A Formalizaion of Ray Casing Opimizaion Techniques J. Revelles, C. Ureña Dp. Lenguajes y Sisemas Informáicos, E.T.S.I. Informáica, Universiy of Granada, Spain e-mail: [jrevelle,almagro]@ugr.es URL: hp://giig.ugr.es Absrac The ray-scene inersecion es is he mos cosly process when a scene is rendered. This process may be improved using any sraegy o be able o speed-up i. Generally, any sraegy used is based on he building of a spaial indexing in he scene domain or in he rays domain. In his paper, an acceleraion echniques formalizaion is proposed. This formalizaion allows an opimizer o be specified according o he spaial index used. Furhermore, a formalizaion of opimizer composiion is presened. Finally, we presen an expression which allows o compare several opimizers, and selec he one wih bes performances. This formalizaion is based on he graphics objecs heory and claims o be a generalizaion o all opimizers which use spaial indexing. Keywords: Graphics objec heory, spaial indexing, ray casing, acceleraion echniques. 1 Inroducion Usually he programs based on ray racing include acceleraion echniques in order o improve he ray-scene inersecion es. Several works have been proposed on his subjec. Glassner [10], Fujimoo and Iwaa [8] presened echniques based on boh uniform and non-uniform spaial subdivision, using a regular 3D grid and an ocree respecively. Boh sraegies are based on subdivision of a 3D region which includes he whole scene. In hese echniques i is necessary o design an algorihm for compuing he inersecion beween a ray and he spaial srucure. Arvo and Kirk [3] proposed anoher echnique called ray classificaion. I consiss of pariioning he five-dimensional space of rays ino small regions which are encoded as 5D hypercubes. Each hypercube is associaed wih a lis of scene objecs ha are oally or parially inside.
Haines and Greenberg [13] proposed a sraegy o improve he shadow es, via he ligh buffer. This echnique is only used in ray racing for spo or direcional ligh sources. Based on he above echniques, some effor has been devoed o develop new algorihms o raverse spaial indexing schemes. In order o improve he raversing process using a regular 3D grid, several works were proposed [1, 8, 7]. Oher papers were presened describing improvemens o he basic ocree raversal algorihm [5, 7, 9, 16, 18, 21]. There are also available echniques which employ oher schemes such as he hierarchical bounding volumes [17, 12] and binary rees [23]. As i has been menioned above, a lo of effor has been employed in he developmen of efficien soluions o he problem of ray-scene inersecion es. In his paper we propose an absrac model of a generic opimizer as a funcion which selecs a se of candidae objecs for a given scene and a given ray. We also propose a model of opimizer based on spaial indexing and wo sraegies o obain new opimizers by composing oher opimizers. The paper is organized as follows: In secion 2 a comparaive wih respec o oher relaed work is presened. In secion 3 several conceps, definiions, and noaions are given. In secion 4, an absrac characerizaion of a sric opimizer and an opimizer are presened. Secion 5 presens several composiions of opimizers, and he recursive opimizers are presened. In secion 6, an efficiency measuremen is proposed. Finally we include our conclusions and planned fuure work. 2 Previous Work Wih respec o our previous work presened in [15], he main differences beween ha work and he one presened here lie in he definiion of an opimizer and is properies. Basically, ha reference inroduced an opimizer as a pair of elemens: he spaial represenaion (a se of graphics objecs called volumeric objecs) obained from he scene, and he raversal funcion which reurns he volumeric objecs inerseced by a given ray. However, in his paper an opimizer is defined as a funcion which reurns a subse of candidae scene objecs from a given scene and a given ray. This definiion of opimizer can model any acceleraion echnique because is more general han previous one, as i focuses more on behaviour and exernal properies han on srucure. Now we view he opimizers based on spaial indexing as a subclass of a more general opimizers class.. oreover, ha reference presens he concep of spaial index funcion as a funcion which reurns a subse of scene objecs which can be inerseced by a ray from a spaial represenaion. Scene objecs obained hrough his funcion are poenialy inerseced by he ray, as hey inersec volumeric objecs also inerseced by he ray. Oher ousanding difference beween hese works is in he concep of composed opimizers. In his work, he same kinds of composed opimizers are presened bu now using a differen formal framework, derive from he new, more general, definiion of an opimizer. Furhermore, a new kind of composed opimizer based on he sequenial composiion of opimizers is presened.
Finally, an opimizaion efficiency measuremen is presened. This measure is presened as a way o compare he ime efficiency of an opimizer wih respec o he null opimizer. 3 Conceps, Definiions and Noaions We focus on opimizers ha improve he ray-objec inersecion es using a spaial index scheme. This scheme is a represenaion of he spaial disribuion of scene objecs. I is composed of a se of voxels, which will be called here volumeric objecs. Each volumeric objec conains informaion of a subse of scene objecs, which are conained in i. The rayobjec inersecion es performs he es wih each volumeric objec and reurns a lis of scene objecs. To formalize he opimizer behaviour, we use he graphic objecs heory [24, 25] and an iniial work abou opimizers which was proposed in [15]. 3.1 Graphic Objec Theory A graphic objec is a pair, in which: is a funcion called presence funcion defined as, where is a presence domain, and is a funcion called aspec funcion wih domain in and range in, where is called aspec domain. We adop as presence domain a subse of, more concreely,. Using his presence domain, a graphic objec is equivalen o an arbirary se of poins in space. The aspec domain is no defined because i is no necessary in he curren framework. We only need he spaial region occupied by a graphic objec. The se of all graphic objecs is denoed by!. For each graphic objec #"! $%&, we define he spaial region Vol as he se of all poins " ( )* such ha. Formally i is:,"! )- Vol " -. ( (/0 (1) where is he presence funcion of The null or empy graphic objec, denoed by 1, is he unique graphic objec saisfying he following propery " ( )2 34 where 1 56. This graphic objec fulfils Vol 1 6 ( denoes he empy se of poins). The presence domain (ha is, he se 77890 ), saisfies he properies of a boolean algebra. This presence domain includes operaors such?> as he union (: ), he inersecion (; ), and he complemen (< ). For any pair of values = " 0 he following expressions are fulfilled: >A =@: >A =@; <%=?> ax = B>C in = ED =
_ R < : ; where in and ax have he usual meaning. Thereupon, he se of graphic objecs! inheris his boolean algebra srucure. The union of wo graphic objecs, F G F F and IH G4 H H 4, is an objec whose presence funcion saisfies he following expression: " ( (% F The graphic objec can be wrien as F(J H. The inersecion of wo graphic objecs, F 74 F? F and H 7 H H whose presence funcion saisfies he following expression: " ( (% F H The graphic objec can be wrien as F(K H. For any graphic objec F F 1 F ha: " ( ) The graphic objec will be wrien as < F H (2), is an objec is complemen is a graphic objec L? & F 4 Opimizer Absrac Characerizaion 4.1 Rays When a opimizer is used ino a rendering sysem, is behaviour can be undersood as a funcion which reurns a subse of candidae objecs for a given ray and a scene. The number of reurned scene objecs mus be lesser han he number of objecs in he scene, in order o reduce he number of ray-objec inersecion ess. We can define a ray as a graphic objec. A ray? & is a graphic objec such ha exiss a unique poin N ", and a unique direcion vecor O "QP such ha: " ( if SUT " V. NXWYTZO (5) V [ oherwise where is he subse of real values sricly greaer han zero, and P is he se of uni lengh vecors in. The poin N is he origin of he ray, and he vecor O is he direcion of he ray. Every ray has associaed a unique origin and a unique direcional vecor. From he above definiion, we deduce ha he volume of a ray is an infinie half-line in. The se of all rays is denoed by \. A ray and a graphic objec may have some poins in common, ha is, hey may inersec. When his happens, we can measure he disance from he origin o he neares common poin, and his will be a posiive real value. When no inersecion occurs, we say ha his disance is infinie. _, where is any elemen ha i is no included in. This value is used o denoe a infiniy disance. By holds `ba ] (]^/ In order o formalize his concep, we define he se as J I_% definiion, any value ` ". (3) such (4)
d u u h d u _ d. h.. h h 4.2 Inersecing Rays and Objecs Le " \, be a ray, and le c"! be a graphic objec, we define d T ". ( (-e NEWYTZO as follows: where is he,f presence -6 funcion of, N is he origin of, and O is he direcion vecor of. When g[6 d, an inersecion occurs beween he ray and his graphic objec. When no inersecion occurs. Funcion d reurns he se of disances from he origin o all poins in he ray which belongs also o he volume of he objec. In fac, we only need he lowes one of hese real values. We define he funcion h wih he same domain of d (] and values in. For each ", and i"! i holds: Rkjmlon pf q6 if d rq6 (7) where inf denoes he infimum of a se of real values, which is always defined even for graphics objec whose volume is no a closed region. The main ineres of he above definiions consiss of deermining which graphic objecs in a given scene are inerseced by a given ray. In wha follows, we will use he symbol s o mean he se of all scenes. 4.3 Objecs Inerseced by a Ray Le " \ if d be a ray, and le " s be a scene, we define u inerseced by, as follows: v w" ẍf _y Ẍwill conain he graphic objecs in inerseced by. (6) as he se of graphic objecs Therefore, he condiion holds. We also wan o know he neares inerseced graphic objecs wih respec o he ray origin. Le " \ be a ray, and le " s be a scene, we define uuz as he se of neares graphic objecs inerseced by as follows: The expression u z u z Ẍ 4.4 Sric Opimizer { c" ;~}I " is also saisfied. ẍf _ I a h (8) (9) An opimizer reduces he number of candidae objecs for he inersecion es. Obviously when a reduced se of scene objecs is obained, we ge an improvemen in erms of execuion ime.
u V z z u ƒ ƒ ƒ V Le ƒ be a funcion wih domain in \3 s and values in s. ƒ is a sric opimizer if and only if i fulfils he following condiion: " \ " 7 s (10) In oher words, a sric opimizer selecs a subse of he scene objecs. The opimizer ƒ yields a se of objecs inerseced by a ray, and possibly, ( oher objecs which are no inerseced. The bes opimizer is one which ) holds u, whereas he worse opimizer is one which always holds ƒ, ha is, i always yields he whole scene. 4.5 Opimizer There are applicaions where we only need he neares objec inerseced by a ray (or he nearess objecs, because i may happen ha here are more han one a minimum inersecion disance). In order o model his requiremen we inroduce he definiion of an opimizer. An opimizer is a funcion of he same class ha a sric opimizer. However he condiion we impose o he se of reurned objecs is weaker, and hus he class of opimizers conains he class of sric opimizers. Le ƒ be a funcion wih domain in \ ˆs and values in s. ƒ is an opimizer if and only if fulfils he following condiion: " \ " s u z 7 V (11) I is easy o prove ha any sric opimizer is an opimizer by using he relaion uuz which always holds. 4.6 Spaial Represenaion A spaial represenaion, from now on SR, is a se of graphic objecs. These graphic objecs will be called volumeric objecs or voxels. The se of all possible spaial represenaions will be called. When he only difference beween wo spaial represenaions is consiss in heir aspec funcions, we consider boh spaial represenaions equivalen. Le Š be a funcion wih domain in s and values in, his funcion Š is a spaial indexing F H 9ŒŒ9ŒC 9Ž z and any F Ž H Œ9ŒŒC Ž mehod (from now on SI) if and only if for any given scene given SR he following equaliy is saisfied: m F F Ž (12) This se of graphic objecs is usually simpler han he original scene, in he sense ha he ray objec inersecion es can be done faser for volumeric objecs han for original scene objecs. This propery is essenial for ray casing speed-up, because we can inersec he ray wih volumeric objecs and discard he scene objecs which are included in volumeric objecs no inerseced by he ray. Ü
. S ƒ Š f 1 In order o deermine he ray-scene inersecion es, a funcion o obain he inersecion beween a ray and a SR mus be defined. Le " \ be a ray, le " s be a scene, and le " be a SR, we consider ha when an inersecion occurs beween an objec w" and a volumeric objec, his volumeric objec is also inerseced by he ray. This se is noed as Obviously, g. This se is more formally defined as follow: i" is always saisfied. Ž " $ Ž K f 4.7 Opimizer Based On Spaial Indexing There are many differen classes of opimizers. Our aenion will be focused on a sub-ype or caegory. This sub-ype will be called opimizers based on spaial indexing. Le ƒ be an opimizer. ƒ is an opimizer based on spaial indexing if and only if he following propery is fulfilled: S Š "ˆš " s " \ Noe ha for each opimizer based on spaial indexing, here is a unique SI associaed o i (as can be deduced from he above condiion). When an opimizer of his caegory is implemened, one SI mus be implemened as well. Tha is, he necessary algorihm o build Š mus be designed and implemened. Normally, a daa srucure residing in memory for he SR Š mus be creaed. Afer ha i is possible o process a se of rays. For each ray in ha se we mus compue which voxels are inerseced. From his se of voxels we obain he se of objecs inersecing hem. The funcion models his algorihm. 5 Composing Opimizers When an opimizer is used he main goal is o obain an efficien SR. Tha is, for a given scene, an opimizer mus be seleced having ino accoun he objecs disribuion in he scene. Due o scene complexiy, i is no always easy o selec he mos appropriaed opimizer. In his case, i would be ineresing o make a pariion of he scene. Each pariion can be processed by using a differen opimizer. In shor, we have several opimizers applied o one single scene. This problem was called by Glassner as he problem of a eapo inside a sadium. Tha is, a very complex and relaively small se of objecs inside a very simple and big one. In hese cases, he available spaial represenaions were no as fas as expeced. A possible soluion was o consider some sraegy o compose wo o more differen spaial represenaions, as was poined ou in [11] as fuure effors. 1 ; Ž K (13) (14)
Š Š z 1 z The main goal is o deermine which opimizers are appropriaed o use for a given scene [4, 2]. In cases for which i is no easy o find he opimizer which has he bes performances, we propose wo ways o compose opimizers: Sequenial: This is very useful when, for a given scene, several opimizers will have beer performances han a unique opimizer. From an iniial SI, he SR is buil. In hose voxels wih a relaively grea number of objecs, a secondary SR is applied. Parallel: We can use his when, for a given scene, here is uncerainy or doub o deermine he bes opimizer. The main goal is o execue in parallel or concurrenly several opimizers (simple or composed sequenially). For each ray and each opimizer, a subse of inerseced objecs are reurned. The final resul is he inersecion of all objecs subse. In he following secions, he above described opimizers are formally defined. 5.1 Sequenial Composiion As i was menioned in he previous secion, he main goal is o separae he complex scene ino simpler subscenes. Ž Wih his purpose in mind, a new opimizer may be applied for each volumeric objec " Š. We will define he sequenial composiion as follow: Le ƒ F be an opimizer based on spaial indexing, le Š be SI associaed o ƒ F, and le ƒ H be any opimizer. For any ray " \ and any scene " s, he resul obained when is applied o is a spaial represenaion Š including œ volumeric objecs or voxels as follow: Ž F Ž H Œ9ŒŒ Ž In hese condiions, we say ha ƒ@ž is he sequenial composiion of ƒ F and ƒ H (noed as ƒ@ž ƒ F. ƒ H ) if and only if he ƒ@ž is he opimizer which holds he following condiion: ƒež Ÿ F ƒ H (16) where is he subscene of Ž including objecs which inersec, ha is: c". K Ž f I is easy o prove ha he sequenial composiion is no commuaive nor associaive in general. However, when ƒ H is also an opimizer based on spaial indexing hen he following wo condiions hold: The opimizer ƒ F. ƒ H is also based on spaial indexing. (15) (17)
Grid Ocree Figure 1: An example of sequenial composiion. For any opimizer ƒ, i holds ha ƒ F. ƒ H. ƒ ƒ F. ƒ H. ƒ (18) This formalism can be used o obain formal models of several opimizers previously proposed by several auhors [19, 20, 14, 4, 2, 22, 6]. One example for a sequenial composiion is shown a figure 1. This figure shows us a composiion of an ocree and a 3D grid. 5.2 Parallel Composiion Parallel composiion is very useful when we have a complex scene and we do no know em a priori which opimizer is he bes one o reduce he cos in erms of execuion ime. In hese cases we can use he concurren or parallel execuion of wo or more opimizers. We will define he parallel composiion of wo opimizers as follow: Le ƒ F and ƒ H be wo opimizers, le " \ be a ray, and le " s be a scene. We say ha ƒ@ is he parallel composiion of ƒ F and ƒ H (noed as ƒ@ ƒ Fi ƒ H ) if and only if ƒ@ is he opimizer which fulfils he following condiion: ƒ Ä ƒ F K ƒ H An example of a parallel composiion is shown a figure 2. Here we show wo opimizers based on spaial indexing. The firs one is based on a 3D Grid (ƒe ), and he second is based on an Ocree (ƒe ). This figure shows us ha ƒ@ reurns less objecs han ƒ@. Tha is: u5= I ƒ u5= ƒ In his case, he parallel composiion is very useful because a reduced number of objecs is reurned for mos of he rays in his scene. (19)
ƒ. Ocree Regular 3D Grid Figure 2: An example of parallel composiion. 5.3 Recursive Opimizers There are many sraegies available o build a SR based on a hierarchical pariioning of he scene. Examples of hese are ocrees [10, 8], binary rees [23], bounding volumes hierarchies [12]. All hese opimizers may be described as recursive opimizers. This definiion is very conneced wih he recursion concep and he sequenial composiion. Wih hese premises, a recursive opimizer can be defined as follows: Le ƒ be an opimizer based on spaial indexing, le be also oher opimizer based on spaial indexing. ƒ is a recursive opimizer if and only if i saisfies: ƒ (20) An appropriaed example of his group of opimizers is an ocree. An ocree can be seen as a sequenial composiion of regular grid (conaining w ˆ w b voxels) wih iself. 6 An Opimizaion Efficiency easuremen Using he above definiions and resuls, an opimizaion efficiency measuremen can be defined. This measuremen consiss of compuing he number of objecs ha an opimizer is capable o rejec for a se of rays and a given scene. A way o ge his measuremen is o use a random disribuion of rays for a given opimizer and scene. For a single ray, we can compue he relaive number of candidae objecs seleced wih respec o he oal number of scene objecs. The relaive gain in efficiency can be compued as he weighed average of his fracion for every possible ray (each ray conribuion mus be weighed by he probabiliy for ha ray o occur during usage of he opimizer). This measuremen of efficiency can be formally expressed by inroducing funcion ª, which is defined for every 3-uple composed of a opimizer, a scene, and89c a probabiliy mea- ) and is defined sure on ray space \. The funcion has real values (in he inerval «
ª ƒ ƒ as: V, 8 C Ve u5= I ± u5= ± where Card is he funcion which reurns he number of elemens which are in a se, and is a probabiliy measure funcion which models he probabiliy disribuion of he rays o be processed. Probabiliy measure depends on he usage of he opimizer in a rendering sysem, ha is, differen disribuions will appear for differen rendering algorihms or, in general, for differen applicaions of he opimizer. For insance, when a simple ray casing is applied, in mos cases he rays sar from he observer and reach a paricular surface poin of he scene. However, for oher algorihms rays may have heir origin on ligh sources. 7 Conclusions and Fuure Work In his work, a formal model of opimizer is proposed. This formal model is shown as a funcion ha reduces he number of candidae objecs for he ray-objec inersecion es. oreover, a model for opimizers based on spaial indexing has been proposed. Two formal models of composed opimizer have been presened: he sequenial and parallel composiion, in addiion o he recursive opimizers. In his formalism, a measure funcion o sudy he performances of any opimizer wih respec o oher one was proposed. As fuure work, we are planning o produce definiions of concree opimizers by applying his formal framework. Wih respec o he efficiency measuremen, we are also planning o sudy his funcion. Normally, his is based on a uniform disribuion. In erms of implemenaion, we have ino accoun he rendering algorihm applied and a reduced sample of he all rays is considered o obain an approximae esimaion of his measure funcion. Obviously, from wo given opimizers and when his measure is known, one of hese opimizers will be more suiable han anoher. In shor, an opimizer will be beer han anoher one when i is capable o reduce he average number of candidae objecs. This measuremen can be useful o selec he bes opimizer. Noice ha in his case, his measuremen does no ake ino accoun wheher he opimizer is based on spaial index or no. I only obains he performance of an opimizer wih respec o a null opimizer. I will also allows o compare he performances of wo opimizers. (21) Acknowledgemens Special hanks o Juan Carlos Torres for his conribuion o his work. This work has been suppored by a gran coded as TIC98-0973-C03-01 from he Commiee for Science and Technology of he Spanish Governmen (CICYT).
References [1] J. Amanaides and A. Woo. A Fas Voxel Traversal Algorihm for Ray Tracing. In EUROGRAPHICS 87, pages 3 10, Amserdam, 1987. [2] J. Arvo. Ray Tracing wih ea-hierarchies. In SIGGRAPH 90, volume 15(6), pages 56 62, 1990. [3] J. Arvo and D. Kirk. Fas Ray Tracing by Ray Classificaion. Compuer Graphics, 21(4):55 64, 1987. [4] J. Arvo and D. Kirk. A Survey of Acceleraion Techniques, chaper chaper 6. An Inroducion o Ray Tracing, pages 201 262. Academic Press, San Diego, 1989. [5] S. Coquillar. An Improvemen of he Ray-Tracing Algorihm. In EUROGRAPHICS 85, pages 77 88, 1985. [6] N.D. Cuong. An Exploraion of Coherence-based Acceleraion ehods Using he Ray Tracing Kernel G/GX. PhD hesis, TU-Dresden, Germany, 1997. [7] R. Endl and. Sommer. Classificaion of Ray-Generaors in Uniform Subdivisions and Ocrees for Ray Tracing. Compuer Graphics Forum, 13(1):3 20, 1994. [8] A. Fujimoo and K. Iwaa. ARTS: Acceleraed Ray Tracing Sysem. IEEE Compuer Graphics & Applicaions, 6(4):16 26, 1986. [9] I. Garganini and H.H. Akinson. Ray Tracing and Ocree: Numerical Evaluaion of he Firs Inersecion. Compuer Graphics Forum, 12(4):199 210, 1993. [10] A.S. Glassner. Space Subdivision for Fas Ray Tracing. IEEE Compuer Graphics & Applicaions, 4(10):15 22, 1984. [11] A.S. Glassner. An Inroducion o Ray Tracing. Academic Press, 1989. [12] J. Goldsmih and J. Salmon. Auomaic Creaion of Objec Hierarchies for Ray Tracing. IEEE Compuer Graphics & Applicaions, 7(5):14 20, 1987. [13] E. Haines and D.P. Greenberg. The Ligh Buffer: A Shadow Tesing Acceleraor. IEEE Compuer Graphics & Applicaions, 6(9):6 16, 1986. [14] D. Kirk and J. Arvo. The Ray Tracing Kernel. In Proceedings of Ausgraph 88, pages 75 82, 1988. [15] J. Revelles and J.C. Torres. Represenación Absraca de Opimizadores para Sínesis de Imágenes. In VI Congreso Español de Informáica Gráfica, CEIG 96, ISBN: 84-921747-0-6, pages 347 360, 1996.
[16] J. Revelles, C. Ureña, and. Lasra. An Efficien Parameric Algorihm for Ocree Traversal. In The 8h Inernaional Conference in Cenral Europe on Compuer Graphics, Visualizaion and Ineracive edia 2000, Plzen (Rep. Chequia), ISBN: 80-7082- 612-6, pages 212 219, 2000. [17] S. Rubin and T. Whied. A Three-Dimensional Represenaion for Fas Rendering of Complex Scenes. Compuer & Graphics, 14(3):110 116, 1980. [18] H. Same. Implemening Ray Tracing wih Ocrees and Neighbor Finding. Compuer & Graphics, 13(4):445 460, 1989. [19] I.D. Scherson and E. Caspary. Daa Srucures and he Time Complexiy of Ray Tracing. Visual Compuer, 3:201 213, 1987. [20] J.. Snyder and A.H. Barr. Ray Tracing Complex odels Conaining Surface Tessellaions. In SIGGRAPH 87, pages 119 128, 1987. [21] J. Spackman and P.J. Willis. The Smar Navigaion of a Ray Through an Oc-Tree. Compuer & Graphics, 15(2):185 194, 1991. [22] N. Sole and R. Caube. Discree Ray-Tracing of Huge Voxel Spaces. In EUROGRAPH- ICS 95, pages 383 394, 1995. [23] K. Sung. Ray Tracing wih he BSP Tree, pages 271 274. Academic Press, 1992. [24] J.C. Torres. Absrac Represenaion of Graphics Sysems. Graphic Objec Theory. PhD hesis, Deparmen of Lenguajes y Sisemas Informáicos, Universiy of Granada (Spain), 1992. [25] J.C. Torres and B. Clares. Graphics Objecs: A ahemaical Absrac odel for Compuer Graphics. Compuer Graphics Forum, 12(5):311 328, 1993.