A Volume Density Optical Model. Center for Supercomputing Research & Development, and. University of Illinois at Urbana
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1 A Volum Dnsiy Opical Modl Pr L. Williams y and Nlson Max z y Cnr for Suprcompuing Rsarch & Dvlopmn, and Naional Cnr for Suprcompuing Applicaions Univrsiy of Illinois a Urbana z Univrsiy of California, Davis, and Lawrnc Livrmor Naional Laboraory Absrac A simpl, bu accura, formal volum dnsiy opical modl is dvlopd for volum rndring scard daa or scalar lds from h ni lmn mhod, as opposd o scannd daa ss whr marial classicaion is involvd. Th modl is suiabl ihr for ray racing or projcion mhods and allows maximum xibiliy in sing color and opaciy. An xprssion is drivd for h ligh innsiy along a ray in rms of six usrspcid ransfr funcions, hr for opical dnsiy and hr for color. Closd form soluions undr svral diffrn assumpions ar prsnd including a nw xac rsul for h cas ha h ransfr funcions vary picwis linarly along a ray sgmn wihin a cll. 1 Inroducion An xac simulaion of ligh inracing wih a volum dnsiy or cloud is qui complx and rquirs h us of Radiaiv Transpor Thory [, 6]. Howvr, for h purpos of scinic visualizaion, lss complx simulaions can b saisfacory. On of h rs compur graphics modls for clouds was rpord by Jams Blinn [1] of h J Propulsion Lab. H dscribd a mhod o synhsiz an imag of h rings of h plan Saurn which consis of clouds of rciv ic paricls in orbi abou h plan. Blinn's modl dals wih h scaring, shadowing and ransmission of ligh propagaing hrough h cloud. I assums ha a ray of ligh is rcd (scard) by only a singl paricl, i.. mulipl rcions ar considrd ngligibl. This simplifying assumpion will b ru if h rciviy (albdo) of ach paricl is small. Blinn's modl also dals wih h shadowing or blocking c of ohr paricls afr a ligh ray has bn scard by a singl paricl. And, i dals wih h ransparncy (ransmianc) of h cloud layr, ha is, h amoun of ligh coming from bhind h cloud no blockd o by paricls. Th brighnss or innsiy of h cloud a ach pixl is calculad by ray ingraion, ha is, by ingraing along a ray from h y passing hrough h pixl. Many subsqun cloud modls ar basd o som xn on Blinn's modl. Kajiya and Von Hrzn [5] giv an alrnaiv modl which dals wih mulipl scaring agains paricls wih high albdo; and hy furhr dvlop Blinn's low albdo modl and giv a ray racing algorihm for i. Ligh propagaing hrough clouds is also discussd by Max [9, 10], Rushmir and Torranc [1] and Ebr and Parn [3]; howvr, hs chniqus ar no dircly applicabl o volum rndring. Up o his poin in h dvlopmn, h ligh sourcs hav bn ousid h cloud and h modl has dscribd how h paricls in h cloud scar, absorb and ransmi his ligh. For h purpos of volum rndring for scinic visualizaion, a slighly dirn volum dnsiy modl is usd in which h cloud islf mis ligh. Two basic modls can b usd for his. In h rs modl, h scalar ld bing visualizd is modld as a cloud of ligh miing paricls. W will rfr o his as h paricl modl. In h ohr modl, h scalar ld is rprsnd as a cloud xprssd as coninuous glowing mdium. Each poin of h mdium boh mis and absorbs ligh. W will rfr o his scond modl as h coninuous modl. Th wo modls ar rally wo dirn xplanaions of h sam physical phnomnon. For a singl ransfr 1
2 funcion, which is all h paricl modl allows, h wo modls giv h sam mahmaical formula for h drivd innsiy. By nglcing shadowing on h way in, Blinn's singl scaring modl urns ou o b h sam as h paricl modl if h phas facor is nglcd. Th opical modl is h mos crucial par of a volum rndrr bu i also can b h mos confusing par. Thrfor i is imporan ha h undrlying modl b clarly undrsood. Earlir modls such as in [13, 17] lackd som gnraliy and/or wr no asy o comprhnd. Our papr prsns a nw coninuous modl which is rigorous and qui gnral, y is inuiiv and asy o undrsand. Th nx scion rviws h paricl modl and also maks som aspcs of is drivaion mor rigorous. Thn, in Scion 3, w prsn h coninuous opical modl for a volum dnsiy. This modl is suiabl ihr for ray racing or for projcion mhods and allows maximum xibiliy in sing color and opaciy. An xprssion for h ligh innsiy along a ray hrough a volum, in rms of six usr-spcid ransfr funcions, hr for opical dnsiy and hr for color, is drivd. Closd form soluions undr svral dirn assumpions ar prsnd, including a nw xac rsul for h cas ha h ransfr funcions vary picwis linarly along a ray sgmn wihin a cll. A mhod is dscribd which allows isosurfac shading wihin a volum rndring. Th Paricl Modl Paolo Sablla [13] rs dscribd a paricl modl for volum rndring which h calld h dnsiy mir modl. I is basd on Blinn's modl bu assums h paricls mi hir own ligh, rahr han scaring ligh from a sourc. Sablla modls h dnsiy of paricls, no h paricls hmslvs. Th siz of h paricl is considrd o b small compard o ohr dimnsions so h dnsiy of h paricls can b rgardd as a coninuous funcion. In Sablla's modl, h dnsiy of paricls a any poin a = (x; y; z) is dnd by considring a volum lmn of h cloud dv cnrd a a. If dv P is h volum occupid by h paricls in dv, hn h dnsiy funcion is dnd o b (x; y; z) = dv P =dv. If v p is h volum of a singl paricl, hn h xpcd numbr of paricls in a rgion R of h cloud is: N R = Z R dv P v p = Z R (x; y; z) dv v p (1) A volum rndrd imag is crad by sing a pixl's color o h innsiy of ligh prcivd by h y along a ray from ha pixl o h y hrough h volum. Considr a cylindr wih cross scion whos axis is a ray o h y, paramrizd by lngh, which nrs h cloud a 1 and xis a. Assum h dnsiy funcion is paramrizd along h ray as (x(); y(); z()), or jus as (). An innisimal sgmn S of his cylindr, cnrd a and of lngh d, has volum dv = d, and conains N dv = paricls. If h () dv v p paricls ar all sphrs wih radius r, hn v p = 4 3 r3, and ach has projcd ara r. So if hy all glow diffusly on hir surfacs wih innsiy, h oal ligh powr crossing h fron surfac of S is r N dv = r () d=( 4 3 r3 ) = 3 () d Th powr is disribud ovr an ara, so h innsiy (powr pr uni ara) conribud by his sgmn is 3 () d. W hav assumd d is innisimal, so ha h paricls do no occlud ach ohr. Bu on h pah from h inrior posiion o h fron dg of h cloud, occlusion can ak plac. To calcula h probabiliy ha his ray from o is unoccludd, ak anohr cylindr C abou i, of radius r, h paricl radius. Th ray will b unoccludd if hr ar no paricl cnrs wihin C. From Equaion 1, h xpcd numbr N C of paricls insid C is: N C = Z C (x; y; z) dv v p = Z (u)r du=( 4 3 r3 ) = 3 Z (u) du If h dnsiy is small nough ha h chancs of muual ovrlap ar small, h paricls can b assumd o b indpndnly disribud. Thn h probabiliy P (0; C) ha hr ar no paricl cnrs in h cylindr is givn by hr Poisson disribuion formula P (0; C) =?NC =? 3 (u) du. Thrfor, h innsiy of ligh raching h y du o dv is: R 3 3 (u) du () d? Th oal innsiy I raching h y du o all conribuions bwn 1 and is: I = 3 Z 1 ()? 3 (u) du d If w assum h innsiy a 1 is zro, ha is, hr is a black background, and w l = 3 and c =, w g: I = c Z 1 ()? (u) du d () For Equaion o hold, h dnsiy mus b small as rquird by h Poisson disribuion. Th dnsiy can ihr b s qual o h scalar ld S which is bing visualizd, or a singl usr dnd ransfr funcion f can
3 + P() n Figur 1: A D viw of a cloud wih a ray P() o h y paramrizd by lngh. b usd, i.. (x; y; z) = f(s(x; y; z)). Sablla uss numrical mhods o sima h ingral in Equaion by sampling along h ray. Whn h indni ingral (u) du can b abulad or calculad analyically, and whn c and ar 0 consans, Max, Hanrahan and Craws [8] show how Equaion can b simplid o a closd form xprssion which can b valuad for ach cll hrough which h ray passs. In addiion, hy ak c and as vcors wih hr componns, rd, grn and blu. This has h c of scaling h singl ransfr funcion dirnly for ach of h hr componns of color. Th imags crad by his mhod ar vry accura rndrings of h volum dnsiy; howvr, h procss is compuaionally innsiv and rally is inndd o b implmnd in microcod. I may b possibl o modify Sablla's modl o includ hr spara ransfr funcions for rd, grn and blu mid ligh by assuming ha a fracion of h paricls mi ligh of a crain color. Thn h dnsiy of h rd-miing paricls, for xampl, will vary wih a dnsiy funcion r (x; y; z), and similarly for h blu and grn paricls. Howvr, hs hr color paricl dnsiis nd o b rlad o h anuaion paricl dnsiy which appars in h xponn in Equaion. I sms asir o mak his gnralizaion in h coninuous modl which is dscribd blow. 3 Th Coninuous Modl W now considr h scond modl, h coninuous modl for a volum dnsiy. This formulaion and dvlopmn is nw and has no bn prsnd bfor. W bn graly from h arlir work of Shirly and Tuchman [14] and Wilhlms and Van Gldr [17]. Th goal is o provid a simpl, bu accura, formal modl on which o bas dirc volum rndring of scalar lds dnd on irrgular mshs and o maximiz h xibiliy of us of ransfr funcions. I is inndd for us wih scalar ld daa from h ni lmn mhod, or scard daa, as opposd o scannd daa ss whr marial classicaion is involvd. Th modl is suiabl ihr for ray racing or for projcion mhods. Th modl is simplid o h bar minimum ndd o clarly display h inrnal srucur of h scalar ld. No amp is mad o produc a highly ralisic simulaion of an acual cloud. 3.1 Modl Dvlopmn In his modl, h volum dnsiy can b hough of as a luminous or glowing gas cloud, such as non or a glowing plasma, ha slcivly absorbs ligh of crain wavlnghs and mis slf-gnrad ligh. Th gas cloud has wo physical propris, opical dnsiy and chromaiciy, boh of which ar funcions of h scalar ld bing visualizd. Th opical dnsiy of h gas a any poin is wavlngh dpndn and is givn by h funcion (x; y; z; ) 0. Th chromaiciy is spcid by a chromaiciy funcion (x; y; z; ) > 0. Ths wo funcions ar d- nd in rms of six usr-spcid ransfr funcions r, g, b, r, g, b, so for xampl, (x; y; z; rd) = r (S(x; y; z)), whr S is h scalar ld bing visualizd. L P () b a ray o h y paramrizd by lngh which nrs h cloud a P ( ) and xis a P ( n ), and l b a poin on h ray cnrd in h inrval ( + ;? ); s Figur 1. L h noaion (; ) sand for (P (); ), and similarly for (; ). Th maning of h opical dnsiy (; ) is ha, in h limi as gos o zro, (; ) is h fracion of ligh of wavlngh nring ha is occludd ovr h disanc. Th chromaiciy (; ) has h maning ha, in h limi as approachs zro, (; )(; ) is h innsiy of ligh of wavlngh (color) mid a h poin P (). Hncforh, I(; ) will rprsn h cumulaiv innsiy of ligh of wavlngh a du o all conribuions up o h poin. 3
4 Th innsiy of ligh raching + is: I( + ; ) = I(? ; )(1? (; )) + (; )(; ) Simplifying, w g: I( + ; )? I(? ; ) In h limi as gos o zro, w g: di(; ) d =?(; )I(? ; )+(; )(;) =?(; )I(; ) + (; )(;) (3) Equaion 3 is insaniad onc for ach of h hr componn wavlnghs of ligh: rd, grn and blu. In ach of hs quaions,, and I ar funcions only of ; hrfor, Equaion 3 is rally a s of hr linar rs ordr dirnial quaions. For xampl, for rd ligh: di r () + r ()I r () = r () r () d Ths quaions can b solvd by numrical mhods, for xampl by h fourh-ordr Rung-Kua mhod, or by linking o a ODE solvr subrouin. Alrnaivly, by us of h ingraing facor, w can rwri Equaion 3 as: hn, di(; ) + (; ) I(; ) = d d d (; )(;) I(; ) = (; )(; ) Ingraing boh sids from o n, using h boundary condiion ha h innsiy a is I( ; ), yilds: and so, I(; ) n n Z n = I(n; )? Z n which simplis o: n (;) d I( n; ) = 0 (; )(; ) d I(0 ; ) = (; )(; ) d Z n n (;) d I( ; ) (; )(; ) d+ (Th limis of ingraion of h ingraing facor wr chosn so as o saisfy h boundary condiion.) By combining h wo xponnials in h rs rm, w g: I( n; ) = Z n?n (; )(; ) d+?n (;) d I( ; ) Equaion 4 can no b solvd in closd form for h gnral cas. Howvr, if h ransfr funcions vary picwis linarly along a ray sgmn wihin a cll, hn his quaion can b ingrad xacly on a cll by cll basis. This soluion is givn in Scion 3.3 afr paramr funcions ar inroducd in Scion 3.. If h chromaiciy is assumd o b consan along a ray hn, by h sam mhod ha Max [8] usd for Equaion or by a simpl ransformaion of Equaion 4, a closd form soluion for Equaion 4 can b obaind: n I( n; ) = ()(1? (u;); du ) + I(0 ; )?n (;) d If w furhr assum ha boh h opical dnsiy and chromaiciy ar consan along a ray, hn from Equaion 5 or by ingraing Equaion 3 by sparaion of variabls, w g: {z } A I( n; ) = () (1??()(n?) ) +I( ; )?()(n?0) ) {z } B Th rm labld B is h ransmianc of h rgion, h fracion of ligh of wavlngh nring h rgion a ha rachs n. Th opaciy is on minus h ransmianc or 1??()(n?0) and rprsns h fracion of ligh of wavlngh ha nrs h rgion a ha is occludd whil passing hrough h rgion. Trm A rprsns h anuaion of h ligh mid wihin h rgion islf. Equaion 6 is h alpha composiing formula dscribd by Porr and Du [11] which hy call h aop opraor. Th rsricion ha h opical dnsiy and/or chromaiciy b consan along a ray is no oo srious sinc h cloud is discrizd ino clls. Th ray ingraion can b valuad by discrizing h ray in xacly h sam way ha is usd in ni lmn analysis and hn ingraing on a cll by cll basis, hus h dnsiy and/or chromaiciy can vary from cll o cll. Equaion 6 can b valuad for ach cll by ling () and () b h avrag chromaiciy and dnsiy rspcivly along h ray bwn 1 and, h poins whr h ray nrs and xis h cll: avg () = (; 1) + (; ) and similarly for h opical dnsiy. Sinc i is common o linarly inrpola h scalar ld wihin a cll, his approximaion is accpabl. Volum rndrrs basd on Equaion 6 ar discussd in [14, 19]; boh rndrrs us h visibiliy ordring algorihm dscribd in [0]. For furhr cincy, h opaciy of h cll along h ray, = 1??()(?1) (7) (4) (5) (6) 4
5 can b approximad by = ()(? 1 ) (8) providd ()(? 1 ) 1:0. Phoographs of imags ha compar h us of Equaions 7 and 8 ar shown in [18]. Th calculaion can b furhr simplid if h opical dnsiy is assumd o b indpndn of wavlngh, so only four usr-dnd ransfr funcions ar rquird. If Equaion 7 is usd, hn h xponnial nd b valuad only onc pr cll. If i is dsird o spcify h opaciy and color in h rang (0; 1), h opical dnsiy and chromaiciy whos rang is (0;1) can b normalizd o h rang (0; 1), as ^ = 1?? and ^ = 1??, whr ^ is h normalizd dnsiy and ^ is h normalizd chromaiciy. 3. Shading, Gradins & Paramr Funcions For shading conour (lvl) surfacs, h innsiy a a poin on h surfac can b mad o vary as a funcion of h angl bwn h surfac normal vcor and a vcor o a poin ligh sourc. On way o incorpora surfac shading ino h modl is discussd a h nd of Scion 3.3. Th surfac normal a any poin p on a lvl surfac of a scalar ld S is h dircion of h gradin of S a p. For rahdra, h gradin will b consan hroughou h cll. For ohr yps of clls usd in h ni lmn mhod, h gradin can b compud from h paramr funcion for h yp of cll involvd. Th paramr funcion S c, somims rfrrd o in visualizaion as h inrpolaion funcion, givs h valu of h scalar ld wihin any givn cll. Th paramr funcion may b linar, quadraic, cubic, c. For xampl, for a 4 nod rahdron, h paramr funcion is linar; and so h scalar ld wihin a cll is givn by h paramr funcion: S c (x; y; z) = c 1 + c x + c 3 y + c 4 z (9) An appndix givs paramr funcions for ohr common clls and also discusss how h paramr funcions rla o h shap or basis funcions. Sinc h scalar ld valu is known a h four vrics of h cll and h coordinas of h vrics ar known, Equaion 9 bcoms a s of 4 simulanous quaions which can b solvd for c 1, c, c 3 and c 4. Th paramr funcions will gnrally no b C 1 - coninuous a h boundary bwn clls; and so h gradin will no b C 0 -coninuous bwn clls. If h chang in h gradin bwn clls is larg hn h shading will show anomalis from cll o cll. This can provid usful fdback o h scinis rgarding h qualiy of his/hr msh. Howvr, if smooh shading is dsird, an avrag gradin a ach vrx may b calculad by avraging h gradins a h cnroids of all clls ha shar h vrx. 1 Th surfac normal a any poin in a cll can hn b calculad by inrpolaing h gradins a h cll's vrics. A mor accura avrag vrx gradin can b calculad by wighing h gradins a h surrounding cnroids by h invrs of h disanc o h cnroid. Ohr mhods ar dscribd in [4]. 3.3 Exac Soluion for Linar Paramr and Transfr Funcions Equaion 4 can b ingrad xacly on a cll by cll basis if h ransfr funcions vary picwis linarly along a ray sgmn wihin a cll. This can b don as follows. If h scalar ld daa has bn gnrad by h - ni lmn mhod, hn, wihin a cll, h scalar ld is givn by h paramr funcion S c (x; y; z). This is shown for a linar rahdron in Equaion 9. W would lik o xprss S c as a funcion of, h ray paramr. In paramric form, a ray hrough h cll is xprssd as: x = 1 + (10) y = 1 + (11) z = 1 + (1) whr ( 1 ; 1 ; 1 ) is h poin whr h ray nrs h cll and ( ; ; ) is a uni vcor along h dircion of h ray. Subsiuing Equaions 10, 11 and 1 ino Equaion 9 givs, for h cas of linar rahdral lmns: S c () = v + w (13) whr v = c 1 +c 1 +c 3 1 +c 4 1 and w = c +c 3 + c 4. Now h ransfr funcions mus b considrd. L s 0 = S c ( ) b h scalar ld valu a h poin whr h ray nrs h cll, and s n = S c ( n ) b h valu a h xi poin. If h ransfr funcions ar picwis linar, as hy ar in Figur, hn hy can b considrd o b composd of m + 1 picwis linar inrvals (s 0 ; s 1 ), (s 1 ; s ),..., (s i ; s i+1 ),..., (s m ; s n ). S Figur whr v inrvals ar shown. Th innsiy of ligh a n can 1 Whn a msh is rcilinar, a ni dirnc schm [15, 7, 16] can b usd o approxima h gradin. 5
6 p,k p = 1.0 k = 1.7 p = 5 3s p = 0.4 k = 10 4s k = s p = 0. s s s s s s n s Figur : Exampl picwis linar ransfr funcions r and r, for rd ligh. s 0 and s n ar h valus of h scalar ld a h poins whr h ray nrs and xis h cll. b calculad by ingraing Equaion 4 ovr ach of h (m+1) inrvals. Th valu of a s 1 ; s ; : : : ; s m can b found from Equaion 13. For xampl: 1 = (s 1? v)=w. For rd ligh, h rms involving r and r in Equaion 4 ar r (s) = a + bs and r (s) r (s) = f + gs + hs or in rms of, using Equaion 13: r () = a + bv + bw r () r () = f + gv + hv + gw + hvw + hw ovr any inrval on which and ar boh linar. Hnc a abl lookup procdur may b usd o rurn h cocins a, b, f, g and h for any givn inrval; v and w ar calculad from h ray nry and xi poins and h paramr funcion for h cll. For xampl, for h rs inrval (s 0 ; s 1 ) shown in Figur, a = 5, b =?3, f = 8:5, g =?5:1 and h = 0. For rd ligh, Equaion 4 can hn b wrin as: I r( i+1 ) = Z i+1 i?i+1 (a i +b i v+b i wu) du (f i + g i v + h i v i+1 +g i w + h i vw + h i w (a i +b i v+b i w) d ) d + I r( i ) i simplifying, w g: I r( i+1 ) = Z i+1 i?i+1 (q 1 +q u) du (q3 + q 4 + q 5 ) d i+1 (q 1 +q ) d +I r( i ) i whr q 1 = a i + b i v, q = b i w, q 3 = f i + g i v + h i v, q 4 = g i w + h i vw and q 5 = h i w. Th scond rm can b ingrad asily o giv: Th rs rm can b valuad by compling h squar of h xponn and hn using ingraion by pars. W vrid our hand calculaions wih Mahmaica, vrsion.0, and rpor hr h Mahmaica oupu for h rs rm: q q 4? q 1 q 5 + q q 5 i+1 q? q q 4? q 1 q 5 + q q 5 i q ( i? i+1 )(q 1 +q i +q i+1 )= + p (q q 3? q 1 q q 4 + q1 q 5? q q 5 ) q :5 (q 1+q i+1 ) =(q ) (Er( q 1 + q i+1 p )? Er( q 1 + q i p )) q q In his xprssion, h argumn o Er is ihr ral, whn q > 0, undnd if q = 0, or pur imaginary, whn q < 0. Er(x) is shorhand for wo funcions, dpnding on whhr x is ral or pur imaginary. If x is ral, Er(x) = p R x 0 u du, whil if x is pur imaginary, of h form x = ib, wih ral b, Er(ib) = i p R b 0?u du. Whn q < 0, h i in h lar xprssion cancls h i in h dnominaor q :5. Thus, w nd only prpar on dimnsional abls for hs wo ingrals, or us subrouins o approxima hm. (Th scond ingral is closly rlad o h ingral of h Gaussian rror funcion, for which subrouins and abls alrady xis.) If q = 0, hn h rs rm aks a dirn form, and ingraion by pars or Mahmaica givs: (q1 q 3? q 1 q 4 + q 5 + q1 q 4 i+1? q 1 q 5 i+1 + q1 q 5 i+1 ) q1 3? (q1 q 3? q 1 q 4 + q 5 + q1 q 4 i? q 1 q 5 i + q1 q 5 i )q 1( i? i+1 ) q 3 1 I r( i )?(q 1 i+1 + q i+1?q 1 i? q i ) This is a nw rsul. I rmains o b sn if hs 6
7 xprssions can b valuad so as o prmi inraciv rndring. Crainly, hy can b usd o gnra bnchmark imags agains which imags gnrad by approxima soluions, such as hos givn in Equaions 5 and 6, can b compard. And, hy can b usd for h gnraion of prsnaion qualiy imags. Accura approximaions nd o b invsigad. Surfac shading is hlpful for undrsanding h orinaion of (iso)surfacs ha may appar in a volum rndrd imag. Howvr, for h rmaindr of h imag shading is no appropria and may vn mak h imag hardr o undrsand. An inrval in h dnsiy map which producs an isosurfac is disinciv, for xampl a all narrow rcangular puls or a dla funcion. If such inrvals ar aggd, surfac shading can b urnd on only for hos inrvals. 4 Conclusion W hav prsnd a coninuum-basd volum dnsiy opical modl spcically for dirc volum rndring of scard daa or scalar lds from h ni lmn mhod, as opposd o scannd daa ss whr marial classicaion is involvd. Th modl has bn dvlopd so as o b inuiiv and also o maximiz h xibiliy of us of ransfr funcions. Six usr-spcid ransfr funcions, hr for opical dnsiy and hr for color, ar prmid. Th modl is suiabl ihr for ray racing or for projcion mhods. Wilhlms and Van Gldr [17] oulin a coninuous modl and dvlop dirnial quaions for cumulaiv innsiy and ransmianc basd on i. Our modl bns from hir dvlopmn, bu also dirs from hir modl in a numbr of rspcs. For xampl, in our modl h opaciy dos no appar as a facor in h dnominaor of h cumulaiv innsiy; and color innsiy is linkd o opaciy, i.. color and opical dnsiy ar muliplicaiv facors. Th valu of h formr propry is ha h cumulaiv innsiy is wll bhavd for vry small or zro opaciy. Th lar propry mans ha if h ransfr funcions ar dnd such ha (a) h opical dnsiy a a poin in h volum is vry small, i.. h mdium is almos oally ransparn a ha poin, and (b) h color innsiy is maximum a ha poin, hn h ovrall conribuion will b minimal. Ths dirncs nd furhr invsigaion. Closd form soluions undr svral dirn assumpions hav bn prsnd, including a nw rsul for h cas whn h ransfr funcions vary picwis linarly along a ray sgmn wihin a cll. This nw xac soluion nds o b implmnd and h rndring im and imag qualiy compard wih hos rsuling from implmnaions of Equaions 5 and 6 and wih rsuls from ohr volum dnsiy modls. Also, h uiliy of h hr dnsiy ransfr funcions inroducd by h coninuous modl nds o b invsigad. Acknowldgmns Hlpful convrsaions wr had wih Rogr Craws, Michal Hah, Paul Saylor and Pr Shirly. John Gray's and Tom DBoni's assisanc wih Mahmaica is apprciad. Th rs auhor is graful o Pr Shirly for ncouraging him o undrak his projc. This work was parially suppord by h U. S. Dparmn of Enrgy undr Gran DE-FG0-85-ER5001 a h Cnr for Suprcompuing Rsarch and Dvlopmn, and conrac numbr W-7405-ENG-48 a Lawrnc Livrmor Naional Laboraory, h Air Forc Oc of Scinic Rsarch Grans AFOSR , and Sun Microsysms Inc. Appndix Th paramr funcion for a 4 nod rahdron is givn in Scion 3.. For h ohr hr mos common 3D lmns (clls), h paramr funcions ar as follows. For an undisord 10 nod (quadraic) rahdron: S c(x; y; z) = c 1+c x+c 3y+c 4z+c 5x +c 6xy+c 7y +c 8yz+c 9z +c 10xz For an 8 nod (rilinar) brick: S c(x; y; z) = c 1 + c x + c 3 y + c 4 z + c 5 xy + c 6 yz + c 7 xz + c 8 xyz For a 0 nod undisord (quadraic) brick: S c(x; y; z) = c 1 + c x + c 3 y + c 4 z + c 5 xy + c 6 xz + c 7 yz + c 8 x + c 9 y + c 10 z + c 11 xyz + c 1 x y + c 13 xy + c 14 x z + c 15 xz + c 16 y z + c 17 yz + c 18 x yz + c 19 xy z + c 0 xyz Whn h lmns ar disord by h us of an isoparamric mapping funcion, hn h mapping funcion nds o b invrd and h inrpolaion prformd in h undisord lmn using h paramr funcion. (In ordr o assur convrgnc in h ni lmn mhod, h mapping funcion mus b invribl.) Th opic of visualizaion of disord lmns is no dal wih hr bu i is an imporan issu ha nds o b facd. Th paramr funcion can in gnral b xprssd in rms of h scalar valus s i a h i vrics of h cll: S c(x; y; z) = X For a linar rahdron, his bcoms: i N i (x; y; z)s i S c(x; y; z) = N 1(x; y; z)s 1+N (x; y; z)s +N 3(x; y; z)s 3+N 4(x; y; z)s 4 Th cocins N i (x; y; z) ar rfrrd o as h shap or basis funcions for h cll. Thy ar polynomials of h form discussd abov; for xampl on h 4 nod rahdron hy ar linar. 7
8 Rfrncs [1] Blinn, Jim. Ligh Rcion Funcions for Simulaion of Clouds and Dusy Surfacs. ACM SIG- GRAPH Compu. Gr (July 198), 1-9. [] Chandraskhar, S. Radiaiv Transfr. Dovr, Nw York, [3] Ebr, David and Parn, Richard. Rndring and Animaion of Gasous Phnomna by Combining Fas Volum and Scanlin A-bur Tchniqus. ACM SIGGRAPH Compu. Gr. 4 4 (July 1990), 357{363. [4] Gallaghr, Richard and Naggaal, Joop. An Ef- cin 3D Visualizaion Tchniqu for Fini Elmn Modls and Ohr Coars Volums. ACM SIGGRAPH Compu. Gr. 3 3 (July 1989), 185{ 19. [5] Kajiya, Jams and Von Hrzn, Brian. Ray Tracing Volum Dnsiis. ACM SIGGRAPH Compu. Gr (July 1984), [6] Krugr, Wolfgang. Th Applicaion of Transpor Thory o Visualizaion of 3D Scalar Daa Filds. Proc. IEEE Visualizaion '90 (Oc. 1990), 73{80. [7] Lvoy, Marc. Display of Surfacs from Volum Daa. IEEE Compu. Graph. Appl. 8 3 (May 1988) [8] Max, Nlson, Hanrahan, Pa, and Craws, Rogr. Ara and Volum Cohrnc for Ecin Visualizaion of 3D Scalar Funcions. San Digo Workshop on Volum Visualizaion, Compu. Gr. 4 5 (Dc 1990), 7{33. [9] Max, Nlson. Amosphric Illuminaion and Shadows. ACM SIGGRAPH Compu. Gr. 0 4 (Aug. 1986), 117{14. [10] Max, Nlson. Ligh Diusion hrough Clouds and Haz. Compu Vision, Graph. and Imag Proc. 33 (March 1986), 80{9. [11] Porr, Thomas and Du, Tom. Composiing Digial Imags. ACM SIGGRAPH Compu. Gr (July 1984), 53{59. [1] Rushmir, Holly and Torranc, Knnh. Th Zonal Mhod for Calculaing Ligh Innsiis in h Prsnc of a Paricipaing Mdium. ACM SIGGRAPH Compu. Gr. 1 4 (July 1987), 93{ 30. [13] Sablla, Paolo. A Rndring Algorihm for Visualizing 3D Scalar Filds. ACM SIGGRAPH Compu. Gr. 4 (July 1988), [14] Shirly, Pr and Tuchman, Allan. A Polygonal Approximaion o Dirc Scalar Volum Rndring. San Digo Workshop on Volum Visualizaion, Compu. Gr. 4 5 (Dc 1990), [15] Upson, Craig and Klr, Michal. V-BUFFER: Visibl Volum Rndring. ACM SIGGRAPH Compu. Gr. 4 (July 1988), [16] Wsovr, L. Inraciv Volum Rndring. Procdings Chapl Hill Workshop of Volum Visualizaion (May 1989), [17] Wilhlms, Jan and Van Gldr, Alln. A Cohrn Projcion Approach for Dirc Volum Rndring. ACM SIGGRAPH Compu. Gr. 5 4 (July 1991), [18] Williams, Pr. Inraciv Dirc Volum Rndring of Curvilinar and Unsrucurd Daa. PhD hsis, Dp. of Compur Scinc, Univrsiy of Illinois a Urbana-Champaign, 199, C.S.R.D. Rpor No [19] Williams, Pr. Inraciv Splaing of Nonrcilinar Volums. To appar in Proc. IEEE Visualizaion '9 (Oc. 199). [0] Williams, Pr. Visibiliy Ordring Mshd Polyhdra. ACM Trans. on Graphics 11 (April 199), 103{16. 8
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