An exhaustive error-bounding algorithm for hierarchical radiosity

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1 An exhaustve error-boundng algorthm for herarchcal radosty Ncolas Holzschuch, Franços X. Sllon To cte ths verson: Ncolas Holzschuch, Franços X. Sllon. An exhaustve error-boundng algorthm for herarchcal radosty. Computer Graphcs Forum, Wley, 998, 7 (4), pp < <0./ >. <nra > HAL Id: nra Submtted on 2 Dec 2009 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Volume 7 (998), number 4 pp An exhaustve error-boundng algorthm for herarchcal radosty Ncolas Holzschuch Franços X. Sllon MAGIS GRAVIR/IMAG - INRIA Abstract Ths paper presents a complete algorthm for the evaluaton and control of error n radosty calculatons. Provdng such control s both extremely mportant for ndustral applcatons and one of the most challengng ssues remanng n global llumnaton research. In order to control the error, we need to estmate the accuracy of the calculaton whle computng the energy exchanged between two objects. Havng ths nformaton for each radosty nteracton allows to allocate more resources to refne nteractons wth greater potental error, and to avod spendng more tme to refne nteractons already represented wth suffcent accuracy. Untl now, the accuracy of the computed energy exchange could only be approxmated usng heurstc algorthms. Ths paper presents the frst exhaustve algorthm to compute fully relable upper and lower bounds on the energy beng exchanged n each nteracton. Ths s accomplshed by computng frst and second dervatves of the radosty functon where approprate, and makng use of two concavty conjectures. These bounds are then used n a refnement crteron for herarchcal radosty, resultng n a global llumnaton algorthm wth complete control of the error ncurred. Results are presented, demonstratng the possblty to create radosty solutons wth guaranteed precson. We then extend our algorthm to consder lnear boundng functons nstead of constant functons, thus creatng smpler meshes n regons where the functon s concave, wthout loss of precson. Our experments show that the computaton of radosty dervatves along wth the radosty values only requres a modest extra cost, wth the advantage of a much greater precson.. Introducton Global llumnaton algorthms now have many applcatons. One of the most promsng felds s n urban and archtectural plannng, where the use of a global llumnaton algorthm allows to vsualze a future buldng, and thus to check for msconceptons. For example, t becomes possble to check Current poston: Invted Researcher, Department of Computer Scence, Unversty of Cape Town, South Afrca. MAGIS s a jont research project between CNRS, INRIA, INPG and Unversté Joseph Fourer Grenoble I. Postal address: B.P. 53, F-3804 Grenoble Cedex 9, France. E-mal: Ncolas.Holzschuch@mag.fr. the ergonomy of the workplace s there enough lght, or too much? or to ensure that the tems n a museum are properly lt. In such applcatons, t s vtal to be able to quantfy the lght arrvng on each pont of the scene, n order to gve the user a precse range n whch the llumnaton s guaranteed to fall. Global llumnaton algorthms generally have at least a parameter that the user can manpulate, choosng ether fast computatons or precse results. For Monte-Carlo ray tracng algorthms, ths parameter can be the number of rays. For herarchcal radosty algorthms, t can be the refnement threshold, used to decde whether or not to refne a gven. Publshed by Blackwell Publshers, 08 Cowley Road, Oxford OX4 JF, UK and 238 Man Street, Cambrdge, MA 0242, USA.

3 2 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty nteracton. Untl recently, however, we had lttle knowledge of the total precson of the result computed, or of the relaton between the parameters and ths precson. Even f t was clear that spendng more tme on the smulaton would produce more precse results, we could not quantfy precsely ths ncrease. y A 2 θ 2 n 2 r 2 θ n In 994, Lschnsk proposed a refnement crteron for herarchcal radosty such that the error on the energy at each pont of the scene could be controlled by the refnement threshold. Ther algorthm used upper and lower bounds on the pont-to-area form factor for each nteracton n order to compute upper and lower bounds for the radosty at each pont n the scene. However, they had no way to compute relable upper and lower bounds for the pont-to-area formfactor on a gven nteracton, and stll resorted to samplng computng a set of values for the form-factor, and takng the mnmum and maxmum of these values. Although Lschnsk s method s easy to mplement, t s not totally relable. In ths paper, we present a method allowng to compute fully relable upper and lower bounds for the pont-to-area form-factor on any nteracton. To acheve ths goal, we use our knowledge of the pont-to-area form-factor dervatves together wth ts concavty propertes. These concavty propertes of the pont-to-area formfactor are descrbed n secton 3. They extend the unmodalty conjecture proposed by Drettaks 2, 3. Lke the unmodalty conjecture, they are only conjectures, and despte ther apparent smplcty, we have been unable to fnd a complete demonstraton for them. However, we also have been unable to exhbt a counter-example. As s explaned n appendx B, we can compute exact values for the dervatves of the pont-to-area form-factor; ether for the frst dervatve, the gradent vector, or for the second dervatve, the Hessan matrx. As we shall also see n appendx B, t s ndeed faster to compute an exact value for the form-factor dervatve than computng approxmate values usng several samples. Usng our knowledge of the dervatves along wth the concavty propertes of the pont-to-area form-factor, we show n secton 4 how to derve bounds for the pont-to-area form-factor n any unoccluded nteracton. We also show an mplementaton of the refnement crteron usng these bounds. When dealng wth partally occluded nteractons we can not use the prevous bounds, as the concavty conjectures do not hold n ths case. But we can exhbt two emtters that are convex and bound the actual emtter, whch we call the mnmal and the maxmal emtter. Usng the prevously defned algorthm, we fnd an upper bound for the maxmal emtter, and a lower bound for the mnmal emtter. The algorthm for fndng these convex emtters s detaled n secton 5. A Fgure : Geometrc notatons for the radosty equaton. 2. Background The radosty method was ntroduced n the feld of lght transfer n 984 by Goral 4. Ths method uses a smplfcaton n order to solve the global llumnaton problem: t assumes that all the objects n the scene are deal dffuse surfaces: ther bdrectonal reflectance s unform, and thus does not depend on the outgong drecton. In ths case, the radosty emtted at a gven pont x can be expressed as an ntegral equaton: cos θ cos θ2 B(x) =E(x)+ρ d (x) B(y) V (x, y)dy πr y S 2 () In ths equaton, S s the set of all ponts y. r s the dstance between pont x and pont y, θ and θ 2 are the angles between the xy vector and the normals to the surfaces at pont x and y respectvely (refer to fgure for the geometrc notatons). ρ d (x) s the dffuse reflectance at pont x,and V (x, y) expresses whether pont x s vsble from pont y or not. In order to solve equaton, Goral 4 suggested to dscretze the scene nto a set of patches [P ], over whch a constant radosty, B s assumed. In ths case, the radosty at pont x becomes: B(x) =E(x)+ρ d (x) B x cos θ cos θ πr y P 2 V (x, y)dy (2) The purely geometrc quantty cos θ cos θ F (x) = V (x, y)dy πr y P 2 s called the pont-to-area form-factor at pont x from patch. It only depends on the respectve postons of pont x and patch. Snce we assume a constant radosty value wthn the patch, we can compute ths value as the average of all the pont values. Ths leads to a matrx equaton: B j = E j + ρ F jb (3)

4 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty 3 where the geometrc quantty F j = cos θ cos θ V (x, y)dxdy A j πr x P j y P 2 s called the form-factor. Schröder 5 showed that there s a closed form expresson for the form-factor n the case of two fully vsble polygonal patches. In the general case, we do not have access to the exact value of the form-factor, but only to approxmate values. Equaton 3 can be solved n an teratve manner, usng Jacob or Gauss-Sedel teratve methods (see Cohen 6 ). The problem s that n order to compute one full bounce of lght across the surfaces n the scene, we have to compute the entre form-factor matrx, whch s quadratc wth respect to the number of patches. A sgnfcant mprovement over the classcal radosty method s herarchcal radosty. In standard radosty, the dscretsaton of one object nto patches does not depend on the objects wth whch t nteracts. In order to model the nteracton between objects that are very close, and exchange lots of energy, we need to subdvde them nto many patches, so as to get a precse modellng of the radosty. On the other hand, an nteracton between two objects that are far away could be modelled wth fewer patches. In herarchcal radosty, ntroduced n 990 by Hanrahan 7, each object s subdvded nto a herarchy of patches, wth each node n the herarchy carryng the average of the radosty of ts chldren. Interacton between objects far away from each other are modelled as nteractons between nodes at a hgh level n each herarchy. On the other hand, nteractons between objects close to each other are modelled as nteractons between nodes at a lower level n the herarchy, thereby allowng more precson n the modellng of radosty. Each nteracton between two nodes s modelled by a lnk, adata structure carryng the dentty of the sender and the recever, as well as the form-factor, and possbly other nformatons on the respectve vsblty of both patches. Ths herarchcal radosty algorthm has later been extended usng wavelets (see Gortler 8 ). The most mportant step n the herarchcal radosty method s the decson whether or not to refne a gven nteracton. Ths decson s deferred to a refnement crteron. Early mplementatons of the herarchcal radosty method used crude approxmatons of the form-factor between two patches. It was known that these form-factor estmates were most mprecse when the result of the approxmaton was large. Hence, nteractons were refned as long as the formfactor estmate was above a certan threshold (Hanrahan 7 ). Ths refnement crteron does not gve the user a full control of the precson on the modellng of the radosty functon. In partcular, t does not gve any guarantee that t wll refne all problematc nteractons, and t can also refne excessvely n places where the soluton has already attaned a correct level of precson (Holzschuch 9 ). Part of these problems can be addressed by usng dscontnuty meshng, where the patches are frst subdvded along the dscontnuty lnes of the radosty functon and ts dervatves (see Heckbert 0, Lschnsk, 2 and Drettaks 3 ). These dscontnuty lnes can be computed usng geometrc algorthms. However, as ponted out by Drettaks, these dscontnuty lnes are not of equal mportance. Some of them do not have a notceable effect on the fnal radosty soluton. Hence t s not necessary to compute all the dscontnuty lnes. Decdng whch dscontnuty lnes are relevant s done by a refnement oracle, usng heurstc methods lke the one descrbed above. Many of the latest research results have dealt wth gvng the user a better control of the level of precson n the modellng of radosty n the herarchcal radosty method. In the most promsng paper on the subject, Lschnsk, suggested to compute for each nteracton an upper and lower bound for the pont-to-patch form-factor between the ponts of the recevng patch and the emttng patch, namely F max and F mn, as well as an upper and lower bound for the radosty of the emttng patch, usng nformaton already avalable n the herarchy. We then know that the radosty on the recevng patch s between F maxb max and F mnb mn. Hence, the uncertanty on the radosty on the recevng patch, due to ths partcular nteracton s: δb recever = F maxb max F mnb mn The naccuracy on the energy of the recevng patch, due to ths partcular nteracton, s: δe recever = A recever (F maxb max F mnb mn) We can then decde to refne all nteractons where ths mprecson on the transported energy s above a gven threshold. The most dffcult part n ths algorthm s fndng relable values for the bounds on the form-factor. Lschnsk suggested computng exact values for the pont-to-area form factor at dfferent samplng ponts on the recever, and usng the maxmum and mnmum value at these samplng ponts as the upper and lower bounds. Although ths algorthm does not gve totally relable bounds, t does provde a close approxmaton, and s qute easy to mplement on top of an exstng herarchcal radosty mplementaton. In the followng sectons we show that t s possble to compute relable upper and lower bounds for the pont-toarea form factor. These bounds can then be used n the precedng algorthm, allowng the refnement of all nteractons where the naccuracy on the transported energy s above the threshold.

5 4 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty Tangent Functon Secant Concave Functon f (x) < 0 Tangents Functon Convex Functon f (x) > 0 Tangent Tangent Xmn Area of Interest Xmax Fgure 3: A functon that remans concave across an nterval les above ts secant, and below all ts tangents on ths nterval..5 Functon Inflecton Pont f (x) =0 Fgure 2: Concavty for unvarate functons. 3. The Concavty Conjectures 3.. Defnton of Concavty Unvarate functons are sad to be concave at a pont when they le entrely below ther tangent at that pont; conversely, they are sad to be convex when they le above ther tangent. When the functon crosses ts tangent, the pont s sad to be an nflecton pont (see fgure 2). Classcally, the concavty of the functon s lnked to the sgn of ts second dervatve: f the second dervatve s postve, then the functon s convex. If t s negatve, then the functon s concave. It s only when the second dervatve changes sgn that we have an nflecton pont. Concavty s often used to fnd upper and lower bounds for functons; f a functon s concave on an nterval, then t s below all ts tangents on ths nterval, and above all ts secants (see fgure 3). Snce concavty allows boundng by affne functons (lke tangents) nstead of constants, t generally provdes bounds that are closer to each other, and hence a better range. Ths noton of concavty extends naturally to bvarate functons, such as radosty defned over a surface. A bvar Fgure 4: A pont where the functon s concave: the functon les below the tangent plane. ate functon s sad to be concave at a pont when t les below ts tangent plane (see fgure 4), convex when t les above ts tangent plane and ndefnte when the functon crosses the tangent plane (see fgure 5). As wth unvarate functons, concavty can be used to fnd upper and lower bounds: f a functon s concave over a trangular area, then on ths area t les below all ts tangent planes, and above the secant plane defned by the three corners of the trangle. A unvarate functon usually crosses ts tangent at an solated pont, the nflecton pont. Contrarly, the set of ponts where a bvarate functon crosses ts tangent plane s a whole regon. The second dervatve of a bvarate functon s a 2 2 matrx, called the Hessan matrx. As wth unvarate functons, the concavty of the functon s lnked to ts second dervatve: f the Hessan matrx s defnte postve, then the functon s convex; f the Hessan matrx s defnte negatve, then the functon s concave; f the Hessan matrx s ndefnte, then the functon s ndefnte. The Hessan can be exc The Eurographcs Assocaton 998

6 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty Fgure 5: A pont where the concavty s ndefnte: the functon crosses ts tangent plane Fgure 6: The C conjecture: the radosty functon has ndefnte concavty everywhere, except over a convex area (hatched), where the radosty functon s concave Concavty Conjectures pressed wth respect to the partal dervatves of the functon: [ 2 f ] 2 f H = u 2 u v = [ ] r s (4) 2 s t 2 f u v 2 f v 2 The Hessan s defnte f rt s 2 s postve. It s defntepostve f rt s 2 s postve and r s postve, defntenegatve f rt s 2 s postve and r s negatve. If rt s 2 s negatve or null, the Hessan s ndefnte, and the functon crosses ts tangent plane. It must be noted that a functon s necessarly concave where t has a local maxmum, and convex wherever t has a local mnmum. Ths property s true both for un- and bvarate functons Concavty of the Pont-To-Area Form Factor Background Let us sngle out an nteracton between an emttng patch and a recevng patch. We seek an upper and a lower bound for the pont-to-area form-factor across the recever. These upper and lower bounds can then be used by a refnement oracle, as ntroduced by Lschnsk. Usng the algorthm descrbed n appendx B, we have access to the form-factor and to ts dervatves at any pont of the recever. However, these values are only vald at ths specfc pont. Snce we seek a result vald across the whole recever, we must exhbt a property of the pont-to-area formfactor that s vald across the recever. A smlar approach was used by Drettaks 2, 3. In the case of a fnte convex emtter, wth constant radosty, and of an nfnte recever, Drettaks made the followng two conjectures: Conjecture U Radosty on the recever has only one maxmum. Conjecture U2 Radosty on any lne on the recever has only one maxmum. These two conjectures are referred to below as the unmodalty conjectures. Lke Drettaks, we consder a fnte convex emtter, wth constant radosty, and we assume the recever s an nfnte plane. We state the followng two conjectures on the concavty of the radosty on the recever: Conjecture C The Hessan matrx of the radosty functon s ndefnte everywhere, except over a bounded area. On ths area, the radosty functon s concave. Furthermore, the area s convex. Conjecture C2 On any lne drawn on the recever, radosty s concave over a bounded nterval, and convex everywhere else. Fgure 6 llustrates the C conjecture: the radosty functon s ndefnte everywhere and crosses ts tangent plane except over a convex regon (hatched). Fgure 7 llustrates the C2 conjecture: the radosty functon defned over a gven lne s convex across [,a] and across [b, + ], and concave across [a, b]. Despte ther apparent smplcty, these conjectures have yet escaped demonstraton. It s obvous that they are true n the smplest case of a pont lght source sendng lght n all drectons. However, even for the case of a dfferental emtter area nstead of a pont lght source, t has not been possble so far to prove the concavty conjectures. Appendx A s a detaled study of the dfferental emtter area Relatonshp between the conjectures Our concavty conjectures are actually an extenson of the unmodalty conjectures: that s, C mples U, and C2 mples U2. Note that we also know that U2 mples U: { U2 = U C2 = U2 C = U U2 = U: Proof Assume U s false. Then there exsts at least two maxma for the radosty functon, M and M2. On the lne jonng M and M2 there are two maxma, whch s n contradcton wth U2.

7 6 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty A B A B The radosty functon on the lne A B and ts second dervatve Fgure 7: The C2 conjecture: the radosty functon on a lne s concave only over a fnte nterval, [AB]. C2 = U2: Proof The functon s concave on the neghbourhood of each local maxmum. If there are two local maxma on a lne, there must be a local mnmum between them. In the neghbourhood of ths local mnmum, the functon would have to be convex, whch s mpossble because of C2. C = U: Proof Assume U s false. Then there exsts at least two local maxma for the radosty functon. On the neghbourhood of each maxmum, the radosty functon s concave. But between the two maxma, there must be a pass-lke pont, where the concavty s ndefnte. Ths s n contradcton wth C. No relatonshp between C and C2: An mportant pont s the ndependence of our two concavty conjectures. C does not mply C2, and C2 does not mply C. 4. Error Control for Unoccluded Interactons In ths secton, we descrbe our algorthm for fndng upper and lower bounds for the pont-to-area form-factor across the recever. These values are then used by a refnement oracle lke the oracle ntroduced by Lschnsk. 4.. Computng Radosty Dervatves Let us call A 2 the emttng patch, A the recever and x a pont on the recever (see fgure ). In ths case, there s an exact formula for the pont-to-area form factor (Segel and Howell 4 ): F (x) = 2π n r 2 A 2 r d l (5) where the ntegral s on A 2, the contour of A 2,andd l 2 s the dfferental element of ths contour. Usng ths expresson of the pont-to-area form-factor, t s possble to compute exact formulae for both ts frst and second dervatves. These formulae for the dervatves are easly mplemented, gvng access to exact values for the functon and ts dervatves (see appendx B, and also Arvo 5 or Holzschuch 6, 7 ). If we compute smultaneously the pont-to-area formfactor and ts dervatves, we can save computaton tme by reusng some geometrc quanttes that appear n several formulae. In ths case, the overall cost of computng the dervatves s reasonable: there s an ncrease of 40% for computng the gradent along wth the form-factor, and an ncrease of 00% for computng both the gradent and the Hesc The Eurographcs Assocaton 998

8 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty 7 san matrx (see appendx B and Holzschuch 6, 7 ). Ths cost must be balanced aganst what t would requre to compute approxmate values for the dervatves usng several formfactor computatons: n ths case, the cost ncrease for the gradent would be of 00%, and that of the Hessan 600%. In our refnement phase, we compute the values of the pont-to-area form-factor and ts dervatves at the vertces of the recevng patch. These values can be reused n the radosty propagaton phase to obtan the radosty values at the vertces.,,,,,,,,,,,,,,,,,,,,,,,,,,,, x All form-factor values n ths half-plane are smaller than F(x). The maxmum les n ths half-plane F(x) 4.2. Computng Bounds for the Pont-to-Area Form-Factor We show here how our knowledge of the pont-to-area formfactor and ts dervatves at the vertces of the recevng patch, used jontly wth our conjectures, gves us access: frst, to the locaton of the maxmum and the mnmum of the pont-to-area form-factor, second, to an exact value for the mnmum, thrd, to an upper bound for the maxmum The Mnmum s at one of the Vertces An mmedate consequence of the unmodalty conjectures (U and U2) s that the mnmum for the pont-to-area formfactor s necessarly at one of the vertces of the recever: If the mnmum was nsde the recevng patch, A, then there would exst several local maxma for the pont-toarea form-factor on the plane supportng A ths s n contradcton wth U. Hence, the mnmum across A must be on the contour of A. The contour of A s made of polygonal edges. If on one of these edges the mnmum s nsde the edge then on the lne supportng the edge the form-factor must have two maxma ths s n contradcton wth U2. Hence, the mnmum can only be at one of the vertces of A An exact value for the mnmum Snce we chose to compute the pont-to-area form-factor at the vertces of the recevng patch, A, we do have access to the exact value of the mnmum across A: t s the mnmum of our computed values for the pont-to-area form-factor at the vertces of A Fndng the Poston of the Maxmum A consequence of U2 s that gven a pont x, gven the pontto-area form-factor F (x) and ts gradent at pont x, F (x), for all ponts p such that xp F(x) < 0, wehavef (p) < F (x). Otherwse, there would be one local mnmum between p and x on the lne passng through p and x, and hence two local maxma, whch s n contradcton wth U2. Fgure 8: Knowledge of the gradent helps fnd the poston of the maxmum. Hence the maxmum of the pont-to-area form-factor can only be n the half-plane defned by: xp F(x) 0 (see fgure 8.) Ths property gves us an algorthm to determne whether the maxmum for the pont-to-area form-factor across the recevng patch A can le nsde the patch, or f t must be at one of the vertces (see fgure 9): For each vertex, there s a half-plane (defned by the formfactor gradent at ths vertex) where the form-factor value can be greater than the value at the vertex. The ntersecton of these half-planes s an area where the pont-to-area form-factor value can be greater than the value at all the vertces. The ntersecton of ths area wth the recevng patch s ether empty or not empty. If ths ntersecton wth the patch s not empty, then there exsts an area nsde the patch where the maxmum can be. If ths ntersecton s empty, then the maxmum for the form-factor across the patch must be at one of the vertces If the Maxmum s at one of the Vertces If the above algorthm tells us that the maxmum can only be at one vertex of the recevng patch, then we know the exact value of the maxmum: t s the value of the pont-to-area form-factor at that vertex If the Maxmum s Insde the Recevng Patch If the above algorthm tells us there exsts an area nsde the recevng patch A where the maxmum can be, then we do not have access to the exact value of the maxmum of the pont-to-area form-factor across A. The only thng we know at ths stage s that the value of the maxmum must be greater than the values computed at the vertces of A. There are three knd of algorthms for fndng an upper bound for the pont-to-area form-factor across A:

9 8 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, x x2,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, The maxmum can,,,,,,,,,,,,, x3 x4,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, x x2,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, x4,,, be n ths regon,,,, x3 The maxmum can be nsde the polygon. The maxmum s at one of the vertces. Fgure 9: Usng the gradent to locate the maxmum nsde or outsde the recevng patch. Heurstc Algorthms: Compute another sample value for the pont-to-area form-factor nsde patch A. The poston of the samplng pont can be arbtrary or can make use of the nformaton gven by the form-factor gradent. Concavty Algorthms: If the pont-to-area form-factor functon on the recevng patch s concave, we use the tangent planes to fnd an upper-bound. Geometrc Algorthms: Usng geometrc tools, buld an emtter that encloses the actual emtter for all the ponts of the recevng patch, and for whch we can fnd the value of the maxmum. Ths value s an upper-bound. Heurstc algorthms nclude gradent descent algorthms, as descrbed by Arvo 5 and Drettaks 2, 3. Gradent descent algorthms make use of the nformaton provded by the gradent to subdvde the recevng patch untl convergence. The gradent can ether be approxmated (Drettaks 2, 3 )oranexact value (Arvo 5 ). In our mplementaton, we use concavty algorthms wherever possble, and resort to geometrc algorthms f the pont-to-area form-factor functon s not concave Concavty Algorthms Accordng to C, the zone where the pont-to-area form-factor functon s concave s a convex one. As a consequence, f the form-factor Hessan s defnte negatve at the vertces of the recevng patch, then t stays defnte negatve across the recevng patch. In ths case, the form-factor functon les below all ts tangent planes at the vertces across the recevng patch. We know these tangent planes snce we know the form-factor gradent at the vertces. Fndng an upper bound for the pont-to-area form-factor s then equvalent to computng the ntersecton of the tangent planes. Ths s manly a lnear programmng problem (see, for example, Preparata 8 ); the computatonal complexty of the problem depends on the dmenson of the problem whch here s always two snce we are dealng wth bvarate functons and on the number of vertces n the recevng patch. Usually, n herarchcal radosty algorthms, we are restrctng ourselves to trangular or quadrangular patches. If ths s the case, we can assume the complexty of computng the ntersecton of the tangent planes s constant Geometrc Algorthms If the form-factor Hessan s not defnte negatve at all the vertces of the recevng patch, then the pont-to-area form-factor functon s not concave across the entre recevng patch. It s therefore not possble to use concavty algorthms. In ths case, we resort to geometrc algorthms: n a plane parallel to the plane of the recever, we construct an emtter wth the followng two propertes: From all the ponts of the recever, t s seen as ncludng the orgnal emtter. It has two axes of symmetry, so that we can fnd the maxmum form-factor due to the emtter. The reason for the second tem les n the symmetry prncple: f the emtter and the recever are left unchanged by a planar symmetry, then so s the pont-to-area form-factor functon on the recever; thus ts maxmum can only le on the ntersecton of the plane of the symmetry and of the plane of the recever. If there are two planes that leave the emtter and the recever un-changed, then the maxmum can only be at ther ntersecton (see fgure 24, n the color secton). To buld ths emtter: select a plane P parallel to the plane of the recevng patch; for each vertex V of the recevng patch, buld the projecton p of the orgnal emtter accordng to ths vertex on P (see fgure 0); ths projecton s totally equvalent to the orgnal emtter for ths partcular vertex;

10 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty 9 any convex regon enclosng all the p projectons s seen from all the ponts of the recever as enclosng the orgnal emtter; as a consequence, the pont-to-area form-factor due to ths convex regon s greater than the pont-to-area form-factor due to the actual emtter; buldng a convex regon enclosng the p s a standard geometry problem (see, for example, Foley et al. 9,Kay 20 or Toth 2 ). Constranng ths convex regon to have two axes of symmetry can ether be a consequence of the boundng object used, lke ellpses and rectangles, or be a property we add afterward. Snce our problem s a two dmensonal geometry problem although we have a set of three dmensonal data ponts we start by projectng our p onto one of the coordnates planes (x, y), (z, x) or (y, z). Once we have bult the result n ths coordnate plane, we wll project t back onto the emtter plane. Several algorthms can be used, ether gvng a faster result, but a greater enclosng emtter, and hence a greater upper-bound, or requrng more tme, but gvng an enclosng emtter that s closer to the p, and hence a smaller upper-bound: Buld the convex hull of the p, then buld a regon wth two axes of symmetry enclosng the convex hull. Ths gves the smaller enclosng emtter, but requres more computaton tme Buld a boundng rectangle enclosng the p nsde the emtter plane, as n Toth 2. Ths s one of the fastest possble algorthm. Furthermore, t naturally gves an enclosng emtter wth two axes of symmetry, so there s no constructon tme nvolved for buldng the symmetres. Buld a boundng ellpse enclosng the p nsde the emtter plane. Ths algorthm s slower, but t also gves an enclosng emtter wth two axes of symmetry, so there s no constructon tme nvolved for buldng the symmetres. A boundng rectangle usng the (x, y, z) axes can gve an enclosng emtter much bgger than the p, thus nducng a greater upper bound. A smple mprovement s to use slabs, as suggested n Kay 20. In ths case, n order to buld an object wth two axes of symmetry, we have to restrct ourselves to two sets of orthogonal slabs. Ths algorthm requres more computatonal tme than the prevous algorthm, but can gve a sgnfcantly smaller enclosng emtter. If n e s the number of vertces of the emtter, and n r the number of vertces of the recever, the total number of vertces for all the p s n en r.inths case, the complexty of the convex hull algorthm s O(n en r log n en r), and the complexty of the other three algorthms s O(n en r). Fgure gves an example of the constructon of an enclosng emtter. P Emtter P Recever Fgure 0: The projecton of the emtter on the plane P from a gven vertex. P Emtter P Vew from above Enclosng emtter Fgure : Buldng an enclosng emtter n order to fnd an upper bound Implementaton and Testng Refnement Crteron Once we have access, for each teraton, to the mnmum and maxmum form-factor, t s possble to mplement a refnement crteron based on ther dfference. Followng the algorthm suggested by Lschnsk, we refne every nteracton such that: A recever (B maxf max B mnf mn) >ε Ths means that we refne an nteracton whenever the uncertanty on the ncomng energy of the recevng patch s above the threshold ε Resultng Mesh Smplfcaton In regons where the Hessan matrx of the form-factor s defnte-negatve, we know that the form-factor can be bounded between the tangent planes and the secant planes. We can use these boundng planes to fnd tghter upper and lower bounds for the form-factor. The form-factor for all the ponts on the recevng patch

11 0 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty les below all the tangent planes for the pont-to-area form factor, and above the secant plane. Therefore, we can say that our uncertanty on the pont-to-area form-factor on the recever s equal to the maxmum of the dstance between the secant plane and these tangent planes. Computng ths dstance s agan a lnear programmng problem (see, for example, Preparata 8 ). The complexty depends on the number of vertces of the recever, n r,whchs usually three or four. Let us denote by E FF ths uncertanty on the form-factor. E FF can be used n our expressons as a replacement for F max F mn. Usng the fact that: (B maxf max B mnf mn) = B max(f max F mn)+f mn(b max B mn) we decde to refne a gven nteracton f A recever (B maxe FF + F mn(b max B mn)) >ε It must be noted that ths new boundng of the form-factor does not ntroduce any uncertanty. We are stll boundng the form-factor by fully relable functons. However, snce these functons are affne nstead of constants, they provde much tghter bounds, and we can expect a smpler mesh n the areas where the pont-to-area form factor s concave. Fgure 25 (n the color secton) shows the result of our refnement crteron on a smple box, wth only drect llumnaton. Notce that the mesh produced s coarser n some areas wth respect to the mmedately neghbourng areas (the dsc-shaped area on the floor, and the drop-shaped areas on the walls). These are the places where the Hessan s defntenegatve. Ths refnement crteron extends, n some ways, the mesh smplfcaton found n prevous work (Holzschuch 9 ). The shape of the mesh produced s qute smlar between our new algorthm and the algorthm n Holzschuch 9. However, our new refnement crteron, whle keepng low memory costs, also gves fully relable upper and lower bounds on the radosty of each patch Dealng wth Sngulartes Relatve Complexty of the Algorthm Our algorthm requres the computaton of the frst two dervatves of the pont-to-area form-factor at the vertces of the recever. Ths mples a 00 % ncrease on the computaton tme for each vertex (see appendx B). That s to say, computng the pont-to-area form-factor and ts dervatves costs twce what t would cost to compute the pont-to-area form-factor alone. Snce vertces are shared by several patches, ths overhead cost s shared by several nteractons. On the average, we are only computng one pont-to-area form-factor and ts dervatves for each patch. Thus, the cost of our algorthm s approxmately the cost of computng two pont-to-area form-factors for each patch, plus the tme needed for the explotaton of the dervatves for computng upper and lower bounds. Exstng heurstc refnement algorthms (see Lschnsk ) compute one form-factor sample for each of the recever vertces, plus one sample at the center of the recevng patch. If we assume that the form-factor values at the vertces are shared wth the neghbourng patches, we are computng an average of two pont-to-area form-factors for each recever. Thus, the cost of the heurstc algorthm and the cost of our algorthm are roughly smlar. The man overhead of our algorthm when compared wth the heurstc algorthm s the tme needed for the actual computatons for fndng the poston of the maxmum and for fndng an upper bound for the maxmum, when necessary. Hence, the relatve costs of our refnement crteron are n fact qute small and can be generally regarded as acceptable, especally wth respect to the complete control t gves on the error carred by each nteracton. Also, our algorthm allows for a sgnfcant mesh smplfcaton (see fgure 25, n the color secton) whch may, dependng on the scene consdered, nduce a smaller computaton tme for the exhaustve refnement crteron when compared to a heurstc refnement crteron. 5. Error Control for Partally Occluded Interactons The above algorthm for fndng upper and lower bounds only works n the case of unoccluded nteractons, and wth a convex emtter. Ths algorthm reles on the concavty and unmodalty conjectures, whch do not hold f there are occluders between the emtter and the recever. However, t s possble to construct, usng geometrcal tools, a mnmal and a maxmal emtter that have the followng qualtes: both are convex; any pont of the mnmal emtter s fully vsble from the recever; the maxmal emtter contans all the ponts of the emtter that are vsble from at least one pont of the recever; Then at any gven pont on the recever, the form-factor due to the mnmal emtter s lesser or equal to the actual form factor, and the form-factor due to the maxmal emtter s greater or equal to the actual form-factor. We apply our prevous algorthm to these emtters, and fnd a lower bound usng the mnmal emtter, and an upper bound usng the maxmal emtter. Fgure 26 (n the color secton) shows an example of mnmal and maxmal emtters for a smple confguraton wth only one occluder: the small red square on the ground s the

12 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty Maxmal Emtter Recever Emtter Recever Occluders Fgure 2: A sngle nteracton wth occluders. Occluders Emtter Complement of "Umbra" Fgure 4: The maxmal emtter can be any convex ncludng the complement of the umbra regon. ver Emtter Our mnmal emtter Occluders Fgure 3: Computng the umbra and penumbra volumes usng the recever as a lght source. Recever Occluders Emtter Complement of "penumbra": several canddates for the mnmal emtter Fgure 5: Several possble canddates for the mnmal emtter. recever; the black square wth a whte border s the occluder, and the brght red area s the mnmal emtter the part of the emtter that s vsble from all the ponts of the recever. The dark red area s the maxmal emtter. The blue lne s the contour of the emtter as t s seen from one of the ponts of the recever. 5.. Computng the mnmal and maxmal emtter Our defnton of mnmal and maxmal emtter bears a strong resemblance wth the defnton of umbra and penumbra, except that the roles of the emtter and the recever are reversed. A smlar algorthm has been used by Teller to computer the antpenumbra of an area lght source 22, and to solve the vsblty problem n a herarchcal radosty algorthm 23,and by Drettaks 3. Drettaks 3 used a specfc data structure, the backprojecton, whch gves to the program the structure of the projecton of the occluders on the emtter plane, from any pont on the recever. Algorthms used for computng umbra and penumbra can be quckly adapted n order to compute the mnmal and maxmal emtter for each recever. Let us consder a sngle nteracton, wth one emtter, one recever, and occluders (see fgure 2). We compute the umbra and the penumbra volume usng the recever as a lght source (see fgure 3). The ntersecton of these volumes wth the emtter plane s a close ndcaton of where the mnmal and maxmal emtter are Computng the maxmal emtter usng the umbra volume The ntersecton of the emtter wth the umbra volume s the set of ponts on the emtter that are totally nvsble from the recever. The complement of ths ntersecton s the set of ponts on the emtter that are vsble from at least one pont on the recever. Snce our crteron only works for convex emtters, we have to buld a convex emtter that ncludes ths complement. Our basc rule s that we must not under-estmate the pontto-area form-factor, only possbly over-estmate t. Hence, the maxmal emtter must be any convex regon ncludng the prevously computed complement for example the convex hull of the complement, or the boundng-box of the complement (see fgure 4) Computng the mnmal emtter usng the penumbra volume Smlarly, the ntersecton of the emtter wth the penumbra volume s the set of ponts on the emtter that are at least partally hdden from the recever. The complement of ths ntersecton s the set of ponts on the emtter that are vsble all the ponts of the recever. Any convex regon that s ncluded n ths complement s a sutable canddate for the mnmal emtter (see fgure 5). Dependng on the poston of the occluders, t s possble

13 2 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty to have several canddates for the mnmal emtter. Ideally, we would lke to pck the canddate that gves the largest estmate for the mnmum, snce ths would gve tghter bounds, and hence reduce the number of un-necessary refnements. However, t s mpossble to fnd ths wthout computng the pont-to-area form-factor for all the canddates, whch would prove very tme-consumng. In our mplementaton, we choose the canddate wth the largest area, snce t s lkely to nduce a larger form-factor Implementaton and testng We have mplemented our algorthm for fndng upper and lower bounds for the pont-to-area form-factor usng the maxmal and mnmal emtter. Fgure 27 (n the color secton) shows the result of our refnement crteron on a smple scene, wth a sngle occluder. Notce that the algorthm detects the shadow boundares and refnes properly n order to model them. Outsde of the shadow, the mesh produced s dentcal to the mesh produced wthout occluders Complexty of the Algorthm and Possble Improvements Our algorthm reles on computaton of the umbra and penumbra volumes for all the nteractons. Ths computaton can be qute costly, f t s mplemented n a nave way. Prevous work by Chn 24 has shown that the use of a BSP-tree can greatly mprove the computaton of umbra and penumbra volumes. Teller 22 showed that by extendng the data structure used to store the nteracton between patches to also store the possble occluders for ths nteracton, the complexty of vsblty computatons could be greatly reduced. Both these mprovements work wth our algorthm. Our algorthm can also be used n a combnaton wth standard dscontnuty meshng, as descrbed n Lschnsk. A prelmnary lght-source dscontnuty meshng wll reduce the complexty of the mnmal and maxmal emtter computatons by provdng occluson nformaton and reducng the number of patches where we have to compute these emtters. The backprojecton algorthm descrbed by Drettaks 3, 3 gves for each patch created durng the dscontnuty meshng step the geometrc structure of the emtter as seen from ths patch. Implementng our algorthm on top of a backprojecton algorthm should be a straghtforward postprocessng step. It has been shown (Lschnsk and Drettaks 3, 3 ) that the boundary of the umbra volume can nclude a quadrc surface, and hence can be qute complex to model. However, our algorthm does not requre a complete computaton of the umbra and penumbra volumes for each nteracton, but only the computaton of a surface ncluded n the umbra volume, and of a surface enclosng the penumbra volume. Two such surfaces can be computed n a straghtforward way: For each occluder: For each recever vertex, compute the projecton of the occluder onto the emtter supportng plane; The ntersecton of these projectons s the umbra volume for ths partcular recever; The convex hull of these projectons s the penumbra volume for ths recever. The unon of the penumbra volumes for all occluders s the penumbra volume for the entre nteracton. The unon of the umbra volumes for all occluders s not equal to the umbra volume for the entre nteracton. However, t s ncluded nto the actual umbra volume (see Lschnsk ). Hence, we can use t for buldng the maxmal emtter. The computaton of the projecton of the occluders onto the emtter supportng plane, and the computaton of the unon of these projectons can be reused for computng the exact value of the pont-to-area form-factor n the radosty propagaton phase. The only extra cost of our refnement crteron s then the computaton of the mnmal and maxmal emtter knowng the projecton of all the occluders on the emtter plane. Ths s a two-dmensonal problem, computng a convex regon that contans the complement of the umbra volume, and another convex regon that s ncluded nto the complement of the penumbra volume. Note that we do not have to explctely construct the umbra and the penumbra volume, only the two convex regons. We can use several methods for computng these convex regons, as descrbed n secton The cost of our algorthm s the cost of fndng two convex regons enclosng n rn polygons, where n s the number of occluders, and n r s the number of vertces of the recever. The heurstc algorthm descrbed by Lschnsk uses the same computaton of the exact values of the pont-to-area form-factor at the vertces of the recever, whch wll be reused n the radosty propagaton phase, plus the computaton of the pont-to-area form-factor at the center of the recevng patch, whch mples the projecton of the occluders on the emtter supportng plane and the computaton of the unon of these projectons. Hence, the cost of the heurstc algorthm s n projectons and the unon of n two-dmensonal polygons. 6. Conclusons and Future Drectons We have ntroduced a new and relable way of computng the maxmum and the mnmum of the pont form-factor on any nteracton. These bounds on the form-factor allow a control of the precson of the herarchcal radosty algorthm,

14 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty 3 precson that can be requred for certan applcatons of the algorthm, such as archtectural plannng. These bounds have been ntegrated n a new refnement crteron for herarchcal radosty. We have also presented another refnement crteron that, whle mantanng control on the upper and lower bounds of the energy transported, allows a coarser mesh to be constructed n some places, thus reducng memory and computaton costs. Ths algorthm s a sgnfcant step n error-control for global llumnaton methods. Although t has been devsed and mplemented n a herarchcal radosty framework, nothng n the algorthm prevents the refnement crteron to be mplemented wth progressve refnement radosty, as descrbed by Cohen 25. Knowledge of the error produced n all the parts of the algorthm allows global llumnaton programs to concentrate ther work on parts of the scene where the error s stll large, and to skp parts where t can be neglected. Thus, our algorthm can be hoped to accelerate global llumnaton computatons by reducng the amount of unnecessary refnement. Our algorthm reles on several conjectures: the unmodalty conjectures (U and U2) and the concavty conjectures (C), as well as on a knowledge of the radosty dervatves. Table recalls, for each part of the algorthm, whch conjecture and whch dervatves are beng used. The concavty and unmodalty conjectures assume that radosty on the emtter s constant, that the recever s dffuse and that there s full vsblty. An extenson of our errorcontrol algorthm to cases where radosty on the emtter s not constant, or to reflectance functons that are not constant would frst requre a careful study of to what extent do our concavty or unmodalty conjectures stll hold. For example, t s clear that they cannot hold for whatever dstrbuton of radosty on the emtter, but only for specfc cases. These specfc cases, once dentfed, can be used as a functonal bass for radosty. We have dealt wth the partal vsblty problem by computng maxmal and mnmal emtter, thereby reducng the problem to two full vsblty problems. However, t s known that t s possble to compute the radosty gradent n presence of occluders (see Arvo 5 ), and t seems possble to compute the radosty Hessan n presence of occluders as well (see Holzschuch 7 ). In ths case, t would be possble to extend our refnement crteron to some partally vsble nteractons wthout havng to compute the maxmum and mnmum emtter. Once agan, ths can be done only n specfc confguratons where the concavty or unmodalty conjectures stll hold. Ths s not the case for generc occluders (see fgure 28, n the color secton), but only for certan specfc, smple occluders (see fgure 29 n the color secton). Although the algorthm descrbed n ths paper makes use of the U, U2 and C conjectures, and of the form-factor gradent and Hessan, table shows that t s possble to buld a smpler algorthm to fnd upper and lower bounds by usng only U, U2 and the form-factor gradent. Ths algorthm would be very smlar to the gradentdescent algorthms descrbed by Arvo 5 and Drettaks 2, 3. The man dfference would be the use of geometrc tools, as descrbed n secton to fnd an upper bound. These geometrc tools wll provde a fully relable upper bound on the recevng patch. Ths smpler algorthm would not allow mesh smplfcaton as descrbed n secton 4.3.2; also, snce ths smpler algorthm would only use geometrc methods to fnd upper bounds t can be expected that t wll gve greater upper bounds, and hence nduce more refnement than our current algorthm. On the other hand, ths algorthm would not requre the computaton of the form-factor Hessan, thus savng computaton tme, and would probably be easer to extend to partal vsblty cases, where C may not hold. Future work wll nclude an mplementaton of ths smpler algorthm, and tmng and memory costs comparsons between our full algorthm, the smpler algorthm and the heurstc algorthm, as well as error measurements. 7. Acknowledgements The frst author has been funded by an AMN grant from Unversté Joseph Fourer from 994 to 996. References. D. Lschnsk, B. Smts, and D. P. Greenberg, Bounds and Error Estmates for Radosty, n Computer Graphcs Proceedngs, Annual Conference Seres, 994 (ACM SIGGRAPH 94 Proceedngs), pp , (994). 2. G. Drettaks and E. Fume, Accurate and Consstent Reconstructon of Illumnaton Functons Usng Structured Samplng, n Computer Graphcs Forum (Eurographcs 93), vol. 2, (Barcelona, Span), pp. C273 C284, (September 993). 3. G. Drettaks, Structured Samplng and Reconstructon of Illumnaton for Image Synthess, CSRI Techncal Report 293, Department of Computer Scence, Unversty of Toronto, Toronto, Ontaro, (January 994). 4. C. M. Goral, K. E. Torrance, D. P. Greenberg, and B. Battale, Modellng the Interacton of Lght Between Dffuse Surfaces, n Computer Graphcs (ACM SIGGRAPH 84 Proceedngs), vol. 8, pp , (July 984). 5. P. Schröder and P. Hanrahan, On the Form Factor Between Two Polygons, n Computer Graphcs Proceedngs, Annual Conference Seres, 993 (ACM SIG- GRAPH 93 Proceedngs), pp , (993).

15 4 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty Parts of the algorthm Conjectures requred Dervatves requred Poston and value of the mnmum U and U2 None Poston of the maxmum U2 Gradent Usng tangents to fnd an upper bound C Hessan Usng geometrc algorthms to fnd an upper bound None None Smplfcaton of the mesh C Hessan Table : Dependences for the dfferent parts of the algorthm 6. M. Cohen and D. P. Greenberg, The Hem-Cube: A Radosty Soluton for Complex Envronments, n Computer Graphcs (ACM SIGGRAPH 85 Proceedngs), vol. 9, pp. 3 40, (August 985). 7. P. Hanrahan, D. Salzman, and L. Aupperle, A Rapd Herarchcal Radosty Algorthm, n Computer Graphcs (ACM SIGGRAPH 9 Proceedngs), vol. 25, pp , (July 99). 8. S. J. Gortler, P. Schröder, M. F. Cohen, and P. Hanrahan, Wavelet Radosty, n Computer Graphcs Proceedngs, Annual Conference Seres, 993 (ACM SIG- GRAPH 93 Proceedngs), pp , (993). 9. N. Holzschuch, F. Sllon, and G. Drettaks, An Effcent Progressve Refnement Strategy for Herarchcal Radosty, n Ffth Eurographcs Workshop on Renderng, (Darmstadt, Germany), pp , (June 994). 0. P. Heckbert, Dscontnuty Meshng for Radosty, n Thrd Eurographcs Workshop on Renderng, (Brstol, UK), pp , (May 992).. D. Lschnsk, F. Tamper, and D. P. Greenberg, Dscontnuty Meshng for Accurate Radosty, IEEE Computer Graphcs and Applcatons, 2(6), pp (992). 2. D. Lschnsk, F. Tamper, and D. P. Greenberg, Combnng Herarchcal Radosty and Dscontnuty Meshng, n Computer Graphcs Proceedngs, Annual Conference Seres, 993 (ACM SIGGRAPH 93 Proceedngs), pp , (993). 3. G. Drettaks and E. Fume, A Fast Shadow Algorthm for Area Lght Sources Usng Backprojecton, n Computer Graphcs Proceedngs, Annual Conference Seres, 994 (ACM SIGGRAPH 94 Proceedngs), pp , (994). 4. R. Segel and J. R. Howell, Thermal Radaton Heat Transfer, 3rd Edton. New York, NY: Hemsphere Publshng Corporaton, (992). 5. J. Arvo, The Irradance Jacoban for Partally Occluded Polyhedral Sources, n Computer Graphcs Proceedngs, Annual Conference Seres, 994 (ACM SIGGRAPH 94 Proceedngs), pp , (994). 6. N. Holzschuch and F. Sllon, Accurate Computaton of the Radosty Gradent for Constant and Lnear Emtters, n Renderng Technques 95 (Proceedngs of the Sxth Eurographcs Workshop on Renderng) (P. M. Hanrahan and W. Purgathofer, eds.), (New York, NY), pp , Sprnger-Verlag, (995). 7. N. Holzschuch, Le Contrôle de l Erreur dans la Méthode de Radosté Herarchque (Error Control n Herarchcal Radosty). Ph.D. thess, Équpe MAGIS/IMAG, Unversté Joseph Fourer, Grenoble, France, (March 5th, 996). 8. F. P. Preparata and M. I. Shamos, Computatonal Geometry An Introducton. New York: Sprnger Verlag, (985). 9. J. D. Foley, A. van Dam, S. K. Fener, and J. F. Hughes, Computer Graphcs, Prncples and Practce, Second Edton. Readng, Massachusetts: Addson-Wesley, (990). 20. T. L. Kay and J. T. Kajya, Ray tracng complex scenes, Computer Graphcs, 20(4), pp (986). Proceedngs of SIGGRAPH 86 n Dallas (USA). 2. D. L. Toth, On ray-tracng parametrc surfaces, Computer Graphcs, 9(3), pp (985). Proceedngs SIGGRAPH 85 n San Francsco (USA). 22. S. J. Teller, Computng the antpenumbra of an area lght source, n Computer Graphcs (ACM SIG- GRAPH 92 Proceedngs), vol. 26, pp , (July 992). 23. S. Teller and P. Hanrahan, Global Vsblty Algorthms for Illumnaton Computatons, n Computer Graphcs Proceedngs, Annual Conference Seres,

16 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty 5 da 0.6 (0,0) θ n (u,v) Fgure 6: A dfferental area emtter and an nfnte recevng plane. Fgure 7: An example of the pont-to-area form-factor functon (θ = π 6 ). 993 (ACM SIGGRAPH 93 Proceedngs), pp , (993). 24. N. Chn and S. Fener, Fast object-precson shadow generaton for areal lght sources usng BSP trees, Computer Graphcs (992 Symposum on Interactve 3D Graphcs), 25(2), pp (992). 25. M. Cohen, S. E. Chen, J. R. Wallace, and D. P. Greenberg, A Progressve Refnement Approach to Fast Radosty Image Generaton, n Computer Graphcs (ACM SIGGRAPH 88 Proceedngs), vol. 22, pp , (August 988). Appendx A: Concavty conjectures: case study of a dfferental area emtter Let us consder the case of an nfnte recevng plane and a sngle dfferental area for the emtter. In ths case, due to the symmetres shared by the emtter and the recever, there s only one parameter: the angle, called θ, between the normal of the emtter and a lne parallel to the recever, (see fgure 6). To express the poston of a pont on the recever, we choose a set of axes related to the emtter: the frst axes shares the drecton of the projecton of the normal of the emtter on the recever, and the second axes s orthogonal to the frst. The orgn of our coordnate system s the projecton of the emttng pont. Usng ths set of coordnates, we have a smple expresson for the pont-to-area form-factor at any pont M(u, v) on the recever (see fgure 7 the aspect of the surface): F (u, v) = da π u cos(θ)+sn(θ) (u 2 + v 2 +) 2 Ths value s only for u cos(θ)+sn(θ) > 0. Ifu cos(θ)+ sn(θ) 0, then of course F (u, v) =0. The C concavty conjecture In ths smple case, t s possble to explctly compute the dervatves of the pont-to-area form-factor. An explct computaton of the Hessan shows that t s defnte f and only f θ = 0 θ = 0.2 θ = π/6 θ = π/4 θ = π/3 θ = π/2 Fgure 8: The areas where the pont-to-area form-factor functon s concave for dfferent values of θ. the expresson S(u, v, θ) s postve, where S(u, v, θ) s: S = 3u 4 8tanθu 3 5tan 2 θu 2 4u 2 v 2 +3u 2 +4u tan θ 8tanθuv 2 5tan 2 θv 2 v 2 v 4 +tan 2 θ Although t s mpossble to fnd an explct soluton of the equaton S(u, v, θ) =0, t s possble to plot these solutons for dfferent values of θ. Fgure 8 shows the contour oftheareawheres(u, v, θ) s postve for dfferent values of θ. Outsde these areas, S(u, v, θ) s negatve, and hence the Hessan matrx s ndefnte. Insde these areas, S s postve, and the form-factor s concave. An nterestng pont s the shape of the zones where the pont-to-area form factor s concave. When θ = π,tsof 2 course a dsc, due to the symmetres n the scene. When θ = 0, t s a shape lke a drop, that tapers to a pont n (0, 0).For ntermedate values of θ, the zone has an ntermedate shape between the drop and the dsc, but ths shape always appears to be convex. The C2 concavty conjecture If we now focus on the radosty on a specfc lne v = au+ b on the recevng plane, we have, for the form-factor as a

17 6 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty functon of u, f(u) = da π u cos(θ)+sn(θ) (u 2 +(au + b) 2 +) 2 The form-factor s equal to f(u) f u cos(θ)+sn(θ) > 0. If u cos(θ)+sn(θ) 0, then the form-factor s null. It must be noted that f(u) goes to zero when u goes to ±, andthatf(u) s equal to zero only for u = u 0 = tan θ. It s possble to compute the frst and the second dervatve of f(u). The frst dervatve, f (u), s of the sgn of a second degree polynomal n u, and the second dervatve, f (u) s of the sgn of a thrd degree polynomal n u. As a consequence, f (u) can change sgn at most twce, and f (u) at most three tmes. Snce the functon f(u) goes to zero when u goes to ±, t must have one maxmum between u 0 and +, and one mnmum between u 0 and. As a consequence, f (u) must change sgn exactly twce. Let us call u and u 2 the ponts where the frst dervatve changes sgn (u <u 0 < u 2). f (u) also goes to zero when u goes to ±. As a consequence, t must have one mnmum between u 2 and +, and another between and u, and t must have one maxmum between u and u 2. So the second dervatve changes sgn exactly three tmes. One of the pont where the second dervatve changes sgn s smaller than u, whch s smaller than u 0, and one of them s greater than u 2, whch s greater than u 0. Then the second dervatve changes sgn at least once and at most twce on [u 0, + ]. Whenu goes to +, f s convex, and f s postve. So we just proved that f can be negatve only over a unque bounded segment on [u 0, + ]. The form-factor on the lne s equal to f(u) for u>u 0, and null everywhere else. So the form-factor on a lne s concave only over a unque bounded segment. Ths proves the C2 conjecture for a dfferental area emtter. Fgure 9 shows an example of such a f(u) functon, along wth ts frst and second dervatves. It can be noted that ths functon s concave over a sngle segment, and convex everywhere else. Appendx B: Effectve computaton of the form-factor dervatves In ths secton, we show how t s possble to compute the dervatves of the pont-to-area form-factor wth lttle addtonal computaton expense. In partcular, t s shown that the computaton of the exact value of the form-factor dervatves s always cheaper than the computaton of an approxmate value usng several form-factor samples. For example, the cost of computng the A 2 E e r E A + r + Fgure 20: Notaton when the emtter s a polygon. γ n F =0 foreach edge [E E + ] r = E x r + = E + x crossprod = r r ( + r r + ) gamma = arccos r r + I = gamma crossprod mxt = n crossprod F = I mxt F = 2π Fgure 2: Pseudo-Code for computng the form-factor. form-factor gradent s 30 %, whle computng an approxmate value of the gradent would requre two form-factor samples, thus ncreasng computaton tme by 00 % The Pont-to-Area Form-Factor Let us recall that the pont-to-area form-factor from a pont x on a patch A to a patch A 2 (see fgure ) can be expressed as a contour ntegral: r 2 d F (x) = n l 2 2π A 2 r 2 2 For the explct dervaton of ths contour ntegral from the equaton 2, see Segel and Howell 4. In the case where the emtter s a polygon, ths expresson smplfes to a fnte sum: F (x) = 2π n γ (6) where γ s the vector of norm γ, and of drecton the crossproduct r r + (see fgure 20). An example pseudo-code for computng the form-factor usng equaton 6 can be found n fgure 2. Ths pseudocode makes use of the standard 3D operatons lke addton, cross-product and dot product. x

18 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty 7 Lne cuttng the functon f(x) A lne cuttng through the radosty functon 2 d(x) The radosty functon on the lne 30 s(x) The frst dervatve of the radosty functon on the lne The second dervatve s negatve only over a segment. Fgure 9: The radosty on any lne on the recevng plane s concave only over a segment. Form-Factor Gradent The pont-to-area form-factor gradent can be easly computed by dervaton of the prevous formula (see Arvo 5,or Holzschuch 7 ): F (x) = n e I 2π Wth: I = I 2 = J 2 = 2e 2 +2 n ( r r +)( r I 2 + e J 2) γ e r ( 2 e r 2 ( r 2 r 2 + e r + r+ 2 ) e r r 2 e r e 2 I 2 + e 2 I ) F =0 G = 0 foreach edge [E E + ]. F = I mxt e = E E + I 2 = e r + r + 2 e r r 2 + e 2 I I 2 / = 2crossprod 2 ( J 2 =0.5 r 2 r + 2 e r I 2 J 2 / = e 2 G+ =( n e )I +2mxt( r I 2 + e J 2 ) F = 2π G = 2π Fgure 22: Pseudo-Code for computng the gradent of the form-factor. ) The code n fgure 2 for computng the form-factor can be extended for computng the gradent. Fgure 22 shows the extenson of the pseudo-code needed for computng smultaneously the pont-to-area form-factor and ts gradent (we dd not nclude the part of the code that s exactly dentcal). As can be seen, most of the costly computatons lke nverse trgonometrc functons have been done for the form-factor, and do not need to be redone for the gradent. The exact extra cost of computng the gradent de- pends on the computer and on the compler used. On an R4000 SGI wth the standard cc compler, t s 30 % (see Holzschuch 6, 7 ). What s fundamental s that t actually costs much less to compute the exact value for the gradent than t would cost to compute two radosty values, and then to approxmate the gradent usng these values.

19 8 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty F =0 G = 0 H =0 foreach edge [E E + ]. G+ =( n e )I +2mxt( r I 2 + e J 2 ) I 3 = e r + r + 4 e r r 4 + e 2 I 2 I 3 / = 4crossprod 2 J 3 =0.25 ( r 4 r 4 + ) e r I 3 the form-factor alone (see Holzschuch 7 ), meanng that the overall cost of computng the pont-to-area form-factor and ts frst two dervatves s 2. tmes the cost of computng the form-factor alone. Notce t s much faster to compute the exact value than t would be to compute an approxmate Hessan matrx whch would requre seven separate formfactor computatons. J 3 / = e 2 K 3 = I 2 r 2 I 3 2 e r J 3 K 3 / = e 2 H+ = mxti 2 I + Q( r I 2 + e J 2, n e ) +2mxt(Q( r, r )I 3 + Q( e, e )K 3 +2J 3 Q( e, r )) F = 2π G = 2π H = π Fgure 23: Pseudo-Code for computng the frst two dervatves of the form-factor. Hessan matrx for the pont-to-area form-factor The pont-to-area form factor Hessan matrx can also be computed by dervaton of Equaton 6 (see Holzschuch 7 ): H = Q ( n e, r I 2 + e J 2) π n ( r e )I 2I +2 n ( r e )(Q( r, r )I 3 +Q( e, e )K 3 +2J 3Q( r, e )) We use the followng notaton: Q( a, b) = a t b + b t a ( ) I 3 = e r + 4 e r 2 r+ 4 e r r 4 +3e 2 I 2 ( ) J 3 = 4e 2 r 4 r+ 4 r e e 2 I 3 K 3 = ( ) I2 e 2 r 2 I 3 2( r e )J 3 The code for computng the form-factor and the gradent can be extended to compute the second dervatve as well. Fgure 23 shows the extenson of the pseudo-code needed for computng smultaneously the pont-to-area form-factor and ts frst two dervatves (we dd not nclude the part of the code that s exactly dentcal). Once agan, recyclng geometrc computatons prevously done reduces the cost of computng the Hessan matrx, even f the cost s stll hgh snce matrx operatons are qute expensve: a sngle matrx addton has the same cost as 9 standard addtons. The exact extra cost of computng the Hessan matrx depends on the computer and on the compler. On a R4000 SGI, wth the standard cc compler, t s 80 % of the cost of

20 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty 9 Common symmetry plane The maxmum les on ths lne Locaton of the maxmum Fgure 24: The symmetres of the scene can help fnd the locaton of the maxmum. Fgure 25: Drect llumnaton wth our refnement crteron, unoccluded scene.

21 20 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty Fgure 26: Mnmal and maxmal emtter for a smple confguraton. Fgure 27: Drect llumnaton wth our refnement crteron, wth one occluder.

22 N. Holzschuch and F. X. Sllon / An exhaustve error-boundng algorthm for herarchcal radosty 2 Fgure 28: Wth generc occluders, the unmodalty conjectures do not hold. Fgure 29: Wth certan occluders, the unmodalty conjectures stll holds.