EUROGRAPHICS 0x / Jan Kautz and Sumanta Pattanak (Edtors) Volume 0 (1981), umber 0 Photorealstc Image Renderng wth Populaton Monte Carlo Energy Redstrbuton Y.-C. Yu-Ch 1 and S.H. Fan 1 and S. Chenney 2 and C. Dyer 1 1 Unversty of Wsconsn at Madson, U.S.A. 2 Emergent Technology, U.S.A. Abstract Ths work presents a novel global llumnaton algorthm whch concentrates computaton on mportant lght transport paths and automatcally adjusts energy dstrbuted area for each lght transport path. We adapt statstcal framework of Populaton Monte Carlo nto global llumnaton to mprove renderng effcency. Informaton collected n prevous teratons s used to gude subsequent teratons by adaptng the kernel functon to approxmate the target dstrbuton wthout ntroducng bas nto the fnal result. Based on ths framework, our algorthm automatcally adapts the amount of energy redstrbuton at dfferent pxels and the area over whch energy s redstrbuted. Our results show that the effcency can be mproved by explorng the correlated nformaton among lght transport paths. Categores and Subject Descrptors (accordng to ACM CCS): I.3.7 [Computer Graphcs]: Raytracng 1. Introducton To generate a physcally correct mage nvolves the estmaton of a large number of ntegrals of path contrbutons fallng on the mage plane. It s well known that the ntegrals have hghly correlated ntegrands. However, a standard Monte Carlo renderng algorthm evaluates the ntegrals ndependently. As a result, even a small but mportant regon n the doman s located durng the process. Ths nformaton s lost to other samples because of the ndependent samplng. Sample reuse s an mportant technque to reduce the varance by explotng the correlaton between ntegrals. Markov Chan Monte Carlo algorthms for global llumnaton, such as Metropols Lght Transport [Vea97] and Energy Redstrbuton Path Tracng [CTE05], enable sample reuse by mutatng exstng samples nto new ones, but the choce of good mutaton strateges s non-trval and has a major mpact on mage qualty. Populaton Monte Carlo (PMC) algorthms provdes us a tool to reuse the nformaton collected n prevous teratons. PMC energy redstrbuton, adapts the framework of PMC to energy redstrbuton algorthm, explots nformaton from mportant samples through reuse wth a mutaton process whose mutaton strategy s adapted on-the-fly. It s self-tunng to a large extent. The PMC energy redstrbuton algorthm terates on a populaton of lght transport paths passng through the mage plane. The populaton paths are created by tracng the vew rays passng through stratfed pxel postons on the mage plane by a general Monte Carlo ray tracng algorthm such as path tracng and bdrectonal path tracng. In our mplementaton, we use a general path tracng algorthm. Any nformaton avalable n the prevous teratons can be used to adapt the kernel functon of each populaton path that produce a new populaton based on the current populaton. The resamplng process elmnates part of the populaton paths and regenerate new paths to acheve ergocty. We carefully desgn the resamplng process to elmnate the well explored or low-contrbuton paths from the current populaton and to generate new paths accordng to the need of explorng the mage plane evenly for achevng unbasedness. As a result, new samples are desgned to explore the mage plane n an even manner. The procedure s then terated: sample, terate, resample, adapt, terate, resample.... The result s a selftunng unbased algorthm whch can explore the mportant paths locally. Our contrbuton s a new renderng algorthm, PMC Energy Redstrbuton(PMC-ER), based on the PMC framework. The algorthm adapts the amount of energy redstrbuton at dfferent pxels and the area over whch energy s rec The Eurographcs Assocaton and Blackwell Publshng 2007. Publshed by Blackwell Publshng, 9600 Garsngton Road, Oxford OX4 2DQ, UK and 350 Man Street, Malden, MA 02148, USA.
Y.-C. Yu-Ch & S.H. Fan & S. Chenney & C. Dyer / Photorealstc Image Renderng wth Populaton Monte Carlo Energy Redstrbuton dstrbuted. For example, pxels near a sharp shadow boundary wll not attempt to wdely dstrbute energy, whle those n a smooth dffuse mage regon wll dstrbute energy over a wde area. The remander of ths paper s organzed as follows: secton 2 revews a number of works related to ths algorthm. Secton 3 presents the generc PMC frame work. Secton 4 presents the PMC-ER n detal. Secton 5 shows the results generated by ths algorthm. Secton 6 dscusses the lmtaton and relaton to the exstng algorthm. Fnally, secton 7 gves the concluson of our algorthm. 2. Related Work Currently, most global llumnaton algorthms are based on ray tracng and Monte Carlo ntegraton. There exst two catefores: unbased methods such as [Kaj86,VG94,LW93]; and based methods such as [WRC88, Hec90, Jen01]. Interested readers can refer to Pharr and Humphreys [PH04] for an overvew of Monte Carlo renderng algorthms. Here we only focus on sample reuse whch s drectly related to ths work. Sample reuse va Markov Chan Monte Carlo (MCMC) algorthms s a powerful means of explotng hard-to-fnd lght transport paths n global llumnaton. Metropols Lght Transport (MLT) [Vea97] was the frst algorthm to use ths approach. MLT replaces the Monte Carlo ntegrator used n path tracng wth a Metropols sampler. The man advantage of the Metropols algorthm over Monte Carlo ntergraton s the ablty to preserve the samplng context. Ths s done by usng path mutaton to explore path space n a localzed way. Thus, when hgh conbrbuton paths are found, nearby paths wll lkely be explored as well. A number of extensons have been ntroduced snce the orgnal 1997 paper. [PKK00] extended MLT to handle partcpatng meda such as smoke and fog. [KSKAC02] made the MLT algorthm more robust by mutatng n an abstract space of random numbers rather that on the expected qualty. The start-up bas of the MLT was analyzed by [SKDP99] and the analyss of the algorthm was presented n [APSS04]. However, the dsadvantage of MLT was and contnues to be that very large numbers of samples are requred, and stratfcaton s dffcult. Energy redstrbuton path tracng (ERPT) [CTE05] attempted to address ths problem by startng wth a wellstratfed set of ntal samples and locally redstrbutng energy usng MCMC. The nose-reducton technques they proposed ntroduce bas. In addton, the extent of redstrbuton was manually set. [FCL05] used the MLT sampler to take vsual mportance nto account wth complete paths from lght to eye when dstrbutng photons accordng to paths contrbuton on the fnal mage. Ther method solved the dffcult path problem such as lght passes through a small hole on the wall. However, the bas nherted from photon mappng methods prevents the usage of advanced convergence test mechansm. Our PMC-ER algorthm automatcally adapts parameters n an ERPT-lke algorthm and uses the adaptaton of the kernel functons to locally explore mportant lght transport paths. In addton, the algorthm s unbased. Ghosh, Doucet and Hedrch [GDH06] appled the framework of Sequental Monte Carlo algorthm to the problem of samplng envronment maps n anmated sequences. Ther work re-uses samples from prevous teraton and s a complementary to our method. However, ther work s lmted to the envronment map. Our algorthm can be appled to more general types of lght transport paths. 3. D-Kernel Populaton Monte Carlo The Populaton Monte Carlo algorthm [CGMR04] provdes us an teratve mportance samplng framework. The dstngushng feature of PMC s that the kernel functons are modfed after each step based on nformaton gathered from pror teratons. The kernels adapt to approxmate the deal mportance functon based on the samples seen so far. Whle ths dependent samplng may appear to ntroduce bas, t can be proven that the result s ether unbased or consstent, dependng on whether certan normalzng constants are known (n our case they are known). The generc D-Kernel PMC samplng algorthm [DGMR05a,DGMR05b] whch s an evoluton of PMC s stated n Fgure 1. 1 generate the ntal populaton, t = 0 2 for t = 1,,T 3 adapt K (t) (x (t) x (t 1) ) 4 for = 1,, 5 generate X (t) K (t) (x X (t 1) ) 6 w (t) = π(x (t) )/K (t) (X (t) X (t 1) ) 7 resamplng process: elmnaton and regeneraton Fgure 1: The generc D-Kernel Populaton Monte Carlo algorhtm. The { algorthm works } on a populaton of samples denoted by X (t) 1,...,X(t), where t s the teraton number and s the populaton sze, to evaluate R D f(x)dx, where s f(x) = π(x)h(x) by samplng accordng to the target dstrbuton π(x). The algorthm frst creates a set of ntal populaton by usng any unbased samplng method. A kernel functon, K (t) (x (t) x (t 1) ), for each member n the populaton s adapted n the outer loop. The responsblty of the member kernel fucton s to take the exstng member sample, X (t 1), as nput and produces a canddate new sample, X (t), as output (lne 5). The resamplng step n lne 7 s desgned to cull canddate samples wth low weghts and promote hghweght samples. The resamplng process conssts of two c The Eurographcs Assocaton and Blackwell Publshng 2007.
Y.-C. Yu-Ch & S.H. Fan & S. Chenney & C. Dyer / Photorealstc Image Renderng wth Populaton Monte Carlo Energy Redstrbuton steps: elmnaton and regeneraton. It s desgned to elmnate the samples wth low contrbuton to the fnal result and to explore new unexplored regons. The weght computed for each sample, w (t), s essentally ts mportance weght. At any gven teraton, an estmator of the ntegral of nterest can be computed and s unbased for π(h): f(x) = π(h) = 1 E[ 1 w (t) h(x (t) )] = 1 =1 =1 = 1 Z =1 D w (t) h(x (t) ) =1 E[w (t) h(x (t) )] π(x)h(x) K (t) (x x (t 1) ) K (t) (x x (t 1) )dx = 1 Z π(x)h(x)dx =1 D Z = f(x)dx D It concludes that π(h) s an unbased estmator of π(h). Before apply PMC to renderng problems, we must frst answer the followng questons: What s the samplng doman and how bg s populaton sze? What s the member functon and what s the adapton crtera? What technques are used for samplng from the kernel functons and resamplng step? The followng sectons descrbe an applcaton of ths framework by mutatng the energy redstrbuton algorthm through answerng each queston properly. Then, we conclude wth a general dscusson on PMC for renderng problems. 4. PMC Energy Redstrbuton (PMC-ER) PMC Energy Redstrbuton (PMC-ER) s an algorthm motvated by energy redstrbuton path tracng (ERPT) [CTE05] that adaptvely selects pxels for redstrbuton, and can also adapt algorthm parameters. ERPT as orgnally proposed traces a path nto the scene from each pxel, usng path tracng to form complete lght transport paths from the eye to the lght. For each pxel, the path s used as the ntal state for a Markov Chan Monte Carlo (MCMC) sample chan that redstrbutes the path s energy to nearby pxels and fnds addtonal lght paths. The ntuton s that dfferent pxels wll fnd dfferent ntal paths, and the nformaton can then be conveyed to neghborng pxels through the Markov Chan. Due to space lmtatons, we cannot dscuss ERPT n detal; readers are referred to the orgnal paper. ERPT uses the estmaton of the energy of the entre mage from the path contrbuton to determne how many constant length chans are needed for every pxel, regardless of how much t dffers from ts neghbors. In addton, the redstrbuton regon s also fxed and manually set. Ths s sub-optmal some pxels that have hgh varance should take more samples and more tme to redstrbute ts energy, whle others are n a neghborhood where most lght transport paths are smlar and redstrbuton acheves nothng. To address the former problem, Clne et al. [CTE05] desgned flters that ntroduce bas nto the calculaton, makng the mage darker than t should be. Our PMC-ER algorthm uses the same basc premse as ERPT: hgh-energy paths should be mutated to dstrbute the nformaton they carry to neghborng pxels. The sample populaton s a set of lght transport paths through the scene. The kernel functons mutate these paths to create new paths. The resamplng step removes low energy or well-dstrbuted paths, keeps hgh-energy paths and generates new paths to evenly explore regons and adapts the kernel functon for each populaton path. The work s focused on the mportant transport paths and correlated samplng of the ntegraton doman. In ths secton, we frst present an overvew of our two energy redstrbuton algorthms. The remanng of the secton s to explore the mplementaton detal needed for these two algorthms. 4.1. PMC-ER Equal Deposton Algorthm Fgure 2 shows the PMC-ER equal deposton algorthm. In the preprocess phase, the algorthm frst generates a pool of stratfed pxel postons used to explore the mage plane evenly. Ths pool of pxel postons s used to generate ntal populaton paths and to generate new stratfed replacement paths durng the resamplng process n each teraton n order to guarantee even exploraton of the mage plane. Then, the algorthm estmates the average energy contaned n the mage, Ẽ, and the deposton energy, e d, for each mutaton whch are dscussed n secton 4.3. An ntal populaton of paths are created by usng the path tracng alogrhtm, the rays of whch shoot from the camera and pass through the pxel poston, (x, y), selected from the stratfed pool. In ths work, a path, Ỹ, s referred to as a lght transport path startng from a lght, L, scatterng dffusely, D, or specularly, S, nsde the scene several tmes, and endng at the camera, E. The path s denoted as L(S D) E. Interested readers can refer to [Hec90, Vea97] for detal. Fgure 4 and 5 shows two examples of such paths. In each nner loop, we do Equal mutatons at each path n the populaton accordng to the path s kernel functon, K (s) (ỹ (t) Ỹ (t 1) ), dscussed n secton 4.4. After mutaton, the acceptablty probablty, A(Ỹ (t) Ỹ (t 1) ), s used to determne whether the path n the populaton swtches to the new generated path, Ỹ (t), or stays as the orgnal path, c The Eurographcs Assocaton and Blackwell Publshng 2007.
Y.-C. Yu-Ch & S.H. Fan & S. Chenney & C. Dyer / Photorealstc Image Renderng wth Populaton Monte Carlo Energy Redstrbuton Ỹ (t 1), before mutaton. Then, e d energy s deposted on the mage plane at the pxel poston of the new populaton path, Ỹ (t). In the outsde loop, the resamplng process whch s dscussed n secton 4.5 s to elmnate well-dstrbuted and lowcontrbuton paths, regenerate paths consderng the stratfcaton, and adapt the weghts for perturbatons wth dfferent raduses. 1 generate a pool of stratfed pxel poston 2 estmate the Ẽ,e d 3 generate ntal populaton of paths n t = 0 4 for s = 1,,T 5 determne α (s) for each perturbaton 6 for = 1,,n 7 f E,reman +U(0,1) > Ẽ 8 for t = 1,, mutatons 9 generate Ỹ (t) K (s) (ỹ (t) Ỹ (t 1) ) 10 Ỹ (t) = (U(0,1) < A(Ỹ (t) Ỹ (t) 11 depost e d energy on Ỹ (t) 12 E,reman = e d 13 w (t) = E,reman ))?Ỹ (t) : Ỹ (t 1) 14 resample the populaton: elmnaton and regeneraton Fgure 2: The PMC-ER equal deposton teraton loop. U(0, 1) generates a random number unformly dstrbuted between 0 and 1, and E,le ft s the energy left n the populaton path,, after the nnter energy redstrbuton loops. 4.2. PMC-ER Balance Energy Transfer Algorthm The PMC-ER balance energy transfer algorthm s presented n fgure 3. The frst step stll generates a pool of stratfed pxel postons. The next step s to generate an ntal populaton of paths. otce that there s no step to estmate the average energy contaned n the mage, Ẽ, and ths saves us a lttle computaton tme. In each nner loop, we do Balance perturbatons at each populaton path accordng to the kernel functon, K (s) (ỹ (t) Ỹ (t 1) ). After mutaton, E d = E,reman A(Ỹ (t) Ỹ (t 1) )/ Balance energy s deposted on the pxel poston of the newly mutated path and the same amount of energy s removed from the populaton path. The resamplng process s smlar to the PMC-ER equal deposton algorthm. 4.3. Energy Estmaton When applyng MCMC method, the count of samples fallng n each pxel s proportonal to the real energy.e. the llumnance of that pxel. Thus, we must estmate the energy contaned n the mage whch s 1 generate a pool of stratfed pxel poston 2 generate ntal populaton of paths n t = 0 3 for s = 1,,T 4 determne α (s) for each perturbaton 5 for = 1,,n 6 for t = 1,, Balance 7 generate Ỹ (t) K (s) (ỹ (t) Ỹ (t 1) ) 8 depost E d = E,reman A(Ỹ (t) Ỹ (t 1) )/ balance on Ỹ (t) 9 E,reman = E d 10 w (t) = E,reman 11 resample the populaton: elmnaton and regeneraton Fgure 3: The PMC-ER balance energy transfer teraton loop. E IP = Z L(ỹ) du(ỹ) (1) I where L(ỹ) s the llumnance deposted by the path, ỹ, on the mage plane.e. the llumnance of the radance, L(ỹ), transported from the lght to the camera. We can estmate the mage energy by computng the expected value of the mage energy, Ẽ, from a set of vald sample paths usng the followng two equatons: E(Ỹ) = L(Ỹ) p IP (Ỹ) = L(Ỹ) A IP (2) Ẽ = 1 E(Ỹ ) (3) =1 where E(Ỹ) s the mage energy estmated from a vald path, p IP (Ỹ) s the probablty for the path to pass through that specfc pxel poston, A IP s the physcal area of the flm. From Monte Carlo theory, we know that lm Ẽ = E IP. As a result, we can also estmate the deposton energy, e d, for the equal deposton algorthm whch s: e d = Ẽ mutatons (4) where mutatons s the expected total number of mutatons whch should be multplcaton of the total number of teratons, the total number of samples n the populaton, and equal. Wth ths value, the PMC-ER equal deposton algorthm can drectly render the fnal mage from the accumulaton of energy wthout the need to calbrate the total energy of the accumulaton mage. 4.4. The Kernel Functon for Each Path The kernel functon for each populaton path s a condtonal kernel, K (s) (ỹ (t) Ỹ (t 1) ), that generates a sample path n c The Eurographcs Assocaton and Blackwell Publshng 2007.
Y.-C. Yu-Ch & S.H. Fan & S. Chenney & C. Dyer / Photorealstc Image Renderng wth Populaton Monte Carlo Energy Redstrbuton Fgure 4: The top s a path wth the form of LDDSSE and used to demostrate the lens perturbaton. We would lke to replace the lens subpath y 5 y 4 y 3 y 2 y 1 of the form of ESSD. We frst perturb the pxel poston of the orgnal path at y 5 by unformly choosng a pont from the perturbng dsk and then cast a vew ray to pass through the new pxel poston as showed n the bottom to get y 4. We extend the subpath through the same specular bounces at y 4 and y 3 as the correspondng y 4 and y 3 to get y 2. Then, y 2 and y 1 are lnked to form a new lens-perturbed path wth the same form of LDDSSE as the orgnal one. teraton t, Ỹ (t), gven sample n teraton t 1, Ỹ (t 1) (see Fgure 2 and 3). we use a mxture dstrbuton: K (s) (ỹ (t) Ỹ (t 1) ) = α (s),d j T(ỹ (t) Ỹ (t 1) : d j ) (5) d j Each component, T(ỹ Ỹ : d), mutates an exstng path to generate a new one for exploraton of the path space accordng to the perturbng radus, d. Snce the ergocty of the algorthm s acheved by tracng paths at stratfed pxel postons, the mutaton s only used for local exploraton. Therefore, T(ỹ Ỹ : d) s only desgned to perform a perturbaton on the member path based on the perturbaton radus, d. Lens and caustc perturbaton are two good canddates for ths job. The followng s smple descrpton of these two mechansms: Lens perturbaton: Fgure 4 shows an example of lens perturbaton. The lens perturbaton s to replace a subpath y n 1 y k of the form EDS (L D). The perburbaton takes the exstng path and moves the mage pont whch t passes. In our case, the new pxel locaton s unformly sampled wthn a dsk of radus, d, a parameter of the kernel component. The remander of the path s reconstructed to pass through the new mage pont and extend the subpath through addtonal specular bounces to be the same length as the org- Fgure 5: The top s a path wth the form of LDSSDE and used to demostrate the caustc perturbaton. We would lke to replace the caustc subpath y 1 y 2 y 3 y 4 y 5 of the form DSSDE. At the head vertex of the caustc subpath, y 1, we perturbed the outgong lght ray drecton by an angle, θ, unformly sampled from [0,θ max] to get y 2 as showed n the bottom. We extend the subpath through the same specular bounces at y 2 and y 3 as the correspondng y 2andy 3 to get y 4. Then, y 4 and y 1 are lnked to form a new complete caustcs-perturbed path wth the same form of LDDSSE as the orgnal one. nal path. The transton probablty for lens pertubaton can be computed as T d,lens (Ỹ Ỹ) = G(y n 1,y n 2 ) A d n k 2 G(y j,y j+1 ) j=n 2 cosθ j,n where G(y j,y j+1 ) s the geometrc term between y j and y j+1, A d s the area of the perturbaton, and θ j,n s the angle between the normal of the surface and the drecton of the ncomng lght ray at y j. Caustc perturbaton Fgure 5 shows an example of caustc perturbaton. The caustc perturbaton s to replace a caustc subpath wth a suffx y m y k of the form (D L)S D + E. To do ths, we generate a new subpath startng from the vertex y m, the head vertex of the caustc subpath. The drecton of the segment y m y m+1 s perturbed by a random amount (θ, φ) unformly sampled from [0,θ max] and [0,2π] where the central axs, θ = 0, corresponds to the drecton of the orgnal ray and extend the subpath through addtonal specular bounces to be the same length as the orgnal one, and θ max s the range of samplng angle computed from correpsondng perturbaton radus, d, by the followng equaton from [Vea97]: θ max = θ(d) y n 1 y n 2 n 1 k=m y k y k 1 where θ(d) s the angle through whch the ray y n y n 1 needs to be perturbed to change the mage locaton by a (6) c The Eurographcs Assocaton and Blackwell Publshng 2007.
Y.-C. Yu-Ch & S.H. Fan & S. Chenney & C. Dyer / Photorealstc Image Renderng wth Populaton Monte Carlo Energy Redstrbuton dstance of d pxels. The transton probablty for caustc perturbaton can be computed as T d,caustcs (Ỹ Ỹ) = G(ym,y m 1) 2πθ max cosθ m,out m k 2 G(y j,y j+1 ) j=m 1 cosθ j,out where θ j,out s the angle between the normal of the surface and the drecton of the leavng lght ray at y j. In orgnal ERPT work, the sze of the pertubaton was a parameter to be fxed at startup. In PMC-ER, we can choose a reasonable set of dfferent szed perturbatons n the mxture whch s three n our case. The large pertubaton s effectve at redstrbutng nformaton over a wde area, whle the smallest s beneft for mage regons where llumnaton s changng quckly. When usng the kernel functon to perturb a path, we frst choose d accordng to the weghts, α (s), where d s the radus of the lens perturbaton and d j α(s),d = 1. And then ether j lens or caustc perturbaton s chosen accordng to γ lens = 0.1 and γ caustc = 0.9 n our case whch s set to prefer caustc perturbaton when t s possble. We can then perturb the current path to generate a new perturbed path. The acceptablty s to determne whether a path swtches to the newly generated path and calculated accordngly as follow: A(Ỹ Ỹ) = mn(1.0, f(ỹ)k(s) (Ỹ Ỹ) f(ỹ )K (s) ) (7) (Ỹ Ỹ ) where f(ỹ) s the path contrbuton defned n [VG97]. When evaluatng the acceptablty probablty, all possble proposals that mght generate Ỹ from Ỹ should be consdered whch s: K (s) (Ỹ Ỹ) = d j α (s),d j ( γ lens T d j,lens(ỹ Ỹ) (8) + γ caustc T d j,caustc(ỹ Ỹ)) However, t s also acceptable to consder only the functon derved from the proposal strategy chosen to generate Ỹ [Te98]: K (s) (Ỹ Ỹ) = T d j, op type(ỹ Ỹ) (9) In ths work, we use Equaton 9 to avod the computaton of other possble transton functons to mprove the effcency of mutaton. 4.5. Resamplng The resamplng step n ths algorthm acheves three purposes: t carres forward to next round samples that have hgh energy remanng wthout flowng out, t provdes an opportunty to add some completely new paths nto the populaton for evenly explorng the mage space, and the nformaton about whch perturbatons are chosen nsde the nner loop gudes the adapton of the kernel functons. The followng decrbes these three steps n detal: Elmnaton: Ths step s to elmnate well-explored and lowcontrbuton samples from the popluaton. When we generate a new populaton path, the energy of the path, E(Ỹ), s computed usng the equaton 3 and set t to E reman. After each perturbaton, we reduce the energy remanng n the path by e d for the equal energy deposton method and by the amount of energy flowng out, E reman A(Ỹ (t) Ỹ (t 1) ) for the balance energy transfer method. The probablty of the paths survvng n the elmnaton process s proportonal to the energy remanng n the path, E reman. Regeneraton: Regeneraton s to mantan the constant number of paths n the populaton. It also gves us the chance to decde where we would lke to explore n the next teratons. For achevng unbasedness, we need to evenly explore the mage plane. Thus, the regeneraton of new paths s accordng to the crtera of stratfcaton. In the preprocess phase, we compute the total stratfed number of pxel postons needed for the entre process. Then a pool of stratfed pxel postons s generated accordng to that number. Durng the regeneraton process, we keep askng the pool to gve us the next unused stratfed pxel poston. A new path s generated by tracng through the new pxel poston wth the path tracng algorthm and the energy of the path, E(Ỹ), s computed usng the equaton 3 and set t to E reman. Adapt α s Values The purpose of α s valuse s to choose a proper perturbaton radus for decdng the area of exploraton accordng to the successes of the perturbatons. Thus, when a new path s generated, the α (s),k s set to be a constant probablty for each component, whch allows us to unformly choose all perturbtatons. After ntalzaton, each perturbaton acceptablty was tagged wth the kernel mxture component that generated t and the ndex of the path n the populaton. At the adaptaton step, we computate the accumulaton of the acceptablty probabltes tagged wth k-th component for each member path and uses t to adjust the mxture probabltes. We can then set: α,k α (s),k = A (t) d j (Ỹ (t) Ỹ (t 1) )δ j,k = ε+ (1 ε)α,k n k =1 α,k where δ j,k = 1 f d k s chosen as the radus of perturbaton c The Eurographcs Assocaton and Blackwell Publshng 2007.
Y.-C. Yu-Ch & S.H. Fan & S. Chenney & C. Dyer / Photorealstc Image Renderng wth Populaton Monte Carlo Energy Redstrbuton If the e d s too small, the algorthm becomes too slow and neffcent but t converges to smooth results. However, f e d s too large, the algorthm generates brght spots because a path must have hgh energy to pass the dstrbuton crtera to run a MC muaton chan, whch dstrbutes ts energy. However, most paths fal to reach the crtera. In addton, the perturbaton radus affects the area where the energy can be dffused to and the success rate of the dffuse operaton. In the smooth lghtng area, we hope that ths radus s large, n order to get a smooth mage as soon as possble. However, n complex lghtng areas such shadow, caustc regons, we hope that t s small or the rate of success declnes largely. Our algorthm automatcally adjusts these two aspects through the process of resamplng and adaptng α s values. Fgure 7: A dragon scene computed usng our PMC-ER equal deposton at the top. The bottom left s the zoom-n of the caustc part computed by PMC-ER equal deposton and the bottom rght s the same part computed by ERPT. PMC-ER has fewer artfacts overall. By sharng more nformaton among paths and by better reusng the hgh contrbuton paths, PMC-ER s an mprovement over ERPT. n step j,.e. j = k 5. Results The results from the PMC-ER balance energy transfer algorthm show that although we can mprove the brght spots caused by the energy remanng n the orgnal path by keepng the energy that fals to be dstrbuted n the path tself for further exploraton at the next teraton, we realze that when fndng a hgh-energy path, the energy beng dstrbuted out at the very frst step s large comparng to the energy beng dstrbuted out n the followng teratons. Ths causes hgh varance, whch s showed as a brght spot, n the fnal result. Ths motvate us to develop the PMC-ER equal deposton algorthm. Thus, the results demostrated n ths secton are generated from the PMC-ER equal deposton algorthm. We observe that the deposton energy, e d, and perturbaton raduses are two mportant factors for ERPT algorthm. We compared our PMC-ER equal deposton algorthm wth the energy redstrbuton path tracng (ERPT) algorthm on the Cornell Box scene, a dragon scene, and a complex room scene usng the crtera of startng wth a smlar number of ntal PT paths. In all three cases we used a populaton sze of 5000. There are three pertubaton raduses: 5, 10, and 50 pxels, respectvely. The caustc perturbaton s computed wth Eqn. 6. In each step nsde the nner loop, each member generates 16 mutatons, and 40% of the populaton s elmnated based on ts remanng energy and regenerated usng the stratfcaton mechansm. We also use 4 spps for estmatng the energy contaned n an mage for both PMC-ER and ERPT algorthms. The Cornell Box mage (Fgure 6) s rendered usng our PMC-ER equal deposton algorthm wth 1000 teratons whch roughly has the same total number of ntal PT paths as the mage rendered usng the ERPT wth 8 spps. We can see that our algorthm removes the brght spot artfacts from ERPT algorthm. When we compare our result wth an mage rendered wth ERPT wth 16 spps, our mage get fewer artfacts. Observng the strategy mage whose brghtness shows the perturbaton count, we see that the probablty of paths stayng n the populaton for next teraton are s proporton to ts energy remanng. In other words, regons such as the caustc area contaned more hgh energy paths get more number of mutatons. In addton, the radus of mutaton near physcal borders and lghtng borders such as, the shadow and caustc area and the lght edge, automatcally adjusts to ncrease the success rate of flowng energy out. However, generally, the average tme for paths stayng n the populaton s short. Thus, our algorthm cannot have enough tme to adjust to the shortest radus at ths area. We can only observe a yellow color around the edge nstead of a red color for the edge. PMC-ER acheves a vsually more converged mage compared to the correspondng mage generated by the ERPT algorthm wth the same number of ntal PT paths. The dragon scene (Fgure 7) was rendered at 900 900 wth 12800 teratons and 20 mutatons for each member n the populaton nsde the loop n comparson wth mage renc The Eurographcs Assocaton and Blackwell Publshng 2007.
Y.-C. Yu-Ch & S.H. Fan & S. Chenney & C. Dyer / Photorealstc Image Renderng wth Populaton Monte Carlo Energy Redstrbuton Fgure 6: The frst mage on the left s a Cornell Box mage computed usng PMC-ER equal deposton algorthm; the second mage s computed usng ERPT wth 9 spps; the thrd mage s computed usng ERPT wth 16 spps; and the fourth mage s the mutaton strategy used durng the process. The strategy mage shows that the mutaton near the physcal border and lghtng border wll automatcally adjust to ncrease the success rate of transferng mage. Fgure 8: A room scene computed usng our PMC-ER equal deposton at the left and ERPT at the rght. PMC-ER has fewer artfacts overall. By sharng more nformaton among paths and by better reusng the hgh contrbuton paths, PMC-ER s an mprovement over ERPT. dered usng ERPT wth 32 spps and 20 mutatons to each ntal PT path. We can see that mage rendered usng PMC-ER has fewer artfacts than the mage rendered usng ERPT. The room scene (Fgure 8) was rendered at 720 405 wth 19200 teratons and 20 mutatons for each member n the populaton nsde the loop n comparson wth mage rendered usng ERPT wth 128 spps and 20 mutatons to each ntal PT path. We can see that mage rendered usng PMC- ER has fewer artfacts than the mage rendered usng ERPT. ote that for all PMC-ER equal deposton and ERPT mplementatons, we dd not use the flter proposed n the orgnal ERPT paper to smooth the fnal mage. The statstcs for three rendered mages s presented n Table 1. We use the mean squared effcency (Eff) metrc for comparng algorthms, computed as: Err = pxels e 2 1, Eff = pxels T Err where e s the dfference n ntensty between a pxel, the ground truth value, T s the runnng tme of the algorthm on that mage and pxels s the overal pxel count. Eff s a measure of how much longer (or less) you would need to run one algorthm to reach the qualty of another [PH04]. We Image Method Tme (s) Err Eff Box1 ERPT(8) 4401.8 0.85 2.7e-4 ERPT(16) 8935.7 0.526 2.1e-4 PMC-ER 5281.2 0.37 5.4e-4 Dragon ERPT(32) 88596.1 1.13 1.0e-5 PMC-ER 97455.7 0.46 2.3e-5 Room ERPT(128) 82656.5 0.052 2.3e-4 PMC-ER 96575.1 0.010 1e-3 Table 1: Measurements comparng energy redstrbuton path tracng (ERPT) wth PMC-ER, for a roughly equal number of sample rays. can see that our algorthm gets better effcency than ERPT algorthm does. 6. Dscusson The most mportant varable parameter n our algorthms s the resample rate. A small resample rate reduces the number of samples kept n the populaton, whch results n a faster exploraton of the sample doman but at the cost of a large amount of teraton nformaton beng lost durng the regeneraton process. On the other hand, a larger resample rate c The Eurographcs Assocaton and Blackwell Publshng 2007.
Y.-C. Yu-Ch & S.H. Fan & S. Chenney & C. Dyer / Photorealstc Image Renderng wth Populaton Monte Carlo Energy Redstrbuton means that more teraton nformaton related to paths s kept durng the teraton. However, the rate to explore the entre sample doman s slow. Many PMC kernels n the lterature are mxture models. Mxtures are typcally formed by combnng several components that are each expected to be useful n some cases but not others. The adapton step then determnes whch component are useful for a gven nput. Mxtures allow otherwse unrelated functons to be combned, such as the perturbaton wth dfferent szed raduses. We would prefer the kernel functon havng many components. However, when the kernel functon contans many adaptable parameters, each teraton would requres hgh adaptve sample counts for gatherng proper nformaton to adapt the kernel functon. Ths prevents us from usng a larger number of dfferent perturbng raduses. Such a strategy would be appealng for effcently renderng a scene wth geometres havng very dfferent szes appearng on the mage plane, but the adaptve sample count requred to adequately determne the mxture component weghts would be too large. Instead we use three perturbaton raduses for all mages rendered. 7. Concluson A new global llumnaton algorthm, PMC-ER, s presented by applyng PMC framework to energy redstrbuton algorthms. PMC-ER learns to become an effectve sampler based on the nformaton collected from early teratons. The algorthm automatcally explores the mportant lght paths found n the prevous teraton, adjusts the area of exploraton accordng to results of prevous muatons, and also uses resamplng to acheve ergocty. There are several future research drectons. The PMC-ER should be able to use the perceptual varance as regeneraton crtera to focus on the hgh perceptual varance area. However, the energy brought by a varance path generated ths way should also be adjusted accordngly. Also, how to dentfy a varance comng from an artfact of renderng not from physcal and lghtng dscontnuty s another queston. In addton, all paths ntalzed the α s values to a constant value. However, we can record the alpha used prevously n an mage because spacal correlaton wll gve us smlar α s values n most places n the mage plane. We can reuse the α nformaton to reduce the process of probng to estmate a proper set of α s values. Based on the framework of Populaton Monte Carlo, PMC-ER can mprove the renderng effcency. PMC should be able to provde further research opportuntes for global llumnaton communty. References [APSS04] ASHIKHMI M., PREMOZE S., SHIRLEY P., SMITS B.: A varance analyss of the metropols lght transport algorthm. Computer and Graphcs 25, 2 (2004), 287 294. [CGMR04] CAPPÉ O., GUILLI A., MARI J.-M., ROBERT C.: Populaton Monte Carlo. Journal of Computatonal and Graphcal Statstcs 13, 4 (2004), 907 929. [CTE05] CLIE D., TALBOT J., EGBERT P.: Energy redstrbuton path tracng. In SIGGRAPH 05 (2005), pp. 1186 1195. [DGMR05a] DOUC R., GUILLI A., MARI J. M., ROBERT C. P.: Convergence of adaptve samplng schemes. Techncal Report 2005-6, Unversty Pars Dauphne, 2005. [DGMR05b] DOUC R., GUILLI A., MARI J. M., ROBERT C. P.: Mnmum varance mportance samplng va populaton Monte Carlo. Techncal report, Unversty Pars Dauphne, 2005. [FCL05] FA S., CHEEY S., LAI Y.: Metropols photon samplng wth optonal user gudance. In Proc. of the 16th Eurographcs Symposum on Renderng (2005), Eurographcs Assocaton, pp. 127 138. [GDH06] GHOSH A., DOUCET A., HEIDRICH W.: Sequental samplng for dynamc envronment map llumnaton. In Proc. Eurographcs Symposum on Renderng (2006), pp. 115 126. [Hec90] HECKBERT P. S.: Adaptve radosty textures for bdrectonal ray tracng. In SIGGRAPH 90 (1990), pp. 145 154. [Jen01] JESE H. W.: Realstc mage synthess usng photon mappng. AK Peters. [Kaj86] KAJIYA J. T.: The renderng equaton. In SIG- GRAPH 86 (1986), pp. 143 150. [KSKAC02] KELEME C., SZIRMAY-KALOS L., A- TAL G., CSOKA F.: A smple and robust mutaton strategy for the metropols lght transport algorthm. vol. 21, pp. 531 540. [LW93] LAFORTUE E. P., WILLEMS Y. D.: Bdrectonal path tracng. In Proceedngs of Compugraphcs (1993), pp. 145 153. [PH04] PHARR M., HUMPHREYS G.: Physcally Based Renderng from Theory to Implementaton. Morgan Kaufmann, 2004. [PKK00] PAULY M., KOLLIG T., KELLER A.: Metropols lght transport for partcpatng meda. In Proc. of the 11th Eurographcs Symposum on Renderng (2000), Eurographcs Assocaton, pp. 11 22. [SKDP99] SZIRMAY-KALOS L., DORBACH P., PUR- GATHOFER W.: On the Start-up Bas Problem of Metropols Samplng. Techncal report, Unv.of Plzen, 1999. [Te98] TIEREY L.: A note on Metropols-Hastngs kernels for general state spaces. The Annals of Appled Probablty 8, 1 (1998), 1 9. c The Eurographcs Assocaton and Blackwell Publshng 2007.
Y.-C. Yu-Ch & S.H. Fan & S. Chenney & C. Dyer / Photorealstc Image Renderng wth Populaton Monte Carlo Energy Redstrbuton [Vea97] VEACH E.: Robust Monte Carlo Methods for Lght Transport Smulaton. PhD thess, Stanford Unversty, 1997. [VG94] VEACH E., GUIBAS L. J.: Bdrectonal estmators for lght transport. In Proc. of the 5th Eurographcs Workshop on Renderng (1994), Eurographcs Assocaton, pp. 147 162. [VG97] VEACH E., GUIBAS L. J.: Metropols lght transport. In SIGGRAPH 97 (1997), pp. 65 76. [WRC88] WARD G. J., RUBISTEI F. M., CLEAR R. D.: A ray tracng soluton for dffuse nterreflecton. In SIGGRAPH 88 (1988), pp. 85 92. c The Eurographcs Assocaton and Blackwell Publshng 2007.