REAL-TIME RENDERING OF CUT DIAMONDS

Transcription

1 Online ID 0459 Page REAL-TIME RENDERING OF CUT DIAMONDS paper caegory: process (auhor ) (auhor ) Absrac We presen a mehod o creae in real ime compuer-generaed images o cu diamonds o convex polyhedral shape The mehod is based on beam racing and models he complee underlying physics o Fresnel s equaions and polarizaion racing To achieve high compuaional eiciency, we linearize he non-linear reracion and ake advanage o he convex shape We analyze he scaling o our mehod wih increasing deph o recursive beam racing We compare resuls produced by our mehod wih hose generaed by ray racing For ypical scenes and image resoluions o approximaely 800-by-800 pixels, he speed-up acor o our mehod compared o ray racing is abou 000 o 5000, depending on recursion deph, wih he obained images being almos idenical Our mehod produces phoo-realisic images, which we demonsrae by comparison o phoographs aken under careully conrolled lighing condiions o images produced by our mehod when simulaing he same condiions Inroducion and Moivaion We describe a highly eicien mehod or real-ime and nearphoorealisic rendering o cu diamonds Figure shows highqualiy images produced by our mehod Our work was moivaed by he desire o develop an ineracive ool supporing he rapid generaion o imagery o cu diamonds The primary objecive was o devise a novel approach ha is as eicien as possible and can suppor ruly ineracive rendering o diamonds, allowing a user o ully explore he parameer space given by geomery, lighing and camera We have achieved his objecive The sysem ha we have creaed allows a user o speciy and change easily he parameers conrolling opical/geomerical diamond characerisics and he seings or our camera viewing model Our main objecives in designing our sysem were o devise an algorihm ha is in accordance wih he physical laws o relecion and reracion (Snell s law and Fresnel s equaions), and leads o a compuaionally highly eicien implemenaion producing high-qualiy renderings We briely review he essenial physical laws underlying our approach; describe how hese laws are implemened in he design o our algorihm; and compare he resuls produced by our prooype wih hose obained by ray racing and wih acual digial phoographs In compuer graphics, ray racing is he proo-ypical mehod used or generaing digial images o hree-dimensional scenes Eecs like relecion and reracion can be modelled well wih a ray racing approach Neverheless, o generae correc ray-raced images i is crucial o embed he whole physics o relecion and reracion coeiciens in one s ray racing algorihm - and in he Fig : Renderings o diamonds obained wih our mehod

2 Online ID 0459 Page deiniion o all objecs consiuing a scene In general, such a rigorous, physics-based approach is currenly no applicable o scenes conaining housands o complicaed objecs wih dieren opical properies Our work was moivaed by he goal o demonsrae ha i is possible o produce correc images o cu diamonds, and o documen ha his goal can be achieved on oday's commodiy PCs using sandard graphics cards Wih our curren implemenaion i is possible or a user o obain highqualiy renderings o cu diamonds a ineracive rame raes, ie he/she can vary rendering or diamond parameers and obain a rendering insanly Our approach o eicien rendering o cu diamonds is based on he principle o beam racing, see [Heckber and Hanrahan 984] Ray racing, being an image-/screen-space approach requires us o shoo rays hrough every screen pixel; beam racing can lead o subsanial acceleraions over sandard ray racing by using enire "bundles" (beams) o rays insead We have adaped beam racing or cu diamond rendering There are numerous ray racing packages ha use Snell s law o compue reracions However, he relecion and reracion coeiciens, given by Fresnel s equaions, are no always implemened, as heir compuaion is expensive To compare he eiciency o he implemenaion o our mehod wih a ypical commonly used ray racing package, we chose o use MegaPOV [MegaPOV], an exension o he popular ray racer Persisence o Vision [POV-Ray], or comparison, as i makes use o Fresnel s equaions We have compared compuaional eiciency and qualiy o rendering resuls Subjec o he chosen image resoluion, i has urned ou ha our implemenaion can produce resuls highly similar o hose creaed wih MegaPOV - wih speed-up acors o several orders o magniudes To validae ha our implemenaion o he physical laws and our mahemaical simpliicaions are also in agreemen wih physical realiy, we have perormed comparison wih digial phoographs Our experimens have conirmed ha our implemenaion can produce highly realisic renderings Mehod We consider a convex polyhedron, in he remainder o his paper reerred o as sone, made o a maerial o isoropic index o reracion This deiniion comprises almos all relevan cases, as wih he excepion o he hear shape, all shapes commonly used or jewellery are convex polyhedra Boh diamond and cubic zirconia, he bes commercial subsiue or diamond, are maerials wih an isoropic index o reracion The sone s aces will be reerred o as aces Due o he polyhedron s convexiy, all aces are convex polygons Firs, we describe he individual componens o our mehod, and hen we show how hey are used in he main algorihm Beam racing Ray racing is known o produce very realisic images Wih he appropriae models or local ineracion o ligh and objecs, many physical eecs can be simulaed wih high accuracy Unorunaely, ray racing is also very slow, as every pixel o he rendered image has o be compued individually Beam racing [Heckber and Hanrahan 984] builds upon he idea o enlarging he sampling area As in he case o ray racing, ligh is raced backwards, rom he observer s eye o he objecs in he scene and inally owards he ligh sources Insead o ininiesimally hin rays (poin sampling), beams wih inie Fig : A beam b is deined by an eye poin v and a polygon p Insead o using p, we use he screen-projeced polygon p o deine he beam b exension are used (area sampling) In his paper, a beam is undersood as a cone wih convex polygonal cross-secion (Fig ) A poin v (he locaion o he observer s eye) and a convex polygon p deine a beam For a given beam, here is no unique polygon ha deines he beam Any inersecion o he beam wih a plane can be used By esablishing a reerence plane (he screen ) wih a coordinae sysem and using he inersecion o screen and beam as deining polygon, we can deine beams in erms o wo-dimensional screen coordinaes In OpenGL [Shreiner 999], he mapping sc OpenGL ha convers hree-dimensional objec poins o wo-dimensional screen coordinaes is sc OpenGL = pd o mv o hg wih hg : ( x y z) a ( x y z ) he ransormaion o homogeneous coordinaes, 4 mv : x a M x, M R R he produc o he modelview and projecion marices and ( ) pd ( ) OpenGL : x y z w a x y z w he perspecive division We use he same kind o mapping, bu in a slighly modiied noaion and omiing he deph componen in he perspecive division, as we will no need i Our mapping sc o screen coordinaes is sc = pd o cm wih cm : x a A x + b, A R R, b R and x pd : x a w R R x w,, c + c y For given OpenGL modelview and projecion marices, we can compue A, b, w and c o yield he same mapping In oher words, we use OpenGL-like screen coordinaes o deine beams and compue inersecions o polygons wih beams However, his is no done by he OpenGL subsysem, bu direcly in he algorihm The concep o beams is well suied or scenes consising o convex polygons, as he inersecion o a beam and a convex polygon yields a convex polygon, which is used o deine a child beam wih he same origin [Ghazanarpour and Hasenraz 998] In his ramework, relecions are easily incorporaed, as hey are linear ransormaions, ransorming polygons ino polygons and beams ino beams 4

3 Online ID 0459 Page Fig : The inersecion o beam wih aces The compuaion o he inersecion is done in erms o screen coordinaes, which reduces he calculaion o he beam-polygon-inersecion o he inersecion o wo convex polygons The beam subdivision As explained in he preceding secion, we deine beams in erms o wo-dimensional screen coordinaes This simpliies he calculaion o he inersecion o a beam wih a ace, as i can be done in screen coordinaes Figure shows he hree-dimensional arrangemen and he corresponding screen-projeced scene As he mapping sc consiss o a linear ransormaion and a perspecive division, he projecion o screen ransorms convex polygons (he aces) ino convex polygons (he projeced polygons in Fig ) As he paren polygon ha deines he beam is convex oo, we have o compue he inersecion o wo convex polygons, which is easy and can be done raher as In order o avoid he projecion o aces ha do no yield an inersecion wih he paren polygon, we sar wih a ace ha is known o yield an inersecion We deermine a suiable sar ace by consrucing a ray in he cener o he beam and deermining he ace where i leaves he sone While acually compuing he inersecion, all aces adjacen o he ace currenly reaed are marked or processing, i hey mus have a non-vanishing inersecion wih he beam This can easily be deermined rom he muual posiion o he polygons edges When all aces marked or processing have been projeced and heir inersecions compued, he subdivision o he beam is inished, as () here are no urher aces wih non-vanishing inersecion, as he sone is known o be convex, and () he sum o he child beams is equal o he beam isel, as he surace o he sone is closed The second saemen can be easily undersood by realizing ha when racing he beam, we are inside he sone, hus here is no way o leaving he sone wihou crossing is surace The relecion mapping r The relecion upon a ace wih plane equaion x n c = 0, n R,c R is given by he mapping ( n n ) x + c n r : R R, x a As i depends only on he plane parameers, i can be rewrien in he more convenien linear orm r : x a A x + b, and he marix A and he vecor b sored or every ace a he sar o he beam race algorihm 4 The reracion mapping On ineraces o maerials wih dieren index o reracion, ligh ges reraced according o Snell s law, given as n sinα = n sinα Fig 4: The reracion o an inciden ray on a plane inerace beween media wih indices o reracion n < n For a viewer a posiion v, ligh emied rom a poin x inside medium seems o come rom a virual image poin ~ x We consider a plane inerace o wo maerials wih indices o reracion n < n (Fig 4) For a viewer a poin v, ligh originaing a poin x inside maerial seems o come rom he virual image poin ~ x For a ixed viewing posiion v and in he limi β 0, ie, or small opening angles around he main ray, he relaion beween x and ~ x is well deined We reer o his mapping as m : R R, x a ~ rer x For non-vanishing opening angles β, here is no single virual image poin In opics, he resuling image auls are called aberraions In his paper, we neglec he eecs o aberraion As mrer is nonlinear and compuaionally expensive, we linearize i by Taylor series expansion Using he Taylor series o m rer a poin x 0 m rer ( x ) = mrer ( x0 ) + mrer ( x0 )( x x0 ) + L we deine a linear projecion ( ransmission ) : R R, x a A x + b wih A : = m rer ( x 0 ), b : = mrer ( x 0 ) mrer ( x 0 ) x 0 The developmen poin x 0 has o be on he inerace in order o asser coninuiy: As poins in medium do no ge reraced (he ligh does no cross he reracing boundary beween medium and ), hey are mapped ono hemselves In order o yield an overall coninuous mapping, mus map he poins in he inerace plane ono hemselves, jus as m rer does This is assered by choosing a developmen poin on he inerace The mos suiable choice or x 0 is he cener o he ace A he inerace, m rer is coninuous bu is derivaive is no coninuous, hus, sricly speaking, he erm m rer ( x 0 ) is a singlesided derivaive For he compuaion o m rer ( x 0 ), we are ree o choose a suiable base ( b, b, b) and consruc m rer ( x 0 ) rom he direcional derivaives along he base vecors For b and b, we use he plane vecors o he inerace plane, as i is very simple o compue he direcional derivaives or hem The choice o b and he compuaion o he direcional derivaive is explained in Figure 4 From he direcion o incidence d inc, deined by v and x 0, and Snell s law, we obain he direcion d rer o he reraced ray We choose a virual image poin ~ x = x 0 + εdinc wih using a small value or ε and a small angle β o deine a second ray, compue he reracion o his second ray and ge he poin x as he inersecion poin o he wo

4 Online ID 0459 Page 4 reraced rays Thus, he direcional derivaive along b : = x x ~ 0 is b ~ = x x0, and we inally ge he derivaive m rer ( x 0 ) as ~ m ( ) ( ) ( ) rer x0 = b b b b b b Boh A and b are hereore ixed, and he mapping is deermined This compuaion mus be done only once or every ace direcly visible rom he viewer 5 Fresnel s equaions Snell s law describes he reracion angle, bu no he inensiies o he releced and he reraced ligh componens These inensiies are given by Fresnel s equaions [Jackson 999] Using he quaniies and sign convenions as shown in Figure 5, or he ligh componen wih an elecric ield orhogonal o he plane o incidence, he relecion and ransmission coeiciens are given by E, orh n n r, orh : = = E0, orh n + n E, orh n, orh : = = E n + n 0 orh For he elecric ield parallel o plane o incidence hese coeiciens are given by E, par n n r, par : = = E0, par n + n E, par n, par : = = E n + n 0, par In he case o oal inernal relecion, he relecion coeiciens become complex Their absolue value is one, as all ligh ges iδorh releced Thus, hese coeiciens have he orm r TIR, orh = e iδ par and r TIR, par = e For our applicaion, we do no need he absolue phase, as we consider only incoheren ligh Only he phase dierence δ = δ orh δ par is relevan The equaion or δ is given in [Born and Wol 999] For he compuaion o he releced and ransmied inensiies, he polarizaion sae has o be known Algorihms ha do no rack he polarizaion sae usually assume unpolarized ligh and use he relecion and ransmission coeiciens r, unpol = r, orh + r, par = +, unpol, orh, par 6 Polarizaion racking As he relecion and reracion coeiciens given by Fresnel s equaions dier or he parallel and orhogonal componen, he polarizaion sae o ligh changes due o he relecions and reracions Compleely unpolarized ligh can ge compleely linearly polarized by a single relecion (he angle o incidence or his case is known as Breweser s angle) As here are many relecions and reracions o ake ino accoun when racing ligh hrough he sone, he polarizaion sae o he ligh mus be racked There are several ways o represen he polarizaion sae o ligh: The Sokes parameers, he coherence marix and he Jones calculus Each o hese ways has is own disinc advanages [Chipman 995, Wol and Kurlander 990] For he applicaion o cu-diamond rendering, he mos suiable represenaion is he coherence marix: I can handle parially polarized ligh and lends isel o easy ransormaion For a given ligh ray, an orhogonal coordinae sysem is esablished wih he Fig 5: Adoped sign convenion or elecic ield used in Fresnel s equaions z-axis corresponding o he direcion o he ligh propagaion The coherence marix J is deined as E x E xex ExEy J = ( Ex Ey ) =, Ey EyEx EyEy wih E x and E y denoing he componens o he elecric ield wih respec o he coordinae sysem, and he angular brackes denoing ime averaging The coherence marix does no include E z, as here is no elecric ield in he direcion o propagaion The ligh inensiy is given by he race o he coherence marix r J = J xx + J yy, while he deerminan gives he degree o polarizaion For deails on how o inerpre he coherence marix, see [Born and Wol 999] In order o apply he relecion and ransmission coeiciens, he coordinae sysem mus be roaed such ha is x-axis is aligned wih he direcions orhogonal o he plane o incidence and he y- axis parallel o he plane o incidence Wih respec o a coordinae sysem ha is roaed around he z-axis by he angle ϕ, he elecric ield is given as Ex cosϕ sinϕ = Ex E y sinϕ cosϕ E y 4444 mro: = Now i is possible o apply he ransmission or relecions coeiciens The releced par is given as Ex ro 0 Ex Ex = = m rel mro E y 0 rp E rel y E y 44 mrel: = or, in he case o oal inernal relecion, iδ e 0 m = rel 0 wih δ he phase dierence deined in secion Fresnel s equaions The ransmied par is Ex o 0 Ex E x = = mrans mro Ey 0 p E rans y E y mrans : = Acually, he cos acor does no belong o he ransormaion o he elecric ield, bu o he ransormaion o he ligh power (energy per ime): The ligh power or he sampling area changes proporionally o he raio / due o he change o he

5 Online ID 0459 Page 5 beam s cross-secion Insead o soring his acor and applying i when compuing he ligh power, we apply is roo in he ransormaion o he elecric ield As a consequence, when compuing he coherence marix rom he ransormed elecric ield, i has already been aken ino accoun 7 Deph o ield The imaging o small objecs ends o go hand in hand wih a considerable deph-o-ield: Due o heir size, small objecs emi lile ligh In order o capure enough ligh or a phoograph he enry lens mus be large, ie, mus have large numerical aperure A large numerical aperure yields a small deph o ield As cu diamonds usually are relaively small, mos phoographs o cu diamonds exhibi a pronounced deph o ield We are used o his deph o ield in diamond picures, and i is his phenomenon ha makes crisp picures look unrealisic o he human observer For added realism, we mus simulae deph o ield There is a sraighorward approach o simulae deph o ield: We can posiion he (pin-hole) camera a several locaions in he lens (wih adjused viewing direcion) and calculae he average o he images This approach amouns o rendering he same scene several imes wih a slighly modiied camera, and compuing he average o all picures In OpenGL, his calculaion is done in he accumulaion buer [Shreiner 999], as is increased color deph avoids rounding errors For a smooh-looking deph o ield, many camera posiions are necessary, ypically 0 o 00 8 Parallel and perspecive reracion The accumulaed projecion ransormaion m acc or every beam is o he orm x a A x + b, wih A R R, b R Aer applicaion o his projecion, he perspecive division pd is applied For every inersecion es, his ransormaion mus be done or all corners o he ace o be projeced Considering overall compuaional cos, his sep requires he execuion o a relaively large number o loaing-poin operaions We can reduce his compuaional cos by using parallel reracion insead o he reracion mapping deined above The idea is depiced in Figure 6 Insead o using, we use he mapping parallel : R R, x a Ax + b, wih n drer drer A : =, b : = c drer n drer n where n and c are he parameers o he ace s implici plane equaion x n c = 0 This mapping projecs all poins in he direcion d rer ono he ace plane Insead o using he overall projecion pd o cm o, we use he projecion pd o zrecons o red d o cm o parallel, aking advanage o he possible reducion o wo-dimensional space, red ( ) ( ) d : R R, x y z a x y, which is compensaed by an ensuing reconsrucion mapping deined as z ( ) ( ( )) recons : R R, x y a x y recons x, y, which re-compues he z-componen ha is los in red d Reconsrucion wih he uncion recons is possible due o he ac ha all poins mapped by cm o parallel lie in a plane deined by he ace plane equaion, ransormed linearly by cm Fig 6: (a) Perspecive reracion and (b) parallel reracion We spli he overall projecion sc ino wo pars, given by sc = pd o o o 44 o zrecons redd cm parallel, : = pdrecons = : mred using a modiied perspecive division pd recons : R R and a linear mapping mred : R R, x a A red x + bred wih A red R R, bred R The crucial idea or he reducion o compuaional cos is o use he mapping m red insead o sc or he inersecion es This approach reduces he number o loaing-poin calculaions or he linear par by one hird and leaves ou he perspecive division compleely Only during he las sage o our algorihm, when we draw o he screen, do we compue he acual screen coordinaes by pd recons As many more coordinaes are projeced or he inersecion es han coordinaes drawn on he screen, he delayed perspecive division reduces he number o loaing-poin operaions as well To summarize: Insead o using screen coordinaes, we use preperspecive-division coordinaes resuling rom m red or he compuaion o he inersecions This approach reduces he number o loaing-poin operaions by more han one hird In principle, a problem can occur during he reconsrucion uncion recons, as i can lead o a division by zero In pracice, he probabiliy o his siuaion o occur is so small ha i can be negleced O course, due o he missing perspecive division, polygons ha are benign in screen coordinaes can be almos degenerae in pre-perspecive-division coordinaes However, he inersecion es or he subdivison o beams mus be able cope wih almos degenerae polygons, and as he opposie siuaion is jus as likely o occur (polygons benign in pre-perspecivedivision coordinaes, bu almos degenerae in screen coordinaes), parallel reracion is numerically no more sensiive han perspecive reracion The only real issue ha remains is his one: Due o he missing perspecive division, someimes he orienaion o projeced polygons is wrong (ie, he polygon s corners are ordered in mahemaical negaive insead o posiive sense) As he inersecion algorihm relies on a ixed orienaion, or polygons oriened wrongly, we use red d wih he y-componen reversed, perorm all compuaions wih his projecion, and inally reverse he y-componen when drawing, ie, jus beore he perspecive division pd recons

6 Online ID 0459 Page 6 Fig 7: Dieren levels o image qualiy Recursion deph: eigh (all our examples), image size 800*800 pixel, using ani-aliasing From le o righ: (a) Rendered wih parallel reracion (655 ms); (b) rendered wih perspecive reracion (809 ms); (c) same as (b), wih addiion o specral sampling (hree wavelenghs) (6 ms); (d) same as (c), wih addiion o deph o ield (9 camera posiions, 65 s) 9 The main algorihm 4 or (every ace direcly visible) { creae beam o deph by projecing ace according o pd o cm, he projecion o screen coordinaes; creae local coordinae sysem and compue change m ro,camera rom local o camera coordinae sysem; compue direcion and coeiciens o relecion; compue polarizaion ransormaion or oubound (releced) beam: pacc,ou = m ro,camera o m rel o m ro,ligh source apply pacc,ou o ligh emanaing rom ligh source, compue and sore coherence marix; or he compuaion o he child beams, add reracion mapping o projecion and sore accumulaed projecion m acc = cm o ; se polarizaion ransormaion o relec change o coordinae sysem and reracion coeiciens: pacc = m ro,camera o m rans ; } or (d= o maximal recursion deph) { or (paren = every beam o deph d-) { projec aces according o paren s accumulaed projecion mapping m acc, paren, ollowed by pd; use projeced aces o spli paren beam ino child beams; or (every child beam) { compue direcion o reracion; creae local coordinae sysem and compue change m ro rom local o paren coordinae sysem; compue polarizaion ransormaion or oubound (reraced) beam pacc,ou = pacc, paren o m ro o m rans o m ro,ligh source apply pacc,ou o ligh emanaing rom ligh source, compue and sore coherence marix; or he compuaion o child beams, add relecion mapping r o projecion and sore accumulaed projecion macc = macc, paren o r ; add change o coordinae sysem and relecion coeiciens o polarizaion ransormaion pacc = pacc, paren o m ro o m rel }}} or (every beam compued) { use lighing model o deermine ligh inensiy or oubound direcion; muliply inensiy wih coherence marix o oubound direcion; use resuling inensiy o render polygon on screen; } To evaluae he perormance o our mehod, we have esed i on a PC running a GHz, having 5MBye o main memory and an NVidia Ti 400 graphics card We have used he graphics card's hardware ani-aliasing (wo-by-wo super-sampling) o reduce aliasing eecs Resuls 4 Perormance and image qualiy Figure 7 shows a ypical image obained wih ixed recursion deph, bu or dieren qualiy levels The ases version is parallel reracion, which yields images ha are qualiaively correc, bu no quaniaively correc (see also Fig 9) To saisy average rendering demands, in erms o image qualiy, he bes choice is perspecive reracion I is nearly correc and has good perormance A urher simpliicaion o he reracion leads o visible deviaion rom he correc image (Fig 9), while reducing rendering imes by jus 5% or small and 5% or large recursion dephs (Fig 8) Using specral sampling, ie, rendering he same scene or dieren wave lenghs, he color eecs due o diamond s dispersion [Edwards and Philipp 985] appear The rendering imes increase linearly wih he number o wavelenghs used The simulaion o deph o ield increases compuaional cos linearly wih he number o sampling poins As a smooh deph o ield usually requires 0 o 00 sampling poins, his opion is very expensive Fig 8: Scaling o rendering imes per rame wih recursion deph or he scene shown in Fig 7, wih parallel reracion (doed line), perspecive reracion (solid line) and perspecive reracion wih hree-wavelengh specral rendering (dashed line)

7 Online ID 0459 Page 7 4 Beam and ray racing The de aco sandard or rendering echniques is usually ray racing In his secion, we compare ray-raced images wih images rendered by our mehod For his purpose, he ray racer used or comparison mus be capable o handling Fresnel s equaions correcly Polarizaion handling is desirable, bu no necessary, as we can disable polarizaion racking in our mehod We chose o use he ray racer POV-Ray as i is powerul, readily available and an exension called MegaPOV 07 is available ha includes Fresnel s equaions As i is open-source, we could veriy he correc handling o Fresnel s equaions in he code Unorunaely, i does no rack polarizaion, so we had o disable polarizaion racking in our mehod in order o asser comparabiliy This was done by using he coeiciens or unpolarized ligh given a he end o secion Fresnel s equaions or boh he parallel and orhogonal componen In order o compare ray racing wih our mehod, we mus use a scene ha can be se up boh by he ray racers s scene language and our mehod While he cu could be convered easily or he ray racer, he lighing had o be simpliied in order o guaranee ha boh MegaPOV and our mehod perormed he same calculaions The lighing used or Figure 9 consiss o hree planes and wo spheres All ive objecs have homogeneous color and are li by pure ambien ligh, hus every poin on an objecs emis ligh wih he same inensiy The inensiies o he ive objecs are all dieren Rendering is done or a single wavelengh, ie, a single index o reracion Thereore, no colors are visible, as hey emerge rom dispersion, ie, a wave lengh dependen index o reracion Beam racing has he advanage o reducing aliasing [Ghanzanarpour and Hasenraz 998], as i is an area-sampling mehod in conras o ray racing, which is a poin-sampling mehod In ray racing, he sampling or he image pixels is done in soware An improvemen o image qualiy by ani-aliasing increases he compuaional cos (Fig 9d), whereas or beam racing, he ani-aliasing can be done in hardware, resuling in a negligible perormance reducion (a) Ray racing (b) Beam racing wih perspecive reracion (c) Beam racing wih parallel reracion (d) Rendering imes or dieren echniques (e) Image (b) minus (a) plus 50% grey () Image (c) minus (a) plus 50% grey Fig 9: Comparison o rendering echniques All images have a size o 800 x 800 pixel and make use o ani-aliasing (a) hrough (c) show he same scene, rendered wih ray racing, beam racing wih perspecive reracion and beam racing wih parallel reracion In order o emphasize he non-linear eecs o he reracion, he camera locaion was chosen close he sone (he disance beween camera and sone being less han six imes he sone s diameer) As all images closely resemble each oher, (e) and () show he dierences beween he beam- and he ray-raced images Fig 9(e) demonsraes ha he linearizaion used or perspecive reracion is a very good approximaion o he correc non-linear reracion Fig 9(d) shows he rendering imes per rame or beam racing wih perspecive reracion (solid line), ray racing wihou ani-aliasing (doed line) and ray racing wih ani-aliasing (dashed line) For beam racing, disabling ani-aliasing changes rame raes so lile ha, in his graph, i would yield he same line as ani-aliased beam racing The perormance advanage o beam racing agains ray racing ranges rom 00 o 900 or high image qualiy (wih ani-aliasing) and rom 600 o 600 or reduced image qualiy (wihou ani-aliasing)

8 Online ID 0459 Page 8 Fig 0: Comparison o phoograph wih resul produced by our mehod (a) Phoograph o a cubic zirconia sone o round brillian shape, aken wih a microscope under well-conrolled lighing condiions (b) Image rendered by our mehod wih he scene parameers maching he physical microscope seup Deph o ield is enabled wih superposiion o 7 individual rames Rendering ime: 00 s (a) phoograph (b) rendered wih our mehod 4 Beam racing and phoography While he comparison wih ray racing is convincing rom a compuer graphics poin o view, he ulimae es is he comparison wih real phoographs The simulaion o a real scene wih our mehod is only easible i he scene is well deined The mos criical par is he lighing We buil a dedicaed box wih a well-deined window or lighing The sone was placed a a ixed posiion in he box and viewed wih a microscope A digial relex camera was used, as i yields a linear connecion beween ligh inensiy and he RGB values o he image pixels The picures aken wih he camera were no alered in any way, hey were only cropped o he relevan size As no real diamond o appropriae size was available, we used a sone made o cubic zirconia, he bes commercial subsiue or diamond Using he index o reracion o cubic zirconia [Wood and Nassau 98], he daa o he microscopes objecive, he dimensions o he lighing window and he posiion o he sone under he microscope s objecive, we rendered he same scene wih our mehod Figure 0 shows he phoograph and he resul obained by our mehod Almos all o he relexes seen in he phoograph can be clearly ideniied in he digially produced duplicae; many o he colored relexes have he same color The dierences can be aribued or he mos par o he ac ha he sone s cu is no perec, as he locaion and size o he individual relexes depend very delicaely on he angles beween he individual aces Though he rendering was done wih jus hree wavelenghs (60, 540 and 445 nm or he red, green and blue componen, respecively), he colors are qualiaively correc Apparenly, his raher coarse approximaion o he ull specral disribuion already yields a good approximaion o he real image 5 Conclusions and possible exensions We have described a mehod or he rendering o cu diamonds I is possible o generae near-phoorealisic images o cu diamonds, a high resoluion and high recursion deph, wih our implemenaion o his mehod Today's commodiy PCs equipped wih conemporary graphics cards are suicien o generae hese images Concerning he rigorous evaluaion o our mehod we have compared our resuling images wih hose obained wih a popular ray racing package and wih acual phoographs o a cu sone In summary, our evaluaion procedures have demonsraed ha our implemenaion compares very avorably wih experimens The key conribuions and speciic srenghs o our novel approach are: Our mehod is exremely eicien, much more eicien in general han exising compeiive echniques we are aware o our mehod does no require any pre-compuaion seps our mehod is physically correc, as i akes ino accoun Snell s law, Fresnel s equaions and polarizaion our mehod can be used o produce highly realisic renderings o cu diamonds Possible exensions are he handling o colored sones by incorporaion o deph racking and he sones exincion coeiciens and he simulaion o phoographic areacs like bleeding due very srong relexes A poenially very ineresing applicaion or our mehod is he mahemaical opimizaion o diamond cus, as i can be used o consruc opimizaion uncions, eg or brilliance and ire, which, due o our mehod being an area sampling mehod, are coninuous and hus lend hemselves o numerous opimizaion algorihms 6 Reerences BORN, M AND WOLF, E 999 Principles o Opics, 7h ed Cambridge Universiy Press, Cambridge, UK CHIPMAN, R A 995 Mechanics o polarizaion ray racing Opical Engineering 4, 6, EDWARDS, D F AND PHILIPP H R 985 Cubic Carbon (Diamond) In Handbook o Opical Consans o Solids Academic Press, Orlando, Florida Palik E D, Ed, GHAZANFARPOUR, D AND HASENFRATZ, J-M 998 A beam racing mehod wih precise anialiasing or polyhedral scenes Compuers & Graphics,, 0-5 HECKBERT, P S AND HANRAHAN, P 984 Beam racing polygonal objecs In Compuer Graphics (Proceedings o ACM SIGGRAPH 84), 8,, ACM, 9-7 JACKSON, J D 999 Classical Elecrodynamics, hird ediion John Wiley & Sons, New York MEGAPOV Version 07 hp://megapovinearne/ POV-RAY hp://wwwpovrayorg/ SHREINER D 999 OpenGL Reerence Manual, hird ediion Addison Wesley Longman, Reading, Massachuses WOLFF, L B AND KURLANDER, D J 990 Ray Tracing wih Polarizaion Parameers IEEE Compuer Graphics & Applicaions 0, 6, WOOD, D L AND NASSAU, K 98 Reracive Index o cubic zirconia sabilized wih yria Applied Opics, 6,