Xn Sun Guofu Xe Yue Dong Stephen Ln Wewe Xu Wencheng Wang Xn Tong Banng Guo Mcrosoft Research Asa State Key Laboratory of Computer Scence, ISCAS GUCAS Fgure : Our method provdes a compact and explct representaton of dffuson curve mages for texture mappng onto a surface. The sharp features and detaled color varatons of textures are well preserved n the renderng results. Abstract We ntroduce a vector representaton called dffuson curve textures for mappng dffuson curve mages DCI) onto arbtrary surfaces. In contrast to the orgnal mplct representaton of DCIs [Orzan et al. 008], where determnng a sngle texture value requres teratve computaton of the entre DCI va the Posson equaton, dffuson curve textures provde an explct representaton from whch the texture value at any pont can be solved drectly, whle preservng the compactness and resoluton ndependence of dffuson curves. Ths s acheved through a formulaton of the DCI dffuson process n terms of Green s functons. Ths formulaton furthermore allows the texture value of any rectangular regon e.g. pxel area) to be solved n closed form, whch facltates ant-alasng. We develop a GPU algorthm that renders ant-alased dffuson curve textures n real tme, and demonstrate the effectveness of ths method through hgh qualty renderngs wth detaled control curves and color varatons. CR Categores: I.3.7 [Computer Graphcs]: Three-Dmensonal Graphcs and Realsm Color, shadng, shadowng, and texture; Keywords: vector mages; dffuson curves; texture mappng and renderng Lns: DL PDF Introducton Dffuson curves [Orzan et al. 008] are powerful prmtves for creatng and edtng smooth-shaded vector graphcs mages. They ACM Reference Format Sun, X., Xe, G., Dong, Y., Ln, S., Xu, W., Wang, W., Tong, X., Guo, B. 0. Dffuson Curve Textures for Resoluton Independent Texture Mappng. ACM Trans. Graph. 3 4, Artcle 74 July 0), 9 pages. DOI 0.45/8550.85570 http://do.acm.org/0.45/8550.85570. Copyrght Notce Permsson to mae dgtal or hard copes of part or all of ths wor for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or drect commercal advantage and that copes show ths notce on the frst page or ntal screen of a dsplay along wth the full ctaton. Copyrghts for components of ths wor owned by others than ACM must be honored. Abstractng wth credt s permtted. To copy otherwse, to republsh, to post on servers, to redstrbute to lsts, or to use any component of ths wor n other wors requres pror specfc permsson and/or a fee. Permssons may be requested from Publcatons Dept., ACM, Inc., Penn Plaza, Sute 70, New Yor, NY 0-070, fax + ) 869-048, or permssons@acm.org. 0 ACM 0730-030/0/08-ART74 $5.00 DOI 0.45/8550.85570 http://do.acm.org/0.45/8550.85570 consst of control curves wth dfferent colors defned along each sde. Images are produced from them by dffusng the colors over the mage space n a manner that resembles radatve heat transport. As wth other vector graphcs models, dffuson curves have a concse representaton that s resoluton ndependent and easy to manpulate. But what maes dffuson curves especally appealng among such technques s ts ablty to model a broad array of mages wth subtle shadng varatons. Because of ts smplcty and representaton power, dffuson curves has become a popular tool for graphc artsts and s consdered a possble addton to the Scalable Vector Graphcs SVG) specfcaton. Dffuson curve mages DCIs) and other forms of vector graphcs have drawn consderable attenton as a representaton for texture maps [Qn et al. 006; Nehab and Hoppe 008; Jesche et al. 009b]. In contrast to conventonal btmap textures that suffer from lmted resoluton, a DCI s resoluton ndependent n addton to beng compact. However, there exst two major challenges n usng them for texture mappng. Frst, dffuson curves provde an mplct representaton that does not support random access. In order to obtan the DCI value at a partcular pont, the full mage must be syntheszed by solvng a Posson equaton. Second, pxels on the control curves need to be specally processed to generate ant-alased results, whch complcates the renderng process and decreases performance. Jesche et al. [009b] desgned a real-tme rasterzaton method for convertng a DCI nto btmaps for ant-alased texture mappng. However, very hgh resoluton pxel grds are requred by ths approach to accurately model fne-scale texture detals, wth a consequent reducton of speed. Recently, Bowers et al. [0] and Pang et al. [0] proposed schemes that support random access but generate only an approxmaton to the DCI. Moreover, t s unclear how to perform ant-alased renderng wth these technques. In ths paper, we propose an explct representaton of dffuson curve mages called dffuson curve textures that effcently supports random access and ant-alased renderng, whle mantanng the favorable characterstcs of dffuson curves. The ey dea of our wor s to formulate the dffuson process of DCIs n terms of Green s functons. We defne weghted ernels based on Green s functons along each of the control curves. These Green s functon ernels are fxed-dstance functons that descrbe the contrbuton of a control pont to each pont n the D mage doman, whle the ACM Transactons on Graphcs, Vol. 3, No. 4, Artcle 74, Publcaton Date: July 0
74: X. Sun et al. ernel weghts are determned from the colors defned along the dffuson curves. Wth ths Green s functon based representaton, we show that the mage value at any gven pont can be drectly evaluated by aggregatng the contrbutons of weghted Green s functon ernels along all of the control curves. A major advantage of the Green s functon based representaton s that the ntegraton of Green s functon ernels for a rectangular mage regon has a closed-form soluton. Dffuson curve texture values for arbtrary szed rectangles e.g. pxel areas) can thus be computed at a constant renderng cost, an mportant feature for fast ant-alased renderng. We develop an effcent algorthm for renderng ant-alased dffuson curve textures that ncorporates a curve cullng scheme and an adaptve samplng strategy to expedte processng. Snce our algorthm renders all pxels n a unform manner, t can be easly mplemented on the GPU. The renderng performance depends only on mage resoluton and s ndependent of texture scales over surfaces and vewng drectons. Wth ths technque, we demonstrate hgh qualty renderngs from dffuson curve textures for varous levels of dffuson curve complexty. In summary, the contrbutons of ths paper are Dffuson curve textures, an explct representaton of dffuson curve mages that supports random access and fast ntegraton. An effcent algorthm for renderng ant-alased dffuson curve textures mapped over 3D surfaces. A GPU mplementaton of the renderng method that provdes real-tme performance. Related Wor Vector Images and Textures A vector mage can be drectly modeled by a set of smple vector prmtves such as ponts, lnes, curves, and polygons assocated wth colors. Qn et al. [006; 008] developed technques for renderng ant-alased vector textures based on the dstance between pxel samples and prmtves. Nehab and Hoppe [008] used a lattce structure for speedng up the random access of vector prmtves and generated ant-alased results by preflterng and supersamplng the vector prmtves n a pxel. In our wor, we am for smlar goals but for smoothly shaded mages defned by dffuson curves. Dfferent vector representatons have been ntroduced for modelng smoothly shaded mages. The gradent mesh s wdely used n ndustry for vectorzng mages that consst of smooth color regons [Lecot and Levy 006; Sun et al. 007; La et al. 009]. Xa et al. [009] segmented mages nto trangular patches and modeled the color varaton nsde a patch wth thn plate splnes. Dffuson curve mages [Elder and Goldberg 00; Orzan et al. 008] and ts extensons [Bezerra et al. 00; Taayama et al. 00] construct mages or volumes by dffusng colors specfed on curves and surfaces. Most recently, Fnch et al. [0] ntroduced controlled thn plate splnes for authorng vector mages wth greater expressve control. These wors all focus on authorng or vectorzng raster mages, rather than on how to use them for texture mappng. Recently, Wang et al. [00; 0] presented a vector representaton for sold textures n whch the regon boundary s defned by mplct dstance functons and the volumetrc color varatons are modeled by radal bass functons RBFs). Ths approach, however, s dffcult to extend to DCIs wth non-closed control curves. Several hybrd representatons combne raster textures wth sharp vector features embedded n texels for ant-alased texture mappng [Ramanarayanan et al. 004; Sen et al. 003; Sen 004; Tumbln and Choudhury 004; Tarn and Cgnon 005]. Most of these methods store a fxed number of prmtves n each texel and need hgh resoluton mages for representng textures wth hgh local complexty. Although storng varous numbers of prmtves n texels [Ramanarayanan et al. 004] could avod ths problem, ths leads to hgh computatonal costs for texture samplng and nterpolaton. We use the compact representaton of dffuson curve textures to model both sharp features and smooth color varatons n the mage and render them n the same way. Dffuson Curve Images Dffuson curve mages were ntroduced by Orzan et al. [008] for modelng vector mages wth smooth color varatons. A sgnfcant dfference of DCI from other vector graphcs representatons s that the mage has to be computed by solvng a Posson equaton defned over the entre D doman. Generally, the soluton at an n n resoluton can be solved by multgrd methods [Bolz et al. 003; Kazhdan and Hoppe 008; Jesche et al. 009a] wthon ) tme complexty. To render a DCI mapped onto a surface, a naïve soluton s to rasterze the DCI as btmaps at specfc resolutons and then render the result va the tradtonal renderng ppelne. However, a hgh resoluton mage s needed to preserve sharp features at close-up vews, whch leads to large computatonal and storage costs. Jesche et al. [009b] allevated ths ssue by warpng the texture space accordng to the current vew and usng a dynamc feature embeddng algorthm to retan sharp features n the generated texture mage. However, for dffuson curve mages wth rch features or complex curves, a hgh resoluton soluton s stll needed. By contrast, our method drectly ntegrates texture values n each pxel regon from the compact Green s functon based representaton and preserves sharp features at any resoluton. Wth the two acceleraton schemes presented n the paper, the tme complexty of our renderng algorthm s proportonal to the screen resoluton and ndependent of model and mage complexty. Bowers et al. [0] desgned a ray tracng based approach to smulate DCIs. By reformulatng the problem to be equvalent to fnal gatherng n global llumnaton, the mage values can be evaluated by GPU based stochastc raytracng. They also extended the curve attrbutes of DCI to support addtonal authorng tass. Most recently, Pang et al. [0] used mean value coordnates nterpolaton on a trangle mesh to generate mages smlar n appearance to DCIs. Both methods provde only approxmate solutons to the DCI. Also, t s unclear how to effcently perform ntegraton over regons wth these methods, whch s a requrement for texture antalasng. Unle these approxmaton technques, our method provdes an accurate soluton for dffuson curve mages and supports random access as well as fast ntegraton. Green s Functons Green s functons have been appled n varous graphcs applcatons. D Eon and Irvng [0] used a quantzed Green s functon for renderng lght dffuson n translucent materals. Lpman et al. [008] presented Green Coordnates for mesh deformaton based on the Green s functon for the Laplacan. An analytc ntegraton of the Green s functon over a closed polyhedral cage s used for Green Coordnates computaton. Farbman et al. [0] modeled the soluton of the Posson equaton n the mage doman as the convoluton of a Green s functon and the mage gradents. They then approxmated the convoluton wth specally desgned convoluton pyramds. Our method adapts the Green s functon for the Laplacan n D space to model and render dffuson curve mages, for whch the underlyng Posson equaton can be accurately modeled by Green s functon ernels defned on control curves. Moreover, we derve an analytc soluton for ntegratng the D Green s functon ernel contrbutons over a rectangular regon and present a method to compute ths n real tme. ACM Transactons on Graphcs, Vol. 3, No. 4, Artcle 74, Publcaton Date: July 0
3 Dffuson Curve Textures Nrx ) Nx ) x Clx )/Elx ) δ xx 0.5 where δ s the Drac delta functon. D I D n x )! n x ) ) where n x) s the normal drecton on the boundary D. The followng expresson for ) can be formulated from Eq. ): D ). 3) From Eq. ) and Eq. 3), we can derve Green s thrd dentty to evaluate ): ) D + I D 3. n x )! n x ). 4) Green s Functon Kernels We utlze Green s functons to solve for values n a dffuson curve mage. Dffuson curves [Orzan et al. 008; Jesche et al. 009a] represent an mage as a harmonc functon ) that satsfes the Laplace equaton: ) 0 c) d) e) f) x D {Cl x, Cr x } x B. Wth Eq. 5) and Eq. 6), we can smplfy Eq. 4) to ) D Cl x Cr x C x B + B B n x ) nr nr x ) x ) nl x ) Fgure : A dffuson curve texture defned by two control curves: an S-shaped curve, and a curve along the mage boundary. Control curves wth boundary condtons 0.75 for the rght S-shaped boundary, 0.5 for the left S-shaped boundary and the mage boundares). Profle of Green s functon ernel G x, x ) from one pont on a control curve. c) Profle of Green s functon ernel Gn x, x ) from one pont on a control curve. d) Dffuson curve mage computed from all of the Green s functon ernels. e) Image generated from only the Green s functon ernels G x, x ). f) Image generated from only the Green s functon ernels Gn x, x ). In the rghtmost two columns, red ndcates postve values, whle blue ndcates negatve values. where C x ) Cl x )+Cr x ), E x ) n x ) nl x ) nr E x x )!! ux ) + nr x ). ) X C x B Gn x, x E x. 8) In our wor, we employ the Green s functon G x, x ) n R : ln π xx 9) wth the correspondng normal dervatve Gn x, x ): Gn x, x n x ) d n x nx + ny 0) π + ) where n x ) nx, ny ) and, ) x x.!! ux ) nl x ) Eq. 7) ndcates that ) at any pont wthn the entre doman D can be evaluated through computatons only on the control curves. Snce the Green s functon s symmetrc wth respect to x and x.e. G x, x) G x, x )), we can vew Eq. 7) n terms of two Green s functon based ernels defned along the control curves, nstead of over the doman D. Therefore, we explctly represent a dffuson curve mage wth the two Green s functon based ernels Gx,x ) G x, x ) and Gn x, x ) nx ), and ther weghts C x ) and E x ) defned along the control curves: 6) As llustrated n Fg., we defne boundary normals to be orented outwards from ther doman.e. toward the other sde of B) and the normal of the control curve B as n x ) nr x ) nl x ). I 4 5) wth boundary condtons Cl, Cr specfed on the two sdes of each control curve B: 56 ) For a scalar functon ) that s twce contnuously dfferentable on regon D n R, Green s second dentty relates a double ntegral over D to a lne ntegral over the regon boundary: ux) x D Green s functons are a useful tool for solvng nhomogeneous dfferental equatons wth boundary condtons [Bayn 006]. For the Laplacan operator, a Green s functon G x, x ) at a pont x satsfes the followng equaton: Nlx ) Crx )/Erx ) Prelmnares on Green s Functons 74:3 Bx ) In ths secton, we brefly revew Green s functons, and then derve a representaton of dffuson curves based on Green s functons from whch dffuson curve mages can be drectly solved. 3., 7) Note that ths explct representaton s dfferent from the orgnal mplct dffuson curves representaton that s based on the Posson equaton. Frst, our method allows evaluaton at only the pont tself wthout consderng other areas, whle a lnear system for the Posson equaton must be solved over the entre doman to compute the value at a sngle pont. Second, unle the explct soluton provded by Green s functon ernels, the orgnal mplct representaton requres dscretzaton of the doman to solve the Posson equaton, whch ntrnscally enforces a tradeoff between renderng qualty and computatonal performance. ACM Transactons on Graphcs, Vol. 3, No. 4, Artcle 74, Publcaton Date: July 0
74:4 X. Sun et al. Fg. c) llustrate the two Green s functon ernels defned at one pont on the control curve, where G s sotropc whle G n s ansotropc. As shown n Fg. e)f), the contrbutons of the two ernels defned along the curve result n the global low frequency color varaton and the hgh frequency color change at boundares, respectvely. To calculate Ex ), we note that for a closed doman D wth Hölder contnuous boundary values ux ) x D, the soluton for ux) s unque. So we solve ux) D u x ) G ) n x,x ) ) nx ) G x,x ),x D ) and compute the values of Ex ) from ux ) and ux ). We n l x ) n rx ) note that to solve the dffuson equaton over an mage, boundary condtons at the mage borders or beyond the mage borders) must be specfed. So n practce we always place a control curve at or beyond the mage borders for a gven dffuson curve mage. 4 Renderng wth Dffuson Curve Textures Gven the Green s functon ernels on the control curves, the harmonc functon ux) can be constructed accordng to Eq. 8). For renderng purposes such as ant-alasng, we wsh to evaluate the ntegral of ux) over a rectangular regon R nstead of at just an ndvdual pontx: φr) ux). ) R Renderng wth ths functon nvolves three components: ntegratng over the regon R, ntegratng along each control curve B, and accumulatng contrbutons from all the control curves. We address each of these components n ths secton. 4. Integraton over a Rectangle The ntegral of ux) n Eq. 8) over a regon R as n Eq. ) can be expressed n terms of ntegratons of the Green s functon ernels Gx,x ) and G nx,x ) over the rectangular regon R {x x 0,x ),y y 0,y )}: φr) C ) x G ) n x,x R B E ) x G x,x ) R B 3) A closed-form ntegral F GR,x ) R Gx,x ) exsts for ths Green s functon over a rectangular regonr: F ) G R,x ) +j H G ˆx,ŷ) 4) 4π,j {0,} H G ˆx,ŷ) 3 ˆxŷ + ˆxŷln ˆx + ŷ ) + ˆx arctgŷ ˆx + ŷ arctg ˆx ŷ. 5) A closed-form ntegral F Gn R,x ) R Gnx,x ) lewse exsts forg n: F ) Gn R,x ) +j H Gn ˆx,ŷ,n x,n y) π,j {0,} 6) ) ˆx ŷ ) H Gn ˆx,ŷ,n x,n y) n yŷarctg ŷ + n xˆxarctg ˆx + nyˆx + nxŷ)ln ˆx + ŷ ) 7) where x x,y), x x,y ), nx ) n x,n y), and ˆx,ŷ) xx. Detaled dervatons of Eq. 4) and Eq. 6) are provded n Appendx A. Eq. 4) and Eq. 6) are derved for an axally algned rectangular regon, but are also vald for rotated rectangles snce the Green s functongx,x ) s rotatonally symmetrc aboutx. We note that though Gx,x ) has a sngular pont at x x, t does not affect the evaluaton of H Gˆx,ŷ) and H Gn ˆx,ŷ,n x,n y). An mportant feature of the closed-form ntegralsf GR,x ) andf Gn R,x ) s that the computatonal load s constant for dfferent szes of R, n contrast to samplng schemes whch generally requre greater computaton for larger regons. Wth the control curve representaton, we can compute Eq. ) as φr) B C x ) F G R,x ) + E x ) F Gn R,x )). 8) To rapdly solve Eq. 8), ntegraton along control curves B and accumulaton of control curve contrbutons need to be performed effcently. These two problems are addressed n Secton 4. and Secton 4.3. 4. Adaptve Samplng on Control Curves To effcently evaluate φr) n Eq. 8), we compute a pecewse constant approxmaton wth respect to a set of sample ponts { x,j } along each control curveb : φr) j ) φc x,j φ )) ) E x,j l x,j where l x,j) s the arc length between x,j and x ) ),j+, and φc x,j and φe x,j are weghted ernels of ) ) ) ) C x,j FGn R,x,j and E x,j FG R,x,j respectvely. Samplng wth a small, unform value of l x,j) would be neffcent over dfferent levels of detal, so we propose an adaptve samplng scheme. Our adaptve scheme ams to sample ponts such that the error n the pecewse constant approxmaton ofb between neghborsx,j and x,j+ falls below a gven bound. We approxmate ths by selectng sample ponts accordng to ) φs x,j+ φ S x,j ))/ φ S x,j) < α, S {C,E} 0) where α denotes a user-specfed bound. For a gven sample pont x,j, the next sample x,j+ cannot be determned from Eq. 0) wthout evaluatng dfferent canddate ponts. To avod ths neffcency, we use the frst order dfferental atx,j to approxmate the value atx,j+: 9) ) φ S x,j+ φ ) S x,j + φ ) ) S x,j l x,j, S {C,E} ) where φ ) ) ) ) S x,j φs x,j / t x,j and t x,j s the tangent drecton along the curve. We then can determne the arc length l x,j) from α: ) { ) ) } l x,j αmn φs x,j / φ S x,s,j {C,E}. ) Snce φ C and φ E can be solved analytcally see Appendx B), we can determne l x,j) through calculatons only at x,j. To sample a control curve B, we start by placng the frst sample x,0 at an arbtrary endpont, then teratvely place subsequent samplesx,j+ based on computed values ofl x,j) untl reachng the end. ACM Transactons on Graphcs, Vol. 3, No. 4, Artcle 74, Publcaton Date: July 0
Fgure 3: Control curve samplng. Unform samplng wth more than 500 samples along the control curve. Adaptve samplng requres only about 0 ponts to produce smlar results at the blue pont α 0.0). Texture Fgure Porcelan ephyr Frog Coffee cup Wood Dragon Fg. Fg. Fg. 4 Fg. 9 Fg. 9 Fg. 9 c) 5 In ths adaptve samplng scheme, we evaluate the samplng nter vals wth respect to φ C x,j and φ E x,j ndvdually nstead of together as the sngle term shown n Eq. 9). Snce the two Green s functon ernels produce dfferent effects as shown n Fg., we wsh to bound each of ther pecewse constant approxmaton errors, so we set the arc length n Eq. ) to the mnmum value computed from among the two. As shown n Fg. 3, ths adaptve samplng scheme greatly reduces the samplng rate whle mantanng the renderng qualty smlar to that of dense unform samplng. 4.3 Cullng of control curves In Eq. 8), φ s computed from all of the control curves, whch can be expensve when there s a large number of curves, even wth the adaptve samplng scheme. However, t s often possble to exclude many control curves wthout compromsng renderng qualty. Shown n Fg. 4, control curves that are enclosed by another control curve do not contrbute to mage values beyond the outer curve. Only the outsde boundary values of the outer curve need to be consdered; the nsde boundary values and the nner curves can be dsregarded. Lewse, regons enclosed by a control curve, as shown n Fg. 4, are unaffected by exteror curves. To facltate cullng, we dentfy areas enclosed by control curves, and organze them herarchcally accordng to contanment relatonshps. Wth ths preprocessng, unnecessary control curves can be effcently removed for a gven ntegraton regon pror to adaptve samplng. Fttng s) 0 0.46 0.96 0.87 660 0.05 Gven a dffuson curve mage, the texture s ux ) generated by solvng Eq. ) wth the normal dervatves nl x ) ux ) and nr x ). We unformly sample a number of ponts {x } along the control curves, wth a dstance of l between each par of adjacent ponts. We denote the left and rght sdes of the curve at pont x as x0 and x respectvely. Then Eq. ) s evaluated as Texture generaton s x XX j6 X Fgure 4: Cullng of control curves. The control curves n blue are enclosed by the control curve n red, so ntegraton regons outsde the red boundary can be computed ndependently of the blue curves and the nner boundary values of the red curve. Areas wthn the red control curve are unaffected by exteror control curves and the outer boundary values of the red curve. Perf. ms) 98 36 39 49 60 43 Expermental Results and Dscussons + Storage KBytes) 96 6 66 66 6 3 74:5 Table : Statstcs and renderng performance for the texture data shown n ths paper. All mages shown n ths paper are rendered wth a resoluton of 800 600 and reszed to ft the pages. u Number of curves 37,00 37 80 3,3,698 60 l u xj Gn s x, xj j n xj ) x s G x, xj u G l). G n s, ) n x ) 3) G and G n are the analytc ntegrals of G x, x ) and Gn x, x ) gven n Eq. 9) and Eq. 0) respectvely, wth x x to avod the sngularty: G l) π l/ l l + l ln ln x π 4) l/ G n s, ) ) s+ lm x 0+ π l/ )s+ x dy. x + y 5) l/ For regons enclosed by a curve wth constant color values, such as n the blue and whte porcelan of Fg., ths evaluaton procedure can be spped. If the texture ncludes regons enclosed by a curve wth color varatons, as exhbted n the coffee cup of Fg. 9, then each of these regons can be evaluated ndependently. To derve a dffuson curve representaton from a natural mage, such as Fg. 9, we trace the edges n the mage to obtan the control curves and then solve for ther colors and normal dervatves from colors sampled throughout the mage: ) X j, l u xj Gn x, xj j n xj ) G x, xj 6) where ponts {x } are unformly sampled over the mage, excludng ponts on the control curves. Snce colors need to be ft n addton to the normal dervatves and {x } {x }, ths process requres much tme. For hgh performance renderng of dffuson curve textures, we developed a GPU algorthm that utlzes the OpenGL ppelne to render each frame, wth texture fetchng and computaton mplemented wth NVda CUDA [NVIDIA 0]. The geometry, lghtng, materal and vewpont are loaded nto the OpenGL ppelne. The control curves are loaded nto the global memory of CUDA. To render a frame, we compute the transformaton, cullng, shadow testng, and materal shadng n the OpenGL Renderng on GPUs ACM Transactons on Graphcs, Vol. 3, No. 4, Artcle 74, Publcaton Date: July 0
74:6 X. Sun et al. c) d) Fgure 5: A vase covered wth dffuson curve textures of dfferent complexty. From left to rght, the numbers of control curves are 5K, 0K, 0K and 37K respectvely. Number of control curves Adaptve samplng wth curve cullng Adaptve samplng w/o curve cullng Unform samplng w/o curve cullng Resultng mage n Fg. 5 5,309 0,747 0,505 37,00 0.00 s 0.043 s 0.076 s 0.098 s 0.600 s.763 s 4.04 s 7.938 s 39.60 s 4.6 s 4. s 444.5 s c) d) Table : Renderng performance wth dfferent numbers of control curves and dfferent renderng algorthms. ppelne. At the begnnng of the fragment shader, we apply a separate pass to extract the texture coordnates and the correspondng DDX and DDY for each fragment. Then we swtch to CUDA processng wth ths data. The texture coordnates represent the center of the ntegraton regon, whle the DDX and DDY represent the orentaton of the regon as well as the length of the two correspondng dmensons. After that, we ntegrate the texture value for each pxel wth the algorthm descrbed above. Fnally, the ntegraton values are transported bac to the OpenGL ppelne as a texture for subsequent shadng n the fragment shader. We mplemented our algorthm on a PC wth two Intel Xeon E560.44 GHz CPUs and an NVda GeForce GTX 480 graphcs card wth GB of graphcs memory. Table lsts the statstcs and renderng performance for all of the texture data. As shown n the table, our method renders all of these examples n real tme. Dffuson curve textures are generated from DCIs at nteractve rates. Wth a natural mage such as the wood n Fg. 9 as nput, dffuson curve texture generaton taes substantally more tme as prevously dscussed. Performance To test the performance of our renderng method for dffuson curve textures wth dfferent levels of complexty, we constructed four dffuson curve textures wth dfferent numbers of control curves 5K, 0K, 0K and 37K respectvely) and mapped them onto the same vase model as shown n Fgure 5. To examne the effects of the two renderng acceleraton schemes, we rendered each texture usng three mplementatons: full algorthm, wthout curve cullng, and wthout both adaptve samplng and curve cullng. Table lsts the performance results for the four textures. It shows that the adaptve samplng scheme provdes a 0 speedup over the mplementaton wthout t, whle the curve cullng scheme further mproves the renderng speed by two orders of magntude. Note that both acceleraton schemes are based on the explct texture representaton proposed n ths paper. Wth these two schemes, the renderng performance of our method decreases sub-lnearly as the number of control curves ncreases. The renderng cost s lnearly dependent on the number of curve ACM Transactons on Graphcs, Vol. 3, No. 4, Artcle 74, Publcaton Date: July 0 Fgure 6: Renderng for close-up vews. The top row rendered wth our dffuson curve texture mantans sharp boundares. The mddle row rendered wth a rasterzed texture of resoluton 4096 4096 exhbts blur when zoomng n. The bottom row shows that the samplng rates wth our renderng method reman almost unchanged through the zoom sequence. Our renderng tmes are 76 ms, 76 ms, 70 ms, and 64 ms from left to rght. samplng ponts, as the most computatonally expensve part n renderng s the evaluaton of Green s functon ernels. Cullng of control curves sgnfcantly reduces ths cost wthout requrng much overhead. The number of samplng ponts s also consderably lessened by our adaptve samplng scheme, whch s based on the observaton that the fnal color of a pxel s nfluenced more by ntersectng or nearby control curves than by dstant ones. The adaptve scheme densely samples along neghborng control curves and sparsely samples along curves farther away. Ths leads to artfactfree textures wthout the prohbtvely heavy renderng cost that would result from dense samplng of all the control curves. For a gven dffuson curve texture, our method can compute the ant-alased texture value for each pxel n almost constant tme. As a result, the renderng performance of our method s proportonal to only the screen resoluton and ndependent of the vewpont and model complexty. The top row of Fgure 6 dsplays renderng results of our method for the vase under close-up vews that magnfy the porcelan texture mapped over the surface. Wth just 3 MB of texture data, our method mantans the resoluton ndependence of dffuson curve mages and preserves the sharp edges of the texture pattern n all of the close-up vews. By contrast, the results rendered from a 4096 4096 raster texture cannot mantan the sharp edges of the texture pattern n the most close-up vew shown n the second row of Fgure 6). The bottom row of Fgure 6 dsplays grayscale maps that encode the computatonal cost for each pxel, whch s determned by the number of control curve samples that contrbute to t. As the vewpont moves closer to the object, the computatonal cost of correspondng pxels s almost unchanged, snce the pxel areas generally receve contrbutons from the same control curve samples. Ant-alasng Fgure 7 compares the renderng results generated by our method and the ant-alased results rendered from a mpmapped raster texture under a seres of zoomed-out vews. In these zoom-outs, each pxel covers a relatvely large regon n the texture doman and thus contans many sharp, detaled features. For ths texture mnfcaton case, our soluton generates results smlar to the conventonal mpmap soluton. However, our method s based on a compact vector representaton that needs no extra storage and compu-
74:7 c) Fgure 8: Close-ups of the blue and whte porcelan texture showng detaled features. Texture reconstructed by the Green s functon ernels. c) Texture reconstructed by Jesche s GPU solver [009a] at a resoluton of 048 048 and 4096 4096 respectvely, then rendered wth the method of [Jesche et al. 009b]. Fgure 7: Renderng for zoomed-out vews. The top row s rendered wth our dffuson curve texture. The mddle row s rendered wth a rasterzed texture of resoluton 89 89, whch generates results smlar to our method. The bottom row shows the samplng rates wth our method. Our renderng tmes are 74 ms, 76 ms, 8 ms, and 88 ms from left to rght. taton for ths renderng tas, whle the raster-based soluton needs a 89 89 btmap to represent all the hgh frequency detals and an extra mpmap for ant-alased renderng. The bottom row of Fgure 7 shows a grayscale map of the computatonal cost. Smlar to the texture magnfcaton case, the computatonal cost of correspondng pxels s farly stable across the four renderngs. Fgure shows two objects mapped wth dffuson curve textures desgned by an artst. The color and texture varatons at dfferent scales e.g. the smooth color varatons and the rch texture varatons) are effectvely modeled and rendered by our method. Results Fgure 9 dsplays renderngs of a coffee cup usng our method. Note that the subtle shadng detals and sharp features n the dffuson curve texture are well preserved for ths and the remanng examples n ths secton. In a dffuson curve mage, boundary colors are taen to be pecewse lnear between the color control ponts, but the correspondng normal dervatves are not. Snce we lnearly nterpolate the normal dervatves between the same set of control ponts, there may be some color leaage along the curve. Ths problem can be allevated by ncorporatng more control ponts. Fndng a more effcent representaton of normal dervatves s a drecton for future wor. Fgure 9 llustrates the renderng results of a sculpture model. The dffuson curve texture of the wood pattern s obtaned by fttng dffuson curves to a rasterzed mage of the wood texture. Our renderng results accurately convey the natural color varatons and sharp boundares of the wood pattern under the dfferent vews. Fgure 9 d) exhbts the renderngs of a dragon model decorated wth complcated color patterns desgned by an artst usng a dffuson curve authorng tool [Orzan et al. 008]. In ths example, both the geometrc shapes and texture patterns are complcated, whch maes t a challenge for exstng dffuson curve renderng methods [Jesche et al. 009b; Jesche et al. 009a]. Our method well models the complcated color patterns and rch color varatons, and generates convncng ant-alased renderng results. Please see the supplemental vdeo for more renderng results wth these models. Dscusson Snce the Green s functon ernels are an explct representatons of DCIs, our method dffers substantally from prevous methods as t can drectly compute mage values at arbtrary postons and support random access wthout a predefned resoluton. Sharpness s mantaned at close nspecton, whle smooth ant-alasng s provded wth mnfed vewng. The storage s both compact and ndependent of the level of zoom. By contrast, prevous mplct solutons obtan results by solvng dffuson equatons over a grd, whch requres prohbtve computaton and storage for hgh magnfcatons. Jesche et al. [009b] solved ths problem by proposng an effcent dynamc feature embeddng technque that produces crsp, ant-alased curve detals. However, t stll needs a reconstructed texture wth suffcent resoluton whch maes sure each pxel only contans at most one control curve. If more than one control curve passes through a pxel, the undersampled detals between the control curves cannot be recovered n magnfcaton. A hgher grd resoluton would be needed to adequately sample these detaled features. Ths problem s llustrated n Fg. 8 wth the method of [Jesche et al. 009b]. Because of the numerous control curves n the texture, t taes 0.3s and 70 MBytes of vdeo memory to solve the DCI at a low resoluton 048 048) that does not fully capture the detaled features n magnfcaton. To generate a result comparable to ours, a hgher resoluton texture of at least 4096 4096 needs to be solved, at a cost of 0.5s and 55MBytes of vdeo memory. In Jesche et al. [009b], the texture s warped accordng to the vew so that t provdes suffcent resoluton n magnfed areas whle reducng the resoluton n mnfed areas. Whle ths wors well for sparsely dstrbuted control curves, alasng problems arse n mnfcaton for pxels that nclude multple curves. Moreover, the warpng scheme requres a good surface parameterzaton for the texture atlas, whch s a challengng tas for surfaces of complex geometry or hgh genus e.g. the dragon and sculpture n Fg. 9). Ours only needs the texture coordnates of each pxel for renderng, and s ndependent of the texture mappng method. 6 Concluson We presented dffuson curve textures based on Green s functons for ant-alased texture mappng of dffuson curve mages. Our technque yelds exact reconstructons of DCIs, and our GPU mplementaton provdes real-tme renderng performance. The most sgnfcant lmtaton of dffuson curve textures s the overhead for calculatng the weghts of the Green s functon ernels, whch depends on normal dervatves along the control curves. Whle dffuson curve mages may be authored usng the system n [Orzan et al. 008], ths overhead n convertng dffuson curves nto our representaton causes a delay before the dffuson curve texture can be used. Our technque nevertheless acheves nteracacm Transactons on Graphcs, Vol. 3, No. 4, Artcle 74, Publcaton Date: July 0
74:8 X. Sun et al. tve performance wth a DCI as nput, and we leave real-tme texture generaton from dynamc DCI nput for future wor. Faster processng may potentally be obtaned by further optmzng the scheme for control curves cullng, whch can greatly reduce the system scale, and by developng a GPU mplementaton. We also wll see to speed up other components of the renderng algorthm. For example, our current method for cullng control curves consders only closed boundares. Addtonally removng open curve segments that are dstant from the renderng pont could lead to further computatonal savngs. We also would le to extend our framewor to other dfferental operators that may be used n vector mage creaton, such as the blaplacan. Dffuson surfaces are another nterestng extenson, as Green s functon ernels can be derved smlarly for them. Acnowledgements We would le to than the revewers for ther valuable comments. The authors also want to than Tan Yuan and Shutan Yan for 3D modelng and texture desgn. 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74:9 c) d) Fgure 9: Some results rendered wth dffuson curve textures. A coffee cup that exhbts both subtle shadng detals and sharp features. A sculpture mapped wth wood texture obtaned by fttng dffuson curves to a rasterzed texture mage. c) A frog model wth a dffuson curve texture desgned usng a dffuson curve authorng tool. d) A dragon model wth abundant geometry detals. Appendx A In the followng, we provde a detaled dervaton of Eq. 4) and Eq. 6). Based on Eq. 9), the ndefnte ntegral of the Green s functon G R, x can evaluated analytcally wth the substtuton of, ) for x x : p ln + d d π + ln + + A0 ) d + arctg 4π 3 + ln + + arctg + arctg 4π HG, ) + A ) + A ). + A ) + A ) 7) 4π From Eq. 9) and Eq. ), we can express φ E x as φ E x E x t x ) FG R, x +E x d FG R, x t x 9) where the dfferental of FG s dependent on the dfferental of HG, whch we compute from Eq. 5) as + ln + HG, ) arctg ; HG, ) arctg + ln + y 30) where x x and y y. Smlarly, we obtan φ C x based on Eq. 6) and Eq. 7): {A } are arbtrary functons because of the ndefnte ntegrals. HG s gven n Eq. 5), and Eq. 4) s obtaned from t accordngly. C x d FGn R, x + C x FGn R, x t x 3) φ C x t x ) Smlarly, the ndefnte ntegral of the normal dervatve Gn x, x from Eq. 0) s solved as where the dfferental of FGn s dependent on the dfferental on HGn computed nx + ny from Eq. 7): Gn x, x d d π + nx nx HGn,, nx, ny ) ny nx + arctg ny arctg + ln + + A0 ) d π ny ny nx arctg ny + + ln + ; ny arctg + nx arctg + ny + nx ) ln + π ny HGn,, nx, ny ) nx ny + arctg HGn,, nx, ny ) + A ) + A ) 8) + A ) + A ) y y π nx nx ny where HGn s gven n Eq. 7) and Eq. 6) s determned from t. arctg nx + + ln +. 3) y y y Appendx B Here, we derve the analytcal form of φ C Eq. ). x,j and φ E x,j used to compute dnx ) Cx ) Ex ) nx ),, and can be obtaned from tx ) tx ) tx ) φ C x and φ E x can be calculated analytcally as above. As the control curves, ACM Transactons on Graphcs, Vol. 3, No. 4, Artcle 74, Publcaton Date: July 0