computer to render even longer. Ths nose makes for most of the cases mpossble to use the pctures wthout any modfcaton or flterng. However, there are many ways to make use of ths process n producton envronment. The generated pctures are great vsual references for the techncal drectors how to lt ther scenes usng local llumnaton. It s also useful to flter out the hgh frequency nose and use the generated pctures as textures for the objects n the scene. The subject of future nvestgaton mght be the effcent reducton of the nose wth some custom flter or other method. The generaton of any mcrofacet structure would be also useful to render non-sotropc objects. 5. Acknowledgements Ths work has been supported by the Natonal Scentfc Research Fund (OTKA ref. No.:T029135). The test scene wth the car has been modelled and rendered by Maya, that was generously donated by Alas Wavefront.

11 6. References [1] Apodaca, T., Grtz, L.: Advanced Renderman Creatng CG for Moton Pcture (chapter 1.), Morgan Kaufmann, [2] Ashkhmn, M., Premoze, S., Shrley, P.: A mcrofacet-based BRDF generator, n Proceedngs of SIGGRAPH 2000, p [3] Campn, E.: Fakng radosty n Maya (tutoral), [4] Glassner, A.: An Introducton to Ray Tracng, Academc Press New York, [5] Greenberg, D., Torrance, K., Shrley, P., Arvo, J., Ferwerda, J., Pattanak, S., Lafortune, E., Walter, B., Foo, S., Trumbore, B.: A framework for realstc mage synthess, n Proceedngs of SIGGRAPH 97, p [6] Kajya, J. T.: The renderng equaton, n Proceedngs of SIGGRAPH 86, p [7] Pearce, A., Sung, K.: Maya software renderng a techncal overvew, Alas Wavefront, 1998, [8] Szrmay-Kalos, L.: Monte-Carlo Methods n Global Illumnaton, [9] Szrmay-Kalos, L., Kelemen, Cs.: A mcrofacet based coupled specular-matte BRDF model wth mportance samplng, Eurographcs Conference, 2001, Short presentatons, pp [10] Woo, A.: Alasng artfacts n Maya a techncal overvew, Alas Wavefront, 1998, [11] Yu, Y.: Image-Based Surface Detals (ntroducton), Course 16 n Course Notes of SIGGRAPH 2000, p. 1-1.