Technical report, November 2012 (Revision 2, July 2013) Implementing Vertex Connection and Merging

Transcription

1 Techncal report, November 22 Revson 2, July 23) Implementng Vertex Connecton and Mergng Ilyan Georgev Saarland Unversty Intel VCI, Saarbrücken Stage : a) Trace lght sub-paths b) Connect lght vertces to eye c) Buld range search structrure VC... VC VM VM Stage 2: a) Connect to a lght source and to correspondng lght vertces b) Merge wth nearby lght vertces c) Contnue eye sub-path Fgure : An overvew of our combned vertex connecton and mergng VCM) algorthm. Renderng an mage s performed n two stages. In the frst stage, we trace sub-paths from the lght sources, store ther vertces, connect them to the eye, and addtonally buld a range search acceleraton structure over them. In the second stage, we trace an eye sub-path for each pxel, and wth each eye vertex we construct full paths va vertex connecto) and vertex mergng VM), evaluate ther assocated multple mportance samplng MIS) weghts, and accumulate the weghted contrbutons. The path weght evaluaton s an mportant mplementaton aspect and s the man focus of ths report. We present a way to effcently evaluate the weght for a path usng only data that s avalable at the two vertces beng connected or merged, avodng access to any other path vertces n memory. Ths also allows us to avod storng the eye sub-path vertces, effectvely makng the second stage of the algorthm an extenson of tradtonal path tracng wth next event estmaton, addng lght vertex connecton and mergng technques. Abstract Bdrectonal path tracng BPT) and photon mappng PM) are probably the two most versatle physcally based renderng algorthms avalable today. It has been acknowledged that BPT and PM are complementary n terms of the types of lght transport effects they can effcently capture. Our recently proposed vertex connecton and mergng VCM) algorthm ams to leverage the advantages of both methods by combnng vertex connecton technques from BPT and vertex mergng technques from PM va multple mportance samplng [Georgev et al. 22]. We showed that ths combned algorthm can effcently capture a wde range of effects, and can be substantally more robust than ether BPT or PM alone, whle preservng the hgher asymptotc performance of BPT. The focus of our orgnal paper s on the formal dervaton, asymptotc analyss, and evaluaton of the VCM algorthm. In ths techncal report, we address the most techncally challengng part of ts e-mal: georgev@cs.un-saarland.de practcal mplementaton the multple mportance samplng MIS) weghtng. Indeed, correctly mplementng MIS s already taxng n BPT, anm ncreases the complexty by addng even more path samplng technques. More mportantly, the effcent lght sub-path reuse wth vertex mergng allows for cheaply constructng large amounts of full paths for each pxel, whch n turn sgnfcantly ncreases the mpact of path weght evaluaton on the overall performance. The tradtonal BPT-style MIS weght computaton that terates over all path vertces can therefore become neffcent. We derve a new scheme to accumulate and store partal weght sums n the vertces of lght and eye sub-paths. Ths allows us to effcently compute the weght for a full path only usng data cached at the two vertces that are connected or merged. The scheme s smlar to the one ndependently developed by van Antwerpen [2a; 2b] for BPT, but n addton accounts for vertex mergng technques. We also dscuss how to handle nfnte and sngular lght sources, cameras and materals wth MIS, how to use per-pxel mergng rad, and how to mplement VCM n a memory effcent way.

2 Symbol Descrpton Appears n x = x... x k Full length-k path: vertex x s on a lght source, x k s on the eye lens y = y... y s Lght sub-path, wth frst vertex y x on a lght z = z... z t Eye sub-path, wth frst vertex z x k on the eye lens p, p Forward.e. actual) and reverse area pdfs for sub-path vertex p = p σ, g and p = p ) σ, g 2), 6) pσ,, p σ, Forward and reverse sold angle pdfs for sub-path vertex 3), 7) g, Forward and reverse pdf converson factors from sold angle measure to area measure 4), 8) p VC,s,t, p VC,s Vertex connecto) pdf for a length-k path wth s lght vertces and t = k+ s eye vertces ), 3) p VM,s,t, p VM,s Vertex mergng VM) pdf for a length-k path wth s lght vertces and t = k+2 s eye vertces ), 3), n VM Number of samples.e. paths) used for vertex connecton and vertex mergng, respectvely ) = n VM πr 2 Shorthand for all constants that appear n the multple mportance samplng path weght 2) 2) Table : A lst of some commonly used symbols n ths document. Fgure 2 llustrates the redundant sub-)path notaton and the pdf notaton. Notaton The techncal nature of ths document requres the extensve use of mathematcal notaton. In ths secton, we ntroduce the notaton for paths and ther samplng probablty denstes. Table summarzes some commonly used symbols n the document. For more defntons we refer to our orgnal paper [Georgev et al. 22]. Paths. Veach s [997] path ntegral formulaton of lght transport defnes the value of a pxel as an ntegral over the space of all paths: [ ] fx) I = fx) dµx) = E = E [ I ]. ) px) Ω A path of length k edges) x = x... x k s a tuple of k + vertces, where the vertex x s on a lght source, and x k s on the eye. A Monte Carlo estmator I for ths ntegral can be constructed by samplng a random path x and dvdng ts measurement contrbuton fx) by ts probablty densty px). Path tracng based methods do not sample the ndvdual vertces on a path ndependently, but sequentally by performng random walks n scene. Sub-paths. Bdrectonal algorthms construct a full path x by jonng the endponts of one sub-path traced from a lght source and another one traced from the eye. We wll denote a lght subpath wth s vertces by y = y... y s and an eye sub-path wth t vertces by z = z... z t. Here, the vertex y s a pont on a lght source, and z s on the eye lens. These notatons, llustrated n Fgure 2, are redundant wth the x-notaton, but convenently ndex the vertces n the order of ther generaton. Ths symmetry wll allow us to perform the same dervatons for lght and eye sub-paths. In partcular, the forward and reverse probablty densty notaton defned for y below also apples to z. Forward vertex pdfs. The probablty densty functon pdf) of a sub-)path descrbes the jont dstrbuton of ts vertces va a chan of condtonal vertex pdfs. We denote these vertex pdfs by: { py ) f =, py) = pσ,y) 2) g y) otherwse, wth pσ,y) = g y) = { p σy y ) f =, p σy 2 y y ) f > 3) cos θ y y. 4) 2 Above, p.) denotes an uncondtonal pdf expressed w.r.t. the area measure, e.g. py ) for samplng y on a lght source. The subscrpt σ denotes a sold angle pdf. The factor g converts the pdf Fgure 2: An llustraton of vertex connecton and mergng path samplng technques, along wth ther assocated vertex pdfs. The dfferent VC an technques for samplng length-k paths are dentfed by the number of lght sub-path vertces s. measure from sold angle to area, wth θ beng the angle between the surface normal at y and the unt vector y y. We call p forward vertex probabltes w.r.t. the random walk drecton). Wth ths notaton, the pdf of a sub-path y wth s vertces s: s p sy) = py). 5) = Reverse vertex pdfs. Analogously to the forward vertex pdf notaton, we defne a reverse notaton: { py k ) f = k, py) = pσ,y) 6) g y) otherwse, wth pσ,y) = y) = { p σy k y k ) f = k, p σy y + y +2) f < k 7) cos θ + y y. 8) + 2 The arrow notaton makes t easy to dstngush between the actual samplng pdf of a vertex, p y), and ts reverse pdf, p y). That s, the latter denotes the probablty densty for samplng y n a drecton opposte to that of the random walk. For example, n 6) y k s a lght sub-path vertex that has landed on the eye lens, and py k ) denotes the probablty for samplng that pont drectly on the lens surface,.e. as a part of an eye sub-path. These reverse pdfs are needed for the multple mportance samplng path weghts. Recall that all notaton above apples to eye sub-paths z as well. 2

3 2 Vertex Connecton and Mergng As we mentoned earler, bdrectonal path tracng and photon mappng complement each other n terms of the lght transport effects they can effcently capture. Amng to combne ther advantages, vertex connecton and mergng VCM) [Georgev et al. 22] utlzes path samplng technques from both algorthms: Vertex connecto) creates an edge between the endponts of a lght and an eye sub-paths. Ths s the samplng technque used n bdrectonal path tracng. Vertex mergng VM) can ntutvely be thought to concatenate the two sub-paths by vrtually weldng ther endponts, f they le wthn a gven dstance r to each other. VM s our reformulaton of photon mappng as a path samplng technque. Fgure 2 graphcally llustrates the VC an technques. In the rest of ths secton, we summarze the mathematcal formulaton of VCM and outlne ts algorthmc mplementaton. 2. Prmary Estmators and Path Denstes Havng a path x sampled va vertex connecton or mergng, we can obtan a prmary estmate for the pxel value ) by dvdng the measurement contrbuton of the path by ts probablty densty: I vx) = fv,s,tx) p v,s,tx), 9) where v s ether VC or VM, and s and t denote the number of vertces on the lght and eye sub-paths, respectvely. Takng nto account that the sub-paths y and z are sampled ndependently, the full path pdfs for vertex connecton and mergng are, respectvely: p VC,s,tx) = p sy)p tz) p VM,s,tx) = p sy)p tz)πr 2, ) where r s the vertex mergng radus. The measurement contrbuton functon f v,s,t has slghtly dfferent shapes for VC an. For ts defnton, as well as for the dervaton of the VM path pdf, we refer to our paper [Georgev et al. 22]. It s mportant to note at ths pont that, unlke the paper, n ths report we adhere to the ntutve defnton of vertex mergng. That s, we consder the two merged vertces to be the endponts of the eye and lght sub-paths, as ths vew convenently algns wth the mplementaton. Wth ths nterpretaton, a full VM path called extended path n the paper) constructed by a VM, s, t) technque s one segment edge) shorter than a VC path constructed by a VC, s, t) technque Fgure 2). 2.2 Combned Estmator The heart of our VCM algorthm s the secondary multple mportance samplng MIS) pxel estmator that combnes vertex connecton estmators I VC and vertex mergng estmators I VM: I VCM = n VC l= s,t n VM n VM l= s 2,t 2 w VC,s,tx l ) I VCx l ) + w VM,s,tx l ) I VMx l ). ) It consders one eye sub-path through the correspondng pxel, whose vertces are connected to the vertces of lght sub-paths and potentally merged wth the vertces of n VM lght sub-paths. Our mplementaton uses =, and we set n VM to the total number of lght sub-paths, whch for symmetry reasons we choose to be equal to the total number of eye sub-paths,.e. the mage resoluton. The power heurstc [Veach 997] weght for technque v, s, t) s w v,s,tx) = = n β VC n β VC n β v s,t s,t n β v p β v,s,tx) p β VC,s,t x) + n β VM p β VC,s,t x) p β v,s,tx) + nβ VM n β v s 2,t 2 s 2,t 2 p β VM,s,t x) p β VM,s,t x) p β v,s,tx) 2), whch takes nto account all possble ways of samplng x wth vertex connecton and mergng. Note that the weght for a technque s amplfed by the number of samples n v,.e. lght paths, t uses. 2.3 Algorthm Snce equaton ) s a straghtforward extenson of Veach s [997, p. 3] bdrectonal path tracng BPT) estmator, we could follow the classcal BPT mplementaton for VCM as well. That s, for each pxel we would frst trace one lght and one eye sub-path, and then connect and merge each par of opposng vertces, thereby reusng the sub-paths to compute multple VC an estmates. However, the VM sub-path concatenaton lends tself to a sgnfcantly more effcent reuse scheme: we can potentally merge each eye sub-path vertex wth the vertces of all lght sub-paths wth a sngle range search, wthout tracng any rays. To be able to do ths, we splt renderng nto two stages, as llustrated n Fgure :. We frst trace all lght sub-paths and connect ther vertces to the eye. Just lke n photon mappng, we then buld a range search structure over the vertces, e.g. a kd-tree or a hashed regular grd. Our mplementaton uses a hashed grd. 2. In the second stage, we trace an eye sub-path for every pxel. Each sampled eye vertex s frst connected to a lght source. We then connect t to the vertces of the lght sub-path assocated wth the correspondng pxel. Fnally, we merge the eye vertex wth the vertces of all lght sub-paths that fall wthn a user-specfed search radus r. The prmary estmate of each constructed full path s multpled by a MIS weght before t s ultmately accumulated to the combned pxel estmate ). In the paper we provde pseudocode for the above algorthm. Note that we do not store the frst vertex of any sub-path.e. y and z ), as we usually can effcently reduce correlaton by samplng a new vertex on a lght source or on the camera lens for each connecton. The last step n the second stage the path weght evaluaton s an mportant mplementaton aspect of the algorthm. Every path requres a MIS weght, and Veach s [997] evaluaton scheme can be neffcent for vertex mergng whch constructs a sgnfcantly larger number of paths than BPT. In the followng secton we wll derve a new weght evaluaton scheme that s more effcent than Veach s scheme for both vertex connecton and vertex mergng. 3 Effcent Path Weght Evaluaton The naïve way to evaluate the weght 2) s to compute all path pdfs n the denomnator ndependently. For bdrectonal path tracng, whch uses the same formula, mnus the VM sum, Veach [997, p. 36] developed a more effcent scheme by explotng the fact that many of the terms n the fractons cancel out when the path pdfs are expanded. He computes the VC sum by loopng once over the lght and eye sub-path vertces, accumulatng the pdf fractons. Whle Veach s scheme can be easly extended to vertex mergng, t has sub-optmal effcency. Frst, t makes many redundant computatons, as every tme a sub-path s reused for connectng or 3

4 mergng) the same terms assocated wth ts vertces are recomputed. Moreover, t requres accessng every path vertex n memory. Weght computaton can therefore become a sgnfcant overhead for vertex mergng, whch reles heavly on sub-path reuse and often constructs a large number of full paths wth a sngle range query. To see how to solve these problems, notce that the unweghted contrbuton of a path s actually computed effcently n BPT, and also M. Each sub-path vertex stores ts throughput,.e. the accumulated contrbuton terms from all precedng sub-path vertces [Veach 997, p. 34]. Upon connectng or mergng two vertces, the unweghted contrbuton,.e. the prmary estmate 9), s quckly evaluated usng only the data stored at those two vertces. We set out to apply the same effcent scheme to path weght evaluaton as well. To ths end, we reformulate the two sums n 2) as recursve quanttes that can be ncrementally computed and cached at the sub-path vertces as we perform the random walks. At ths pont, readers not nterested n the formal dervaton of our new scheme can skp to Secton 4. There we dscuss ts practcal mplementaton, descrbng the quanttes stored at each sub-path vertex and how to compute the full path weght from ths data. 3. Partal Sub-Path Weghts In order to keep the notaton smple, n ths secton we wll consder paths x of length k wth s lght sub-path vertces, often omttng the redundant subscrpt t. Also, wthout loss of generalty, we wll assume β =,.e. that the balance heurstc s used. We now rewrte the path weght 2) more compactly: w v,s,t = n v k+ j= p VC,j p v,s + nvm n v k j=2 p v,s, 3) where p v,j s the pdf for samplng a length-k full path usng a lght sub-path wth j vertces, and v {VC, VM}. We now wrte the path weght n the form w v,s,t = w lght v,s, + + wv,s eye 4) where we have rearranged the sums n 3) to terate over the lght and eye sub-path vertces, respectvely, and have extracted the term p v,s/p v,s = from the approprate sum. The partal lght and eye sub-path weghts wv,s lght and wv,s eye have slghtly dfferent shapes dependng on the value of v. Vertex connecton. For paths sampled wth v = VC we have: w lght VC,s = w eye VC,s = s j= k+ j=s+ p VC,j p VC,s + nvm p VC,j p VC,s + nvm s j=2 k j=s+ p VC,s 5) p VC,s. 6) Vertex mergng. For paths sampled wth v = VM we have: s w lght VM,s = nvc p VC,j + n VM p VM,s j= k+ w eye VM,s = nvc p VC,j + n VM p VM,s j=s s j=2 k j=s+ p VM,s 7) p VM,s. 8) To summarze, the weght for a full path M s gven by equaton 4). Its terms are gven by equatons 5)-6) or 7)-8), dependng on the technque used to construct the path. 3.2 Recursve Formulaton We now reformulate the path weght 4) n a form that s sutable for evaluaton n a forward manner,.e. n the order of the generaton of the sub-paths, nstead of teratng backwards from ther endponts as Veach [997, p. 36] suggests. We wrte the weght as w v,s,t = w v,s y) + + w v,t z), 9) where w v,s y) and w v,t z) are recursve formulatons of the partal lght and eye sub-path weghts wv,s lght and wv,s. eye These two quanttes can be tracked and updated as we trace the sub-paths y and z, and be readly avalable for summng up when we connect or merge the sub-path endponts. We now derve these recursve quanttes separately for VC an. In the followng, we wll use a shorthand notaton to group all constants that appear n the path weght nto a sngle term: = nvm πr 2. 2) Vertex connecton. Usng the path pdf defntons ), we follow Veach [997, p. 36] to expand the pdf fractons n 5): s w lght VC,s = j= s =j py) py) + ηvcm s j=2 s py) pj y), 2) py) where we use the forward and reverse vertex pdf notatons from 2) and 6). Next, we reformulate the VC an sums as two recursve quanttes omttng the y arguments for readablty): w VC VC, = p p w VM VC, = =j w VC p ) VC, = + w VC VC, w VM VC, = p + p w VM VC, p and wrte the partal lght sub-path weght 5) as ) 22), w lght VC,s = w VC,s y) + w VM VC,s y). 23) We can further combne w VC VC, and w VM VC, to formulate 23) as a sngle recursve quantty agan omttng the y arguments): p w VC, = p w VC, = p + + ) p w VC, p 24) 25) Fnally, wth the eye sub-path notaton z Fgure 2) the above recursve formula apples wthout any modfcaton to 6) as well, allowng us to plug w VC,s y) and w VC,t z) nto 9). Vertex mergng. We now expand equaton 7) as above: w lght VM,s = s s 2 py) η VCM ps y) + s s py) j= =j j=2 =j p y) py). 26) Usng the same methodology as for vertex connecton, we obtan: w VM, = + ) p p η 27) VCM p w VM, = + p + ) p w VM, 28) p Agan, usng the z notaton for eye sub-paths, we can apply the above formula to 8) as well, whch allows us to plug w VM,s y) and w VM,t z) nto 9) for the weght for a vertex-merged path. 4

5 4 Practcal Implementaton Equatons 24)-25) and 27)-28) defne the partal weght quanttes w VC, and w VM, assocated wth the sub-path vertces. Ideally, we want to cache these at every lght and eye vertex as we trace the sub-paths. Then, upon connectng or mergng any two vertces, the full weght would be obtaned by smply summng up ther respectve w VC, or w VM, quanttes, as postulated by equaton 9). Unfortunately, the above scheme cannot be drectly mplemented, snce w VC, and w VM, requre reverse probabltes that are not yet known at the pont of samplng sub-path vertex. More specfcally, py) depends on the next two vertces va p σy y + y +2). Smlarly, p y) depends on the next sub-path vertex, y +. Luckly, an effcent mplementaton of the scheme s made possble by splttng up ts computatons. To ths end, at y we cache only and all) the terms that depend on the sub-path vertces sampled up to and ncludng y. We extract three vertex quanttes from w VC, and w VM,, postponng the evaluaton of the remanng terms untl we have nformaton about the requred vertces, e.g. when we sample the next sub-path vertex or couple y wth another vertex: w VC, = p + w VM, = p }{{} M p }{{} M + p w VC, }{{} pσ, ) 29) + ) p σ, + wvm,, 3) p }{{} where we have used the expanson p = p σ, g, wth pσ, beng the sold angle reverse pdf for sub-path vertex. Note that M appears n the path weght for both VC an, whereas and are specfc to VC an, respectvely. Also, recall from Secton 3.2 that the recursve formulas above apply to both the lght sub-paths y and the eye sub-paths z. 4. Sub-Path Vertex Data As we trace a sub-path, we update and store the quanttes M, and at each vertex. Ther formulas are the same for all lght lght and eye vertces, wth the only excepton for y and z. The reason s that we do not consder path samplng technques wth zero eye sub-path vertces, as the probablty of httng the camera lens s usually very low or even zero n the case of a pnhole camera model). Also, recall from Secton 2.3 that we do not store y and z. The precse data we store at every sub-path vertex are: y : M = pconnect = = p p p z : M = pconnect n lght p 3) = 32) = 33) y, z : M = p 34) = + M + ) p σ, 2 p = p + dvcm + ) p σ, 2 35) 36) In the equatons above, p connect and account for the fact that dfferent technques may be used for samplng a vertex on a lght source or on the eye lens, dependng on whether t wll be connected to a sub-path or used to start a new sub-path. The total number of lght sub-paths, n lght, s the number of samples the eye connecton technque VC, s, ) takes. We have obtaned and by recursvely expandng w VC, and w VM, n equatons 29) and 3). Note that the three floatng pont quanttes d are the only path weght related data that we need to store wth the sub-path vertces. 4.2 Full Path Weght The weght for a full path constructed from a lght sub-path y wth s vertces and an eye sub-path z wth t vertces s gven by w v,s,t = w v,s y) + + w v,t z). 37) Ths equaton s the same as 9), but we also nclude t here, for easy reference. We wll next show how to compute w v,s y) and w v,t z) from the vertex quanttes d, dependng on the technque v {VC, VM}. The general-case formulas for VC an.e. for s > and t > ) follow drectly from 29) and 3). Note that due to the symmetry n the notaton for lght and eye sub-paths, these general-case formulas are the same for y and z. Vertex mergng s >, t > ). A path s constructed by mergng lght sub-path vertex y s and eye sub-path vertex z t : w VM,s y) = dvcm s + p σ,s 2 s 38) w VM,t z) = dvcm t + p σ,t 2 t. 39) Vertex connecton s >, t > ). A path s constructed by connectng the lght vertex y s to the eye vertex z t : w VC,s y) = p s ηvcm + M s + ) p σ,s 2 s 4) w VC,t z) = p t ηvcm + M t + p σ,t 2 t ). 4) Vertex connecton s = ). The eye sub-path vertex z t s sampled on a lght source,.e. the lght sub-path has zero vertces: w VC,s y) = 42) w VC,t z) = p connect t M t + t p σ,t 2 t. 43) Vertex connecton s = ). The eye sub-path vertex z t s connected to vertex y on a lght source a.k.a. next event estmaton): w VC,y) = w VC,t z) = p p connect ptrace p connect y) y) 44) pt ηvcm+m t + p σ,t 2 t ). 45) Vertex connecton t = ). The lght sub-path vertex y s s connected to vertex z on the eye lens a.k.a. eye/camera projecton): w VC,s y) = ptrace p connect z) z) ps n lght ηvcm+m s + p σ,s 2 s ) 46) w VC,z) =. 47) Recall that n lght s the total number of lght sub-paths, whch s the number of samples ths eye connecton technque uses. 5

6 sampled path sampled path reverse pdf evaluaton reverse pdf evaluaton Fgure 3: Reverse pdf evaluaton. Left: The VM pdf for the undrectonally sampled segment s evaluated by nterpretng y as the pont where two merged vertces concde. Rght: The drectonal samplng pdf for a vertex-merged segment s evaluated at the eye sub-path vertex z wth drectons y y and z z. 4.3 Reverse Pdf Evaluaton Havng constructed a path segment undrectonally or va an explct vertex connecton, evaluatng the pdf for vertex mergng s straghtforward, as t can also sample the segment. Ths s llustrated n Fgure 3 left, where the merged vertces concde at y. The opposte case, shown n Fgure 3 rght, s slghtly more complcated, as vertex mergng evaluates lght transport along approxmate paths that cannot be sampled any other way. Nevertheless, we can stll evaluate the pdf for drectonally samplng a smlar segment along the same path edges, as shown n the fgure. The pdf for explct vertex connecton s evaluated by consderng the actual connecton edge between the correspondng lght and eye vertces. That s, n the example n Fgure 3 rght, for a connecton between y and z we have p VC, = p y) p z) p z), and for a connecton between y and z we have p VC,2 = p y) p y) p z). 5 Specal Cases In ths secton, we dscuss how to handle nfnte lght sources and orthographc cameras wth multple mportance samplng, and how to deal wth materals and lght sources whose defntons nvolve delta dstrbutons. We also show how to apply our recursve weght evaluaton scheme to bdrectonal path tracng and to bdrectonal photon mappng. 5. Infnte Lght Sources Lght sources that are located very or even nfntely) far away from the rest of the scene geometry are usually not defned va emttng surfaces but va drectonal ncdent radance dstrbutons at the recevng ponts. Ths prevents the straghtforward use of nfnte lghts M, as they cannot be handled by the path ntegral ) whch only consders area ntegraton [Veach 997, p. 222]. A common approach s to turn nfnte lghts nto fnte area lghts by mappng ther drectonal emsson onto a large sphere that surrounds the scene and emts nwards wth properly scaled power). These area lghts are naturally handled by the path ntegral, but they only approxmate the ncdent radance dstrbuton of the orgnal nfnte lght. Enlargng the sphere mproves the approxmaton accuracy, but can cause numercal ssues due to the large dstance between the lght and the scene and also due to the small sold angles nvolved n the emsson samplng for lght sub-paths see Fgure 4). We can avod the problems by handlng nfnte lghts n ther orgnal formulaton,.e. usng sold angle ntegraton. Paths nvolvng an nfnte lght source now have the form x = x x... x k, where we have replaced the pont x wth a drecton x. We start a lght sub-path y by frst samplng y determnstcally for drectonal Fgure 4: Left: Turnng an nfnte lght source to a fnte sphercal lght allows for expressng path pdfs w.r.t. a product area measure, but s only approxmate and can lead to numercal naccuraces. Rght: We avod the ssues by handlng nfnte lghts va sold angle ntegraton, and derve the correspondng path pdfs for use n MIS. lghts). We then sample a pont ŷ on a plane perpendcular to y and project t onto the scene along y to obtan y see Fgure 4 rght and also Pharr and Humphreys [2, p. 74]). We preserve the path pdf notaton used so far, but accommodate for the changes n the path geometry and samplng procedure va the followng modfcatons: p connect y) = p connect σ y ) and y) = σ y ) are now expressed w.r.t. the sold angle measure, p y) = p σ,y) = p σy y y 2) s now a sold angle probablty densty as well, y) =, as no pdf measure converson s needed anymore a consequence of p y) = p σ,y) above), p y) = pŷ ) cos θ, where θ s now the angle between the normal at y and y. The pdf modfcatons for z k y and z k y are symmetrc. 5.2 Orthographc Cameras When an orthographc camera model s used, the same modfcatons we made to the path geometry and the sub-path pdfs for nfnte lghts apply on the eye sde as well. That s, vertex x k n x s frst replaced wth a drecton x k, and then the pdf modfcatons above also apply to z y k and z y k. 5.3 Pont and Drectonal Lght Sources Omndrectonal and spot lghts have nfntely small areas, and drectonal lghts emt n a sngle drecton. Such lght sources cannot be ht by random rays, as ther emsson s defned va a delta dstrbuton,.e. we have p VC, =. The lght sub-path quanttes d then become dentcal to the eye sub-path quanttes d, mnus the n lght factor n M : y : M = pconnect p = p 48) = 49) =. 5) Recall that the eye sub-path quanttes were orgnally defned dfferently because we do not allow randomly httng the eye lens. Also, equaton 44) smplfes to w VC,y) =. Fnally, snce emsson s defned only at a sngle pont or n a sngle drecton, we set the connect and trace probabltes n equatons 43) and 45) to, snce the lght source samplng s determnstc. 6

7 5.4 Specular Materals Drectonal scatterng from materals lke mrror and glass s defned usng delta dstrbutons that cannot be handled by Lebesgue ntegraton. In practce, ths means that path vertces wth such delta BSDFs cannot be connected to or merged wth other vertces. Hence, certan path samplng technques become mpossble, and ther zero probabltes need to be accounted for n the MIS weghts. Specfcally, f specular scatterng s sampled at vertex x n a random walk, then: p VC, = p VC,+ = p VM,+ = 5) Also, snce the scatterng drecton at x s determnstc, we set: pσ,+ = p σx x x +) = pσ, = p σx x x +) =. 52) When the BSDF s a mxture of dstrbutons e.g. reflecton and refracton), the pdfs n 52) are equal to the probablty of choosng the partcular specular scatterng component at x n the random walk. Note also that snce we do not allow randomly httng the camera, purely specular paths,.e. LS E paths usng Heckbert s [99] regular expresson notaton), can only be sampled undrectonally, and thus have the trval MIS weght of. We can account for the zero-probablty technques n the path weght as we trace the sub-paths. If the scatterng event at vertex s specular, then the d quanttes for vertex smplfy to: y, z : M = 53) = pσ, 2 p 54) = pσ, 2. 55) p Note that wth these modfcatons, our scheme correctly handles the cases where the BSDF s a mxture of specular and non-specular components. Even f we sample specular scatterng at vertex n the random walk, we do not modfy ts weght quanttes d. The vertex s stll stored and used for connecton and mergng to construct other paths, for whch the weght formulas from Secton 4.2 apply as usual. 5.5 Bdrectonal Path Tracng The path weght evaluaton scheme from Secton 4 can be easly appled to tradtonal BPT by restrng the formulas to only account for vertex connectons. Ths s acheved by smply settng = n equatons 35), 4)-47), and also elmnatng the vertex quantty. Wth these modfcatons, our scheme becomes nearly dentcal to the one proposed by van Antwerpen [2a; 2b]. 5.6 Bdrectonal Photon Mappng Bdrectonal photon mappng [Vorba 2] s a specal case of VCM that uses multple mportance samplng for vertex mergng only. Analogously to the bdrectonal path tracng case above, restrctng the weghtng to vertex mergng technques s as smple as settng the terms nvolvng n equatons 33), 36), 38) and 39) to zero, and also elmnatng the vertex quantty. 6 Extensons In ths secton, we frst dscuss how to accommodate per-pxel vertex mergng rad n our MIS weght accumulaton scheme, and then go on to descrbe a progressve VCM varant that s much more memory-effcent than the one presented n Secton Per-Pxel Mergng Radus The mergng radus r s an mportant VCM parameter that controls the performance of the vertex mergng technques and ther relatve weghts n the combned estmator ). The optmal radus sze may vary across the scene, however our MIS weght accumulaton scheme from Secton 4 mplctly assumes that all mergng queres use the same global radus r. Ths s because the cumulatve subpath vertex quanttes and depend on r va. If we want to use dfferent mergng rad at dfferent locatons n the scene and stll obtan correct MIS weghts, we need to make the vertex quanttes ndependent of r. To do ths, we avod computng and storng and at every vertex durng sub-path constructon, and c VM and nstead keep track of two new quanttes, c VC y : M = pconnect c VC = p p z : M = pconnect : n lght p 56) c VC = 57) c VM = c VM = 58) y, z : M = p 59) For any vertex, c VC = M + ) p σ, 2 c VC p c VM = p and + ) p σ, 2 c VM can be computed as follows: = c VC The evaluaton of, and thus of 6) 6) + c VM 62) = cvc + c VM 63) and, can now be postponed to the pont when an actual pxel estmate s constructed usng a partcular eye vertex. Ths allows us to choose a dfferent radus r for every eye sub-path, e.g. derved from the pxel footprnt. Note that for any gven full path we must use the same r n the MIS weghts of all possble technques, so they sum up to one. 6.2 Memory-Effcent Implementaton In Secton 2.3 we descrbed a two-stage mplementaton of the VCM algorthm, where we frst trace all lght sub-paths and store ther vertces, smlarly to photon mappng. Snce we also use these vertces for connectons n the second stage, we need to cache ther assocated BSDF structures as well. In some renderers, the shadng structure at a surface pont can be as large as KB, requrng ggabytes of storage for mllons of lght vertces. Ths memory ssue does not appear n bdrectonal path tracng BPT), where pxels are rendered ndependently and every par of lght and eye Shadng structures often store the results from multple texture queres, local geometrc and shadng coordnate frames and dervatves, as well as reflectance dstrbuton parameters. 7

8 sub-paths s mmedately dscarded upon connecton. Photon mappng can also mantan a low memory footprnt, as t always evaluates the BSDF at eye vertces, thereby avodng the need for storng shadng data at lght vertces. To reduce the memory footprnt of lght vertces, we modfy progressve VCM to operate n a sngle stage as follows. At every teraton, for each pxel we frst trace one lght sub-path. We then store ts vertces n a separate lst wthout ther BSDF structures, just lke n photon mappng, but wth the addton of the three cumulatve weght quanttes. After that, we trace an eye sub-path through the pxel. Every eye vertex s connected to the lght sub-path and also merged wth all nearby lght vertces from the prevous teraton. Fnally, we dscard both sub-paths. After processng all pxels, we buld a range search acceleraton structure over the lght vertex lst and dspose the structure from the prevous teraton. Renderng begns by tracng an ntal set of lght sub-paths that are only used for mergng at the frst teraton. Alternatvely, we can skp mergng at the frst teraton, and scale the VM contrbutons at the second teraton by a factor of two. Wth the above modfcatons, the VCM algorthm becomes very smlar to tradtonal BPT, wth the extenson of cachng all lght vertces at every teraton and then usng them for mergng at the subsequent teraton. By delayng vertex mergng by one teraton, ts memory footprnt becomes almost as low as that of photon mappng, wth only three addtonal floats per lght vertex. 7 Conclusons References DAVIDOVIČ, T., AND GEORGIEV, I., 22. SmallVCM renderer. and GEORGIEV, I., KŘIVÁNEK, J., DAVIDOVIČ, T., AND SLUSALLEK, P. 22. Lght transport smulaton wth vertex connecton and mergng. ACM Trans. Graph. 3 December). Proc. SIGGRAPH Asa 22). HECKBERT, P. S. 99. Adaptve radosty textures for bdrectonal ray tracng. In SIGGRAPH 9, ACM, New York, USA. PHARR, M., AND HUMPHREYS, G. 2. Physcally Based Renderng: From Theory To Implementaton, 2nd ed. Morgan Kaufmann Publshers Inc., San Francsco, CA, USA. VAN ANTWERPEN, D. 2. Recursve MIS computaton for streamng BDPT on the GPU. Tech. rep., Delft Unversty of Technology. VAN ANTWERPEN, D. 2. A survey of mportance samplng applcatons n unbased physcally based renderng. Tech. rep., Delft Unversty of Technology. VEACH, E Robust Monte Carlo Methods for Lght Transport Smulaton. PhD thess, Standford Unversty. VORBA, J. 2. Bdrectonal photon mappng. In Proc. Central European Semnar on Computer Graphcs CESCG ). In ths techncal report, we addressed mportant aspects of the practcal mplementaton of the vertex connecton and mergng VCM) algorthm [Georgev et al. 22]. We frst derved a computatonally effcent scheme for evaluatng the multple mportance samplng MIS) path weghts. Ths scheme avods redundant computatons, and more mportantly, sgnfcantly reduces memory access, makng the algorthm well suted for GPU mplementaton. We then descrbed how to handle specal cases such as nfnte and sngular lght sources, cameras and materals. We fnally dscussed how to accommodate per-pxel mergng rad n the MIS weghts and how the progressve VCM algorthm can be restructured to sgnfcantly mprove ts memory-effcency. Lmtatons. One constrant that our recursve path weght evaluaton scheme poses s that no terms n the weght can depend on the sub-)path length. For example, we cannot base the Russan roulette path termnaton probablty or the number of vertex connectons on the eye sub-path length. The reason s that we need to be able to compute these quanttes when we accumulate reverse probabltes durng the lght sub-path tracng, wthout any knowledge of the full path length. Van Antwerpen [2a] also ponts out ths lmtaton. One way to mtgate t s to consder such terms constant or completely exclude them from the MIS weght. Alternatvely, pervertex storage can be augmented wth addtonal partal weghts that are specalzed for connectons to sub-paths of dfferent lengths. Reference mplementaton. We provde an mplementaton of the VCM algorthm that uses our path weght evaluaton scheme n the open source renderer SmallVCM [Davdovč and Georgev 22]. Ths mplementaton covers all specal cases dscussed n Secton 5. The renderer also ncludes tradtonal path tracng and lght tracng. Acknowledgements. The author would lke to thank Tomáš Davdovč for mplementng SmallVCM, and Mloš Hašan and Jaroslav Křvánek for the nsghts nto handlng nfnte lghts. Also thanks to Anton Kaplanyan, Tomáš Davdovč and Jaroslav Křvánek for proofreadng the text. 8