Plane Sampling for Light Paths from the Environment Map

Transcription

1 jgt 2009/5/27 16:42 page 1 #1 Vol. [VOL], No. [ISS]: 1 6 Plane Samplng for Lght Paths from the Envronment Map Holger Dammertz and Johannes Hanka Ulm Unversty Abstract. We present a method to start lght paths on an emttng envronment map n the context of Monte Carlo global llumnaton wth mage-based lghtng or physcal sky models. Wth ths technque we can effcently render unbased caustcs from these knds of lghts, for example usng b-drectonal path tracng. Addtonally t s now possble to use algorthms lke multple mportance samplng, photon mappng and nstant radosty correctly wth envronment maps. 1. Introducton When smulatng lght transport usng Monte Carlo methods, all knds of paths connectng lght sources and the sensor must be generated. Even though all paths wll eventually be found by a smple path tracer (startng at the sensor and recursvely prolongng the path untl a lght source s found), some effects are best captured wth some specalzed samplng technque. A lght tracer for example, startng the random walk at the lght sources and determnstcally connectng to the sensor, s well suted to render caustcs. Other global llumnaton methods, such as photon mappng [Jensen 96], bdrectonal path tracng [Veach and Gubas 94] or pont lght source-based methods as nstant radosty [Keller 97] rely on paths started from the lght sources. Whle startng paths on geometrc lght sources s common practce, t s not as easy to start a path from an emttng envronment map wth a correct samplng densty. A K Peters, Ltd /06 $0.50 per page

2 jgt 2009/5/27 16:42 page 2 #2 2 journal of graphcs tools ω y ω y ω o ω o x Fgure 1. Two ways to sample dstant llumnaton: on the left, ω s obtaned by prolongng an eye path endng n x. On the rght, a pont y outsde the scene s needed to trace a drecton ω towards x. x 2. Background Analytc sky models or envronment maps n the form of captured mages [Debevec 98] approxmate dstant llumnaton by an nfntely large sphere around the scene. Thus the radance L(ω) emtted from the envronment only depends on the drecton ω for any pont n the scene. In a classcal path tracng settng, the drect llumnaton s evaluated by samplng a drecton at a pont x and evaluatng the vsblty of the envronment map (see Fgure 1, left). In a Monte Carlo renderer ths drecton s pcked accordng to a carefully chosen probablty densty functon n order to mnmze varance. In a b-drectonal algorthm ths s not as straght forward as we need to start a path from outsde the scene wth a correct and known probablty densty functon. For example n an nstant radosty-based renderer the ndrect llumnaton s approxmated by dstrbutng pont lghts va a random walk from the lght sources. When usng envronment maps the drect llumnaton can be evaluated as descrbed above. For ndrect llumnaton, sample ponts to start the lght paths have to be generated. Ths s llustrated on the rght hand sde of Fgure 1 where the desred samplng pont s marked as y. The problem s smlar to the samplng of parallel lght sources wth a correct samplng densty that was descrbed n [Pharr and Humphreys 04]. 3. Startng Lght Paths on the Envronment Map Envronment maps can be nterpreted as an nfnte collecton of parallel lght sources (one for each drecton). To start a partcle path from the envronment we dvde the problem nto frst samplng a drecton ω and then samplng an approprate startng pont y outsde the scene for the drectonal lght source assocated wth ω. Algorthmcally, we sample the drecton ω on the envronment map, ac-

3 jgt 2009/5/27 16:42 page 3 #3 Dammertz and Hanka: Plane Samplng for Lght Paths from the Envronment Map 3 mn u ω y max u Fgure 2. Illustraton of the sampled plane n 2d (left), and vsualzaton of sx sampled planes for a 3d scene. cordng to a probablty densty p(ω), for example p(ω) = 1/(4π). Then we create an orthonormal bass (u, v, ω) around ω, and project the eght corners of the boundng box of the scene onto the plane through the orgn spanned by (u, v), as llustrated n Fgure 2. From these projectons, we only keep a 2-dmensonal boundng box n the coordnate system of the plane (mnu, maxu, mnv, maxv). Ths quad s pushed outsde the boundng box of the scene. The exact dstance does not matter, as the quad s used as a parallel lght source. In practce, we project one vertex of the scene boundng box (or the center) onto ω and shft ths pont n the drecton ω by the sum of the three boundng box wdths. Pseudo-code for the plane generaton can be found n Fgure 3. Such a quad s easy to sample. The startng pont y can be chosen unformly on ths quad, whch results n the smple probablty densty p(y) = 1/ A y, wth A y beng the area of the quad. See Fgure 4 for pseudo-code of ths operaton. The estmator remans unbased as long as the quad s large enough to cover all of the scene. In order to maxmze the number of rays actually ntersectng the scene, the quad should be as small as possble. Apart from enablng the use of algorthms relyng on lght paths (nstant radosty, photon maps, etc.), the advantage of the new technque becomes apparent when the envronment map has a hgh dynamc range and mportance samplng s used,.e. p(ω) L(ω). We use sample warpng [Clarberg et al. 05] to sample ω accordng to the lumnance of the envronment map. 4. Comparson Wth the new technque t s now possble to create caustc paths from the envronment effcently. Other unbased technques generate these paths as well, but at a much lower convergence rate. For an llustraton of ths sgnfcant dfference, see Fgure 5. The envronment map contans a sun wth pxels 1000

4 jgt 2009/5/27 16:42 page 4 #4 4 journal of graphcs tools omega = envmap_sample(); nv_p = 1/p(omega); // account for p(omega) L = envmap_radance(omega); (u,v) = get_onb(omega); // create orthonormal bass (u,v,omega) for(nt c=0;c<8;c++) // project scene boundng box { dotu = dotproduct(aabb.corner[c], u); dotv = dotproduct(aabb.corner[c], v); mnu = mn(mnu, dotu); maxu = max(maxu, dotu); mnv = mn(mnv, dotv); maxv = max(maxv, dotv); } float dst = dotproduct(aabb.center, -omega) + aabb.wdth[0] + aabb.wdth[1] + aabb.wdth[2]; Fgure 3. Pseudo-code for the generaton of the quad from a sampled envronment drecton. Ths computaton can be reused for several samples. y = - dst * omega + u*(mnu + (maxu-mnu)*rand1) + v*(mnv + (maxv-mnv)*rand2); nv_p *= (maxu-mnu)*(maxv-mnv); // compensate for p(y) = 1/A partcle.energy = L*nv_p; partcle.y = y; partcle.omega = omega; Fgure 4. quad. Random samplng of the partcle startng pont on the precomputed tmes brghter as n the rest of the mage. All three path samplers use the same mportance samplng [Clarberg et al. 05] to evaluate drect lght, thus the same sharp shadow appears n all mages. However, usng ths knd of mportance samplng for drect lght only fals to capture caustcs. Metropols samplng helps, but wll waste a lot of samples n the sun, as the probablty s proportonal to the radance contrbuted to the mage. Of course Metropols samplng wll perform better when a b-drectonal path sampler s used and when the sun s not drectly vsble, especally n dffcult areas of the mage such as caustcs from multple nner reflectons seen through a mrror.

5 jgt 2009/5/27 16:42 page 5 #5 Dammertz and Hanka: Plane Samplng for Lght Paths from the Envronment Map 5 Fgure 5. Comparson of plan backward path tracng from the eye (left), the same path tracer wth Metropols samplng (mddle), and b-drectonal path tracng wth a lght tracng pass usng plane samplng (rght). Each example on the top row has equal computaton tme (about 5 mnutes on an AMD opteron). The path tracng acheves 342 samples n that tme, the Metropols verson can only evaluate 105, the b-drectonal method has 231 samples per pxel. The bottom row shows a comparson of equal sample count (500 samples per pxel). 5. Dscusson The plane samplng technque s a specalzed soluton to effcently capture promnent effects, such as caustcs, when mage-based lghtng s used. It s also most useful when b-drectonal algorthms are used whch rely on paths started from the lght sources, such as nstant radosty or photon mappng. A b-drectonal path tracer could work around ths by employng path tracng for these effects, but effcency wll be much lower f the envronment s nondffuse. However, there are cases n whch other technques are better. For example a mostly dffuse envronment such as a cloudy sky s best sampled by a path tracer, as mportance samplng by surface reflecton can be done. If only drect lghtng s of nterest to you, combned samplng of ncomng radance and surface reflecton [Clarberg and Akenne-Mo ller 08] s the way to go. The smple method descrbed above to sample a quad, whch s the 2dboundng box of a projected 3d-boundng box, can easly be extended to choose (u, v) to mnmze the sze of the quad, or be replaced by samplng for example a dsc (the projected boundng sphere) as was done n [Pharr and Humphreys 04]. We chose quads because most render systems have the axsalgned boundng box of the scene readly avalable, and t s then straghtforward to extend the technque to projecton maps to be used wth photon mappng or nstant radosty.

6 jgt 2009/5/27 16:42 page 6 #6 6 journal of graphcs tools Acknowledgments. The authors would lke to thank the revewers for ther helpful suggestons. Addtonally they want to thank mental mages GmbH for fundng parts of ths research, and Matthas Raab and Leonhard Grünschloß for many dscussons. References [Clarberg and Akenne-Möller 08] Petrk Clarberg and Tomas Akenne-Möller. Practcal product mportance samplng for drect llumnaton. In Computer Graphcs Forum (Proc. of Eurographcs 2008), pp , [Clarberg et al. 05] Petrk Clarberg, Wojcech Jarosz, Tomas Akenne-Möller, and Henrk Wann Jensen. Wavelet mportance samplng: effcently evaluatng products of complex functons. ACM Transactons on Graphcs 24:3 (2005), [Debevec 98] Paul Debevec. Renderng synthetc objects nto real scenes: brdgng tradtonal and mage-based graphcs wth global llumnaton and hgh dynamc range photography. ACM Transactons on Graphcs (Proc. SIGGRAPH 1998), pp [Jensen 96] Henrk Wann Jensen. Global llumnaton usng photon maps. In Renderng Technques 96 (Proc. of the Seventh Eurographcs Workshop on Renderng), pp , [Keller 97] Alexander Keller. Instant Radosty. ACM Transactons on Graphcs (Proc. SIGGRAPH 1997), pp [Pharr and Humphreys 04] Matt Pharr and Greg Humphreys. Physcally Based Renderng: From Theory to Implementaton. Morgan Kaufmann Publshers Inc., [Veach and Gubas 94] Erc Veach and Leondas Gubas. Bdrectonal estmators for lght transport. In Renderng Technques 94 (Proc. of the Ffth Eurographcs Workshop on Renderng), pp , Web Informaton: Hgh-resoluton mages, the Blender fle used to synthesze the mages n Fgure 5, and a vdeo can be found at Holger Dammertz, Ulm Unversty, James-Franck Rng, Ulm (holger.dammertz@un-ulm.de) Johannes Hanka, Ulm Unversty, James-Franck Rng, Ulm (johannes.hanka@un-ulm.de) Receved [DATE]; accepted [DATE].