Computer Sciences Department

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1 Computer Scences Department Populaton Monte Carlo Path Tracng Yu-Ch La Charles Dyer Techncal Report #1614 September 2007

2 Populaton Monte Carlo Path Tracng Yu-Ch La Unversty of Wsconsn at Madson Graphcs-Vson Lab 1210 W. Dayton St., Madson, WI53706 Charles Dyer Unversty of Wsconsn at Madson Graphcs-Vson Lab 1210 W. Dayton St., Madson, WI53706 Abstract We present a novel global llumnaton algorthm whch dstrbutes more mage samples on regons wth perceptually hgh varance. Our algorthm terates on a populaton of pxel postons used to estmate the ntensty of each pxel n the mage. A member kernel functon, whch automatcally adapts to approxmate the target dtrbuton by usng the nformaton collected n prevous teratons, s responsble for proposng a new sample poston from the current one durng the mutaton process. The kernel functon s desgned to explore a proper area around the populaton sample to reduce the local varance. The resamplng process elmnates samples located n the low-varance or well-explored regons and generates new samples to acheve ergocty. New samples are generated by consderng two factors: the perceptual varance and the stratfcaton of the sample dstrbutons on the mage plane. Our results show that the vsual qualty of the rendered mage can be mproved by explorng the correlated nformaton among mage samples. 1. Introducton To generate mages that are close to realty has become more and more mportant n several dfferent applcatons. Monte Carlo (MC) ntegratons provde us a general soluton to solve ntegraton problems nvolved n renderng. However, the effcency of Monte Carlo methods s stll the man concern when applyng t n practce. Generally, dfferent regons may requre dfferent numbers of samples to render a converged mage. In standard MC estmators, pxel samples are unformly and evenly dstrbuted on the mage plane. Ths s neffcent because low-varance regons only need a small number of samples, and hgh-varance regons may need a large number of samples n order to generate a converged mage. As a result, n order to guarantee generatng a converged mage, MC needs to use a large number of samples among all regons even n low-varance regons. If we can shft those extra samples n low-varance regons to hgh-varance regons, we can mprove the renderng qualty. Our Populaton Monte Carlo path tracng (PMC-PT) algorthm automatcally dstrbute more mage samples to explores hgh varance regons. Our algorthm terates on a populaton of pxel postons on the mage plane. The ntal populaton of samples are evenly dstrbuted on the mage plane. Any nformaton avalable n the prevous teratons can be used to adapt member kernel functons that produce a new populaton based on the current populaton. The resamplng process elmnates part of the populaton samples and regenerates new samples to acheve ergocty. We carefully desgn the resamplng process to elmnate the well-explored samples from the current populaton and to generate new samples by consderng two factors: the perceptually-weghted varance among the samples n each pxel and the need to stratefedly explore the mage plane. As a result, new regenerated samples are desgned to locate n the perceptually mportant areas or to dstrbute on the mage plane n an even manner. The procedure s then terated: sample, terate, resample, adapt, terate, resample.... The result s a self-tunng unbased algorthm whch can locally explore the mportant vsual areas on the mage plane. All pxel postons generated by mutaton and regeneraton are used to estmate the ntensty of each pxel by usng a general MC ray tracng algorthm such as path tracng and bdrectonal path tracng n order to generate an mage. In our mplementaton, we use a general path tracng algorthm. Our contrbuton s a new renderng algorthm, PMC Path Tracng(PMC-PT), based on the PMC framework. Ths algorthm adapts the kernel functons to determne the radus of local exploraton wth the nformaton collected n prevous teratons. In addton, the resamplng process dstrbutes the new samples over the entre mage plane accordng to the perceptual mportance and stratfcaton to acheve ergocty. Samples kept durng the elmnaton process are located n regons wth hgh varance.

3 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 present the PMC-PT 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 algorthms. 2 Related Work Currently, most global llumnaton algorthms are based on ray tracng and Monte Carlo ntegraton. There exst two categores: unbased methods such as [?,?,?]; and based methods such as [?,?,?]. Interested readers can refer to Pharr and Humphreys [?] for an overvew of Monte Carlo renderng algorthms. Here we focus on three specfc areas related to our work: adaptve mage-plane samplng, perceptual metrcs, and sample reuse. Typcally, adaptve mage-plane algorthms ntally render the mage wth a small number of samples per pxel. The ntal mage s analyzed to label pxels as adaquately sampled or n need of further refnement. Then, the algorthms terate on pxels requrng more samples [?,?,?,?,?,?]. However, the label on the pxels based on an ntal samples ntroduces bas [?] nto the fnal result, whch s a problem when physcally accurate renderngs are requred. We carefully desgn the sample dstrbuton probablty wth respect to the perceptual mportance and at the same tme, avod bas durng the resamplng process. Many metrcs have been proposed for the test to trgger addtonal samplng. Lee et al. [?] used a sample varance based metrc. Dppé and Wold [?] estmated the change n error as sample counts ncrease. Panter and Sloan [?] and Purgathofer [?] used a confdence nterval test, whch Tamstorf and Jensen [?] extended to account for the tone operator. Mtchell [?] proposed a contrast based crtera because humans are more senstve to contrast than to absolute brghtness, and Schlck [?] ncluded stratfcaton nto an algorthm that used contrast as ts metrc. Boln and Meyer [?], Ramasubramanan et al. [?] and Farruga and Péroche [?] used models for human vsual percepton, of whch we use a varant. Most recently, Rgau et al. [?,?] ntroduced entropy-based metrcs. Our PMC-PT algorthm uses the adaptaton of the member kernel functon to locally explore perceptual mportant regons and uses resamplng to acheve ergocty and exploraton of hgh percetual varance regons. In addton, t s unbased. 3 D-Kernel Populaton Monte Carlo The Populaton Monte Carlo algorthm [?] s an adaptve algorthm that calbrates the proposed dstrbuton to the target dstrbuton at teraton by learnng from the performance of the prevous proposal dstrbutons. The generc D-Kernel PMC samplng algorthm [?] whch s an evoluton of PMC. Our algorthm, an adaptaton of the generc D-Kernel PMC algorthm, s stated n Fgure 1. Our algorthm adapts the kernel functon for each populaton path nstead of a sngle kernel functon for the entre populaton. 1 generate the ntal populaton, t = 0 2 for t = 1,,T 3 adapt K (t) (x (t) X (t 1) ) 4 for = 1,,N 5 generate X (t) 6 w (t) = π(x (t) (x (t) X (t 1) ) )/K (t) (x (t) K (t) X (t 1) ) 7 resamplng process: elmnaton and regeneraton Fgure 1. The generc D-Kernel Populaton Monte Carlo algorhtm. { Assume we have } a populaton of samples denoted by X (t) 1,...,X(t) N, where t s the teraton number and N s the populaton sze, and we wsh to sample accordng to the target dstrbuton, π(x), where s f(x) = π(x)h(x) n order to evaluate D f(x)dx. We start the algorthm by creatng the ntal populaton wth any method that can generate these samples provded that any sample wth non-zero probablty under π(x) can be generated, and the probablty of dong so s known. The outer loop s responsble for adaptng a member kernel functon, K (t) (x (t) X (t 1) ), for each member n the populaton, (lne 3) usng nformaton from the prevous teratons. The kernel fucton s used to generate a new populaton sample, gven the current one. The nner loop takes an exstng sample, X (t 1), as nput and produces a canddate new sample, X (t), as output (lne 5). The resamplng step n lne 7 conssts of two 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 can be computed and s unbased for π(h): f(x) = π(h) = 1 N N =1 w (t) h(x (t) ) (1) Before applyng PMC to renderng problems, several decsons must be made: Decde the samplng doman and populaton sze. Defne kernel functons and ther adapton crtera. 2

4 Choose the technques for samplng from the kernel functons and resamplng step. The followng sectons descrbe the applcaton of ths framework by mutatng the general path tracng algorthm. Ths algorthm uses kernel functons wth metrcs to accumulate, elmnate, and regenerate samples. Then, we conclude wth a general dscusson on PMC for renderng problems. 4 Populaton Monte Carlo Path Tracng (PMC-PT) To render an mage, the ntensty, I,j, of each pxel must be computed usng Equaton 2. MC } n Equaton 3 uses a set of path samples, { X 1,..., X N, sampled from an mportance functon p( x) to estmates the ntensty, I,j. I,j = Î,j = 1 n = 1 n I W,j ( X)L( X)du( X) (2) N W,j ( X k )L( X k ) k=1 p( X k ) N W,j ( X k )E[L( X k )] (3) k=1 where I s the mage plane, W,j ( X) s the measurement functon for pxel (, j) non-zero f the lens edge, x n 2 x n 1 passes through the support of the reconstructon flter at (, j) where n s the total number of vertces n the path and L( X) s the radance brngng on the path, X. MC can prove that lm N Î,j = I,j. The only dfference among all pxels s the term of W,j ( X k ). Provded p( X) s known n each path whch passes through the vald mage plane, the global nature of p( X) s not mportant. Thus, the renderng equaton can be transformed to evaluate the expected radance, E[L( X)], carred by each sampled path and then accumulate ths radance n pxels whose support of reconstructon ncludes the path-passng pxel poston. At the fnal step of renderng, the accumulaton n each pxel s averaged by the total number of samples dropped n the support of the pxel. Therefore, the unbasedness can be acheved under two condtons: frst, the estmator of expected path radance s unbased; and second, there are nfnte samples fallng n the support of each pxel s recontructon flter as the total number of samples goes to nfnty. When applyng Equaton 3 to render an mage, the sample dstrbuton on the mage plane s unform. However, the complexty of lghtng vares from regon to regon on the mage. Thus, each regon requres dffrent number of samples to acheve a converged values. It s neffcent to put a large number of samples on regons wth low varance due to the need of regons wth hgh varance. PMC-PT s desgned to dstrbute more samples for further exploraton of regons wth hgh varance. The varance of the llumnance among a populaton sample and ts kernel-proposed descendants s used to determne the need of exploraton. The hgher the varance s, the hgher the need of samples s. In ths secton, we frst dscuss the algorthm tself followng wth the detaled dscusson of the adaptaton and resamplng process. 4.1 PMC-PT Algorthm Fgure 2 shows the steps usng PMC-PT algorthm to render an mage. In the preprocess phase, the algorthm frst generates a pool of stratfed pxel postons used to dstrbute the populaton samples evenly on the mage plane. Ths pool s asked to gve a populaton of ntal samples and to generate new stratfed replacement samples durng the resamplng process n each teraton n order to guarantee that every pxel has the chance to be explored. 1 generate a pool of stratfed pxel postons 2 generate ntal populaton of samples n t = 0 3 for s = 1,,T 4 determne α (s) 5 for = 1,,n 6 for t = 1,,T R 7 generate X (t) 8 Compute w (s) K (s) = σ 2/N used (x (t) X (t 1) ) 9 resample: elmnaton and regeneraton 10 adapt β (s) x,y Fgure 2. The PMC-PT teraton loop. T R s the number of kernel teratons per resample step. σ 2 computes the varance of the th populaton sample and all mutated descendent samples after t has been regenerated and ths value s used to calculate the weght, w (t), for elmnaton, and N used s the number of resamplng loop ths sample has been used snce t has been regenerated. The resamplng step n lne 9 s desgned to cull canddate samples located n perceptually low-varance regons and keep samples located n perceputally hgh-varance regons. It takes the canddate populaton, { X (s) 1,...,X(s) n }, and produces a new populaton ready for the next teraton. The kernel adapton (lnes 4 and 10) need not be done on 3

5 every teraton. Our examples demonstrate such cases. After explorng several values for T R, we found a wde range of values to be effectve. The optmal value depends on the populaton sze and the relatve cost of kernel perturbatons compared to resamplng. The am of addng new samples n the resamplng process s to elmnate the possblty of overexplorng a few regons wth very hgh varance durng the mutatng process n order to guarantee the unbasedness of our algorthm. Addng new samples n ths way does not add bas, because nether the mutated populaton nor the new samples are based, so ther unon s not based. After decdng the new pxel poston ether n regeneraton or n mutaton process, we use a path tracng algorthm to evaluate the expected radance along the ray from eye to the pxel poston, I L = E[L(PT(X)] where PT(X) generate a vald path, X, passng through the pxel poston, X, by usng the path tracng algorthm. We only needs the probablty, p(p T(X)), to evaluate the expected path radance, I L, T(x X : d) s not mportant and s only used to dstrbute the samples. 4.2 Kernel Functons The kernel functon for each populaton member s a condtonal kernel, K (s) (x (t) X (t 1) ), that generates a sample poston n teraton t, X (t), gven sample n teraton t 1, X (t 1). we use a mxture dstrbuton: K (s) (x (t) X (t 1) ) = d j α (s),d j T(x (t) X (t 1) : d j ) (4) Each component, T(x X : d), mutates an exstng sample to generate a new one for exploraton of the mage space accordng to the perturbng radus, d. There s a set of perturbng raduses, d, gven as parameters to the algorthm and each s good for dfferent occasons. α s values are used to choose a radus for the current sample path and ther values are adapted through teratons for each path. Once we choose a d, the perburbaton takes the exstng pxel poston and moves to a new pxel postons unformly sampled wthn a dsk of radus, d, a parameter of the kernel component. 4.3 Resamplng The resamplng step n ths algorthm acheves three purposes: samples that locate n regons wth hgher varance are carred forward to the next round, t provdes an opportunty to add some completely new samples nto the populaton for explorng unexplored or perceptually hgh-varance regons, and the nformaton about whch perturbatons are chosen nsde the nner loop gudes the adapton of the kernel functons. The followng sectons gve the detal of three steps Elmnate samples from the populaton The weght, w (s), for each sample s used to determne the probablty of survval of the path durng the elmnaton phase. At each nner loop, we tag the expected radance computed by the PT algorthm, I (t) L, for each member and then n elmnaton process, we evaluate the varance of these llumnances, σ 2. The hgher varance a member has, the hgher the chance t s kept n the populaton. Ths can acheve the goal of contnung to explore hgh-varance regons. However, we would lke to avod over-explorng the samples located n hgh-varance regons n order to gve more chance for other samples. Thus, we wegh the varance down by the total teratons whch t has beng used for perturbatons. Thus, the fnal weght s w (s) = σ 2/N used Regenerate new samples nto the populaton Regeneraton s to mantan the constant number of samples n the populaton. It also gves us the chance to decde where we would lke to explore n the next teratons. Our algorthm consders two aspects: the stratfcaton of pxel postons and the perceptual varance of the ntermedate result. 1. Stratfcaton: Pharr [?] demostrates how mportant samplng evenly on the mage plane s n reducng the varance. Thus, n 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. Ths s also used to guarantee that every pxel gan ts chance to be explored to acheve the second requrement of unbasedness. 2. Perceptual varance: In order to generate new sample paths n regons wth hgh perceptual varance. We use the value β (s),j to ndcate the degree of requrement for more samples at pxel (, j). Pxels that requre further exploraton should have hgher β (s),j. An approprate crtera assgns β (t),j proportonal to an estmate of the perceptually-weghted varance at each pxel. The algorthm tracks the sample varance n llumnance 4

6 seen among samples that contrbute to each pxel. To account for percepton, the result s dvded by the threshold-versus-ntensty functon tv(i) ntroduced by Ferweda et al. [?]. β,j = β (s),j = σ 2,j tv(i,j ) β,j,j IP β (,j ) To get a pxel poston, we frst choose a pxel,(, j) accordng to the weght of β (s),j and then perturb the poston by one pxel dstance n each drecton Adapt α s Values to Propose a New Path When explorng a regon wth lttle change n lghtng such as the dffuse wall far from a lght, we would lke to expand the explorng area,.e. have hgher probablty to choose a larger perturbaton radus. On the other hand, when explorng a regon wth complex change n lghtng, such as a glossy surface wth a lght located near the reflecton glare drecton, we would lke to shrnk the explorng area, that s, have a hgher probablty to choose a smaller perturbaton radus. When a new sample s generated n the regeneraton process, the α (s),k s set as a constant probablty for each component whch allows us to unformly choose any of the perturbatons. Snce the goal s to decde the exploraton accordng to the lghtng detal. After ntalzaton, the expected llumnance, I (t) L, of each new mutated path was tagged wth the kernel mxture component that generated t and ts ndex n the populaton,. At adaptaton step, we uses the tagged llumnance to evaluate the perceptual varance for each component of each member n the populaton to adjust the α s values accordng to: α,k = α (s),k { 0 f σ,k = 0 f σ,k 0 1 σ 2,k = ǫ + (1 ǫ)α,k n k =1 α (,k ) where σ 2,k s the varance of a set of llumnances tagged wth the k-th mxture component for -th member n the populaton. 5 Results We compared PMC-PT wth the path tracng algorthm on two Cornell Box scenes and a room scene wth roughly Fgure 3. A Cornell Box mage computed usng PMC-PT wth 250 teratons on the left, and the mage represents the mutaton strategy (red represents perturbaton of radus 5 pxels, green represents pertubaton of radus 10, and blue represents the perturbaton of 50 pxels) used durng the process on the rght n the frst row. To compute a converged value at the caustc part s dffcult. Hence, we show the cropped mage of the caustc regon. The left mage n the second row s the cropped mage of the mage rendered by our algorthm. The rght mage s the cropped mage of an mage computed wth a PT algorthm wth 64 spps. The strategy mage shows that our algorthm wll adjust the perturbed radus accordng to the lghtng change and physcal edges. The mage also shows that our algorthm wll put more samples on the hgh perceptual varance regons such as the lght s edge and the caustc and shadow regons under the glass ball. the same number of rays shootng out from the camera. One Cornell Box scene s wth a glass ball and other surfaces such as walls, beng Lambertan; and the other Cornell Box scene s wth a glass ball n the front, a mrror ball n the back, one mrror surfaces on the rght sde of the box, and the remanng surfaces beng Lambertan. The room scene contans several complex objects and a glossy table to demostrate the usage of the algorthm for a complex scene. In all three cases we used a populaton sze of There are three pertubaton raduses: 5, 10, and 50 pxels, respectvely. In each step nsde the nner loop, each member generates 16 resamplng perturbatons, and 40% of the populaton s elmnated and regenerated. 50% of regenerated samples are created usng the stratfcaton mechansm, and 50% are generated usng the perceptual varance mech- 5

7 Fgure 4. Another Cornell Box mage wth complex pattern of specular lght transport paths. The left mage s computed usng PMC-PT wth 1000 teratons; the mddle mage s generated usng a PT algorthm by 256 spps; the rght mage represents the mutaton strategy used n our PMC-PT algorthms. Generally, our algorthm gets a better performance at the caustc and shadow regons. The strategy mage shows that our algorthm almost evenly explores all caustc and shadow regons, and t adjusts to shorter raduses whch s showed as a yellow color but the paths do not stay n the populaton to adjust to the shortest radus whch s showed as a red color lke the result of Fgure 3 ansm. The Cornell box scenes were rendered at resoluton and the room scene were rendered at resoluton. The frst Cornell Box scene s rendered wth 250 teratons and the α s and β s values are updated every 100 teratons. The second one s rendered wth 1000 teratons and the α s and β s values are updated every 200 teratons. The room scene s rendered wth 3400 teratons and the α s and β s values are updated every 200 teratons. The mages (Fgure 3) demonstrate that PMC-PT expends more effort on the dffcult regons the regon n the celng near the lght, the caustc and shadow regon under the glass ball and hence has a lower varance n those regons, at the expense of a slghtly hgher varance n other parts of the mage. Ths s a recurrng property of the PMC- PT algorthm: PMC produces a more even dstrbuton of nose, wth lower nose levels overall but hgher n some parts of the mage that are over-sampled wth non-adaptve technques. The strategy mage shows that our adaptaton strategy automatcally adjusts the perturbed radus near the hghly changng lghtng boundary such as the caustc and shadow regon or the physcal edges, such as the corners. These regons also generate hgher perceptual varance. As a result, the algorthm puts more samples on these regons showed as brghter llumnace n the strategy mage. The overall result s a more perceptually pleasant mage compared to the correspondng mage rendered by the PT algorthm usng 64 spps, whch s roughly the same total number of sample paths shootng from the eye. The second Cornell Box contans complex specular tranport paths and has several caustc regons on the mage. The mages (Fgure 4) demonstrate that PMC-PT evenly explores these dffcult regons by dstrbutng more samples on these areas. However, the drect lghtng change areas, such as the caustc regons under the glass ball and the regon n the celng near the lght, have hgher perceptual varance than ther correspondng reflecton and get more samples. Snce hgh proporton of the populaton has a hgh perceptual varance, there s a smaller chance to stay n the vald populaton to adapt to the shortest radus whch s showed n the strategy mage as a red color. However, our algorthms stll adjust the perturbed radus to the shortest two of the three whch s showed n the strategy mage as a yellow color, whch s a combnaton of red and green. The prmnary lghtng regons are an excepton because they have a hgher perceptual varance than the rest of the reflectng hgh varance regons. The paths stll stay longer and have chance to adjust to the shortest radus whch can be observed at the edge of the specularty of the mrror ball. Our algorthm mproves the mage qualty at the shadow and caustc regons, the reflectons of these regons and even the dffuse wall n the back. The strategy mage shows that there are more hgh varance regons, and thus the samples are more evenly dstrbuted around the mage plane. Arround the caustc regon and the lght source, we choose a small radus algorthm to render the mage. PMC-PT acheves a more perceptually pleasant mage compared to the correspondng mage rendered by the PT algorthm usng 256 spps, whch s roughly the same total number of sample paths shootng from the eye. Snce the algorthm shfts extra samples from the low varance regons to hgh varance regons. If the scene contans a large number of hgh varance regons, such as ths example, the number of extra samples can be moved s 6

8 Fgure 5. A room scene computed usng our PMC-PT at the left and PT at the rght. PMC-PT has fewer artfacts overall. By gvng more samples to the hgh varance regons, PMC-PT s an mprovement over PT. relatvely small. Although we stll can gan mprovement but the mprovement s less obvous than those whch have fewer hgh varance regons. Fnally, Fgure 5 demonstrate that PMC-PT can be used to render a complex scene. We notce that the celng and the wall near the long lamp contan hgher varance n the mage rendered by PT algorthm. However, the PMC-PT dstrbutes more samples on these regons and thus, the regons seem to be smoother. The overall artfact s smaller n the mage rendered by PMC-PT than n the mage rendered by PT usng 1024 spps. 6 Dscusson The most mportant varable parameter n our algorthms s the survval rate n the resamplng process. A small survval 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 resamplng process. On the other hand, a larger survval rate means that more teraton nformaton related to paths s kept durng the teraton. However, the rate to explore the entre sample doman s slow. The populaton sze and resample rate can also further affect the renderng speed and the fnal result. If the total number of varance-crtera resamplng samples for the T R -loop (.e T R N Populaton r Resamplng r V arance ) s large enough, we can reduce the cost of generatng a sample from the varance mage, β s, by usng a determnstc samplng method. In addton to effcency, wth determnstc samplng, the sample destrbuton s relatvely stratfed accordng to the samplng probablty. Thus, we expect that a large sze of populaton can further mprove the renderng effcency. 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. The most notable lmtaton of PMC s the hgh adaptve sample counts requred for each teraton when the kernel has many adaptable parameters. 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 We have presented a novel approach, PMC-PT, by showng how to adapt PMC method to dstrbute the samples for a general path tracng algorthm. PMC-PT automatcally dstrbutes more samples over regons wth hgh perceptual varance found n the prevous teraton. It adjusts the mportance samplng densty to generate better samples for reducng perceptual varance of a hgh varance regon and also use resamplng process to elmnate samples locatng n low-varance or well-explored regons and regenerate seed samples for achevng ergocty by the crtera of perceptual varance and stratfcaton. There are several future research drectons. Our PMC-PT only uses the nformaton related to the regons on the mage plane but does not use the correlaton among smlar paths. We would lke to further explore the possblty of the correlated samplng n ths aspect. In addton, all samples ntalzed the α s values to 7

9 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. Populaton Monte Carlo s new to the graphcs socety. We beleve that t can provde further research opportuntes n Computer graphcs communty. 8