A High Accuracy Volume Renderer for Unstructured Data

Transcription

1 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 4, NO. 1, JANUARY-MARCH A High Accuracy Volume Renderer for Unsrucured Daa Peer L. Williams, Member, IEEE Compuer Sociey, Nelson L. Max, and Clifford M. Sein, Member, IEEE Absrac This paper describes a volume rendering sysem for unsrucured daa, especially finie elemen daa, ha creaes images wih very high accuracy. The sysem will currenly handle meshes whose cells are eiher linear or quadraic erahedra. Compromises or approximaions are no inroduced for he sake of efficiency. Whenever possible, exac mahemaical soluions for he radiance inegrals involved and for inerpolaion are used. The sysem will also handle meshes wih mixed cell ypes: erahedra, bricks, prisms, wedges, and pyramids, bu no wih high accuracy. Accurae semiransparen shaded isosurfaces may be embedded in he volume rendering. For very small cells, subpixel accumulaion by splaing is used o avoid sampling error. A revision o an exising accurae visibiliy ordering algorihm is described, which includes a correcion and a mehod for dramaically increasing is efficiency. Finally, hardware assised projecion and composiing are exended from erahedra o arbirary convex polyhedra. Index Terms Volume rendering, unsrucured meshes, high accuracy, finie elemen mehod, isosurfaces, splaing, cell projecion, visibiliy ordering, deph soring. 1 INTRODUCTION T ²²²²²²²²²²²²²²²² P.L. Willaims is wih he IBM T.J. Wason Research Cener, H0-C10, 30 Saw Mill River Road, Hawhorne, NY p.williams@compuer.org. N.L. Max and C.M. Sein are wih he Lawrence Livermore Naional Laboraory, Mail Sop L-307, 7000 Eas Avenue, Livermore, CA {max, sein}@llnl.gov. For informaion on obaining reprins of his aricle, please send o: vcg@compuer.org, and reference IEEECS Log Number YPICALLY, unsrucured meshes have a complex geomeric configuraion and he mahemaics of he absorpion-emission inegral are quie complex. Therefore, mos exising volume rendering sysems for unsrucured daa [5], [6], [7], [8], [10], [1], [13], [15], [16], [3], [9], [30], [31], [33], [34], [35], [37], [41] inroduce various simplifying assumpions and approximaions ino he algorihm in order o cope wih hese complexiies in an efficien manner. Anoher aspec of unsrucured meshes is ha, ypically, hey are adapively refined, so ha, in areas where he field is changing rapidly, he cells are smaller han in oher areas of he mesh. I is no uncommon for such cells o be several orders of magniude smaller han he larges cells. The behavior of he field on hese smalles cells is ofen of grea ineres o he simulaion scienis. However, all volume rendering sysems ha we are aware of are liable o miss hese smaller cells due o sampling error. This paper describes a high accuracy (HIAC) volume rendering sysem for unsrucured daa, especially finie elemen daa, ha, for a given mahemaical opical model [17], creaes images wih very high accuracy. Compromises, or approximaions, are no inroduced for he sake of efficiency. Whenever possible, exac mahemaical soluions for he differenial equaions involved and for inerpolaion are used. Subpixel accumulaion by splaing is used o avoid sampling error. Accurae semiransparen shaded isosurfaces may be embedded in he volume rendering. In addiion, a modified version of he accurae visibiliy ordering algorihm for unsrucured meshes, repored by Sein e al. [31], is used. Several imporan revisions o he original soring algorihm, including a correcion and a mehod for dramaically improving is efficiency, are described herein. Our goal was o design a volume rendering sysem o creae benchmark images for use as a sandard of comparison. The benchmarks can be used o compare resuls from oher volume rendering sysems for unsrucured daa ha use approximaions and simplifying assumpions, and can serve as a validaion suie for verifying he correcness of new algorihms and implemenaions. The HIAC volume rendering sysem is based on he absorpion plus emission opical model [17], [7], [38] and uilies he cell projecion mehod o accumulae he image. A ray inegraion is performed individually for every pixel ono which a cell projecs. The sysem will correcly render images in boh parallel and perspecive projecion, provided he ransfer funcions for color and opaciy are piecewise linear. I is inended primarily for daa ses from he finie elemen mehod, bu will render any unsrucured daa se whose cells are erahedra, bricks, prisms, or pyramids, or any combinaion hereof; see Fig. 1. The meshes may be nonconvex or even disconneced; he faces of adjacen cells may mee on only par of heir common adjacen face, i.e., sliding inerfaces are permied. However, he cells are expeced o be convex and noninersecing, and he visibiliy ordering graph should no conain cycles. The sysem will accuraely render daa ses where he scalar field varies linearly along he edges of he cells, called linear cells or linear elemens. For linear erahedra, he sysem uses he exac soluion o he radiance inegral described in [38]. This paper shows how he exac soluion can be implemened uiliing he Dawson inegral [4], raher han he able-based mehod described in [38] /98/$ IEEE

2 38 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 4, NO. 1, JANUARY-MARCH 1998 Fig. 1. Examples of differen ypes of cells used in he finie elemen mehod. The HIAC sysem will render daa defined on meshes wih any of hese cell ypes. In addiion, he HIAC sysem will accuraely volume render quadraic erahedra (erahedra where he scalar field varies quadraically along he edges of he cells, and, in fac, along any ray hrough he cell). In Appendix C, we describe how he sysem could be exended o accuraely render oher higher-order elemens, such as quadraic bricks, as well as linear bricks and prisms. This paper is an amalgamaion and exension of he resuls in [16], [31], [38]. The main new addiions are he use of he Dawson inegral in he compuaion of he exac soluion of he radiance inegral for linear erahedra, he mehodology for accurae radiance inegraion and isosurface generaion for quadraic erahedra, he use of splaing for subpixel accumulaion, he revision o Sein s visibiliy ordering algorihm, and he exension of hardware-assised projecion and composiing o arbirary convex polyhedra. The nex secion discusses relaed previous work. Secion 3 discusses he geomery of he cells used in he finie elemen mehod, he relaed inerpolaion equaions, and relevan erminology. Secion 4 gives a broad overview of he sysem and, hen, in Secions 5 and 6, we presen he deails of he rendering sysem and of he visibiliy ordering algorihm. Secion 7 discusses hardware assised polyhedron projecion. Secion 8 presens iming resuls and example images. PREVIOUS WORK A mehod for approximaing he volume rendering inegral wih bounded error is described by Novins and Arvo in [1]. By bounding he magniude of he derivaives of he inegrand, hey are able o obain remainder erms ha provide bounds on he approximaion error. They apply his o he rapeoid rule, Simpson s rule, and a power series mehod. The firs wo mehods are more suied o low o medium accuracy approximaions. The power series mehod, on he oher hand, is preferable for very high precision resuls. The echniques developed by Novins and Arvo are very valuable for bounding he error in he evaluaion of he inegral. However, here are oher sources of error in he volume rendering process, e.g., sampling error, which may miss small bu highly imporan cells in he accumulaion process, ha may be even more significan han inegraion error. The HIAC sysem addresses some of hese oher sources of error. For example, i uses subpixel splaing and a high accuracy visibiliy ordering algorihm. For high accuracy inegraion, he HIAC sysem uses a closed-form soluion o he inegral when possible; oherwise, high accuracy Gaussian numerical inegraion is used. This approach appears o be more efficien han he power series mehod, since he power series error bounds are loose. (Novins did no provide iming daa for comparaive purposes.) The error bounds are no easy o calculae for Gaussian quadraure, bu i is known o be very accurae, and i is he quadraure mehod generally used in he finie elemen mehod. When a guaraneed error bound is required for inegraion on higher-order elemens, he Novins and Arvo power series approximaion may be valuable. I is an open quesion wheher Gaussian quadraure or he power series mehod described by Novins is preferable for high accuracy inegraion. Silva and Michell [30] describe a very efficien and ineresing sweep plane volume rendering mehod ha accuraely raverses all ypes of erahedral meshes wih noninersecing cells, even hose wih cyclically overlapping cells. They claim i can be exended in a sraighforward way o more complex convex cells. The real value of he sweep plane algorihm is ha i provides a very efficien and accurae deph ordering of he cells of an irregular mesh along any given ray o he eye; i does no ry o give a global visibiliy ordering of he cells. The mahemaics of he volume rendering inegral is no addressed in heir paper, nor is sampling error. The inegraion mehods described in our paper could be uilied in he sweep plane algorihm. Gallagher and Nagegaal [4] describe mehods for rendering 3D conour surfaces of finie elemen daa, as well as mehods for smooh shading hese surfaces. They render he conour surfaces, which may be curved wihin a cell, as a polygonal approximaion o a parameric bicubic surface fi o each conour in a cell, whereas we render hese same surfaces on a pixel-by-pixel basis o reproduce he exac implici curved surface and use Phong shading calculaed a each pixel. Cline e al. [1] also reproduce his curved surface by recursive subdivision of he volume cells conaining he conour surface. 3 CELL GEOMETRY AND INTERPOLATION FUNCTIONS The cells used for 3D modeling in he finie elemen mehod (FEM) have many differen shapes, bu only a few are in widespread use [14]. We will focus our aenion on he more commonly used 3D cells (also called elemens): he erahedron, brick, and prism. See Fig. 1. In addiion o he verices (also referred o as nodes) used o define he endpoins of a cell s edges, which we will call he convenional verices or nodes, a cell may have addiional verices, which we will call inerior nodes, see, for example, he quadraic erahedron in Fig. 1. The inerior nodes, along wih he convenional nodes, may be used: 1) o define a nonlinear field inside he cell by he use of wha we will refer o as an inerpolaion funcion; and/or

3 WILLIAMS ET AL.: A HIGH ACCURACY VOLUME RENDERER FOR UNSTRUCTURED DATA 39 ) o define curvilinear faces by he use of a parameric mapping funcion. In his paper, we will no deal wih elemens whose geomery is defined by a parameric mapping, since hose elemens may have faces ha are highly curved, and he parameric mapping mus be invered before he scalar funcion can be evaluaed. We will limi our consideraion o he firs caegory of cells. For hose cells, he scalar field value is specified a all verices, convenional and inerior; bu he geomery of he cell is deermined from is convenional verices. The number of erms in a cell s inerpolaion funcion is equal o he number of nodes ha he cell has. So, a erahedron wih four nodes will have an inerpolaion funcion wih four erms. In mos applicaions, he inerpolaion funcion is a polynomial whose erms are elemens of he hree-dimensional power series. Those erms hrough hird degree are: 1 x, y, x, y,, xy, x, y x 3, y 3, 3, x y, x, xy, x, y, y, xy. (1) The inerpolaion funcion for a four-node erahedron is: f(x, y, ) = c 1 + c x +c 3 y + c 4. The scalar field varies linearly along any ray hrough a fournode erahedron, hence, i is called a linear erahedron. A brick wih eigh nodes has he eigh-erm inerpolaion funcion: f(x, y, ) = c 1 + c x + c 3 y + c 4 + c 5 xy + c 6 x + c 7 y + c 8 xy. () The paricular erms of he 3D power series ha are chosen for a given inerpolaion funcion are dicaed by he need of he FEM for cerain desirable properies, such as symmery, nonsingulariies, ec. Here, he scalar field varies linearly along he edges of he brick and, so, i is someimes called a linear brick. However, he field inside he brick varies rilinearly, so i is also called a rilinear brick. Ohers refer o i as an eigh-node brick, or a hexahedron. Ofen i is he case, in he FEM, ha nonriangular faces are slighly nonplanar. Alhough he four-node erahedron and he eigh-node brick are boh referred o as linear elemens, he higherorder erms in he inerpolaion equaion for he linear brick give i exra degrees of freedom ha allow i o solve some problems much more accuraely han could be done wih erahedra alone. From he perspecive of visualiaion, i should be noed ha a conour surface inside an eigh-node brick is curved and no planar, as i is inside a four-node erahedron. The erahedron, brick, and prism are he basic cells. They are ofen referred o as linear cells, since he field varies linearly along he edges of he cells. By adding inerior nodes o he basic cells, we ge cells wih higher-order inerpolaion funcions. We refer o his class of cells as higherorder cells. There are hree imporan higher-order cells. The firs is he 10-node erahedron, also referred o as a quadraic erahedron, whose inerpolaion funcion is: f(x, y, ) = c 1 + c x + c 3 y + c 4 + c 5 x + c 6 y + c 7 + c 8 xy + c 9 x + c 10 y. (3) This funcion is complee hrough he quadraic erms of he 3D power series in (1), herefore, he field varies quadraically along any ray hrough he volume. In he FEM, he six inerior nodes may be specified in differen configuraions; however, he mos common configuraion is for he inerior nodes o be locaed on he edges of he cell, usually a he midpoins. Elemens where all of he nodes lie on he boundary of he elemen are called serendipiy elemens. Serendipiy elemens are he mos common 3D elemens. The remaining higher-order cell ypes, he cubic erahedron and he quadraic brick, as well as he prism, are discussed in Appendix C. 4 OVERVIEW OF THE HIAC SYSTEM The HIAC volume rendering sysem for unsrucured meshes uses he cell projecion mehod and is based on he absorpion plus emission volume densiy opical model [17]. Eiher he Williams and Max [38] or he Wilhelms and Van Gelder [33] reamen of glow energy may be specified for use. The sysem reads in an image specificaion file [39], generaes he specified volume rendered image, and, hen, wries o disk eiher an image file in SGI RGBA forma or separae floaing poin R, G, B, and A files. Transfer funcions for color and opaciy are specified in a piecewise linear mehod, as in [39]. The radiance inegraion along a ray may be specified so as o use exac inegraion [38], which is appropriae when he cells are linear erahedra, or five-poin Gaussian inegraion, which is appropriae for quadraic erahedra. A faser, bu somewha less accurae, mehod, which we call he approximae mehod, assumes he opaciy varies linearly along he ray segmen and assumes he color is consan, equal o he average of he color a he fron and he back of he ray segmen. This is no exacly correc, since he opaciy along he ray segmen hides he far color more han he near one, bu is much quicker o evaluae. The daa ranges on which he ransfer funcions are acually linear are separaed by daa values which we call breakpoins. For he exac inegraion and he approximae mehod, a cell is sliced ino slabs a each ransfer funcion breakpoin ha occurs wihin a cell; in addiion, cells are sliced a each user-specified isosurface value. For quadraic erahedra, he cell is sliced concepually a all breakpoins and conour surfaces as a par of he inegraion procedure. This ensures ha he color and exincion coefficien are smooh polynomials wihin a slab. Wihin a slab from a linear erahedron, we can linearly inerpolae eiher he color and exincion coefficien, or he scalar field. I would no be correc o inerpolae he color or exincion coefficien if he cell conained a breakpoin in a ransfer funcion. Images may be generaed in eiher perspecive or orhographic projecion, wih any specified view ransform, and o any resoluion. Near and far clipping planes parallel o he screen may be specified, in wha we call -clipping, in order o selec a volume slab of ineres. Any number of

4 40 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 4, NO. 1, JANUARY-MARCH 1998 Fig.. Schemaic diagram of he HIAC volume rendering sysem. If cells have neiher ransfer funcion breakpoins nor conour surfaces in heir inerior, hen he cells go direcly from he sorer o he rendering engine. Oherwise, he cells are sliced ino slabs, wih nonerahral cells firs being pariioned ino erahedra. Quadraic erahedra go direcly from he sorer o he rendering engine. as described in Secion 5.. illuminaed Phong-shaded semiransparen colored isosurfaces may be specified for inclusion in he volume rendering. For linear erahedra, he conour surfaces are polygonal, and he surface normal for each polygon is used o shade he surfaces. We do no smooh-shade he conour surfaces, because his migh obscure imporan informaion in he visualiaion ha is useful for deermining wheher he mesh has been properly refined. For higher- order cells, he isosurfaces are creaed and shaded on a pixel-by-pixel basis, and no as a se of polygons; herefore, hese conours are smoohly curved surfaces wihin each cell. Subpixel splaing may be specified, so ha conribuions from small cells ha fall beween pixel ceners are included in he image. Any background color can be specified, as well as any one of a selecion of background paerns. Fas previewing of images is faciliaed by hooks o Williams splaing sysem [37] based on Shirley and Tuchman s rendering algorihm [9] and Williams MPVO visibiliy ordering algorihm [36], and an exension of he echniques of [9] o arbirary convex polyhedra. The HIAC sysem will sor and render all he ypes of linear cells described in Secion 3, as well as quadraic erahedra and oo meshes (meshes which include a combinaion of he various cell ypes). Daa srucures for dealing wih mixed cell ypes and higher-order elemens are described in Appendix A. The HIAC sysem will also sor and render curvilinear daa, provided he faces of he cells are only slighly nonplanar (disored), as is usually he case in curvilinear grids. In he finie elemen mehod, cells can be disored by a parameric mapping funcion, in which case he faces of he cells can be highly curved. How o volume render hese nonconvex cells wih curved faces is an imporan and ineresing open quesion. We discuss i briefly in Secion 9. For linear cells, if he color or densiy is linear hroughou he cell, i.e., no breakpoins nor isosurfaces occur wihin he cell, hen he cell is rendered as a whole. For nonerahedral linear cells, including hexahedral cells from a curvilinear grid, if a breakpoin or conour value occurs wihin a cell, ha cell is firs erahedralied and, hen, processed as described above, slicing he resuling erahedra as necessary. The sysem will also deal wih quadraic erahedra, which i slices concepually during he inegraion process whenever a conour lies wihin he cell. Afer he image specificaion file is parsed, he daa se is read in, he view ransform and he perspecive ransform (if applicable) are applied, and he daa se is clipped o he view volume. Nex, he cells are sored in visibiliy order from back o fron, sliced (if necessary) ino slabs bounded by conour levels and ransfer funcion breakpoins, and, hen, he slabs (or cells) are scan convered and he resuls of he ray inegraion hrough he slab (or cell) for each pixel are composied ino he image buffers. The image buffers hold he red, green, blue, alpha, and values in floaing-poin forma. The buffer is used as a winess o verify he correcness of he visibiliy ordering. The alpha buffer is used o permi posprocessing accumulaion of more han one semiransparen image. A schemaic diagram of he sysem is shown in Fig.. The rendering engine does he scan conversion, radiance inegraion, subpixel splaing, -clipping, and composiing. 5 THE RENDERING ENGINE Max [17] describes several heoreical opical models for ligh ineracing wih a volume densiy, each wih differing degrees of realism. A volume rendering sysem can be creaed based on any one of hese models. If he sysem is consruced faihfully according o is model, wihou he use of approximaions, hen ha sysem will creae accurae images. (If he differenial equaion for radiance can only be solved by numerical mehods, hen he sysem will creae images o some predeermined degree of precision.) A volume rendering sysem can eiher inegrae he radiance over rays cas ou from each pixel hrough he enire volume densiy, or projec each cell in he volume densiy ono he screen in visibiliy order and inegrae he radiance over each projeced cell for each pixel covered by i. When he cell projecion approach is used, a visibiliy ordering of he cells is required in order o composie he semiransparen volume cells ino he image in back-o-fron order. The HIAC volume rendering sysem, which uses he cell projecion approach, is an evolved version of he sysem repored by Max e al. in [16]. Tha sysem used he isoropic densiy emier opical model of Sabella [7] for he volume effecs, and allowed Phong shading of seleced conour surfaces a mos, one conour surface could pass hrough a erahedron. We have now improved he slicing algorihm o allow any number of conour surfaces.

5 WILLIAMS ET AL.: A HIGH ACCURACY VOLUME RENDERER FOR UNSTRUCTURED DATA 41 In he HIAC sysem, he absorpion plus emission opical model [17], [38] is used. In his model, every poin in he cloud absorbs ligh and also emis ligh (glows). The differenial equaion for he radiance along a ray owards he eye hrough he volume is: di af = g af - afaf I, (4) d where is a lengh parameer along he ray, and I() is he radiance a. The opical densiy or exincion coefficien of he volume a, (), is considered o be a physical propery of each poin in he cloud, and defines he rae ha ligh is absorbed or occluded a ha poin. The remaining erm g() is he glow energy emied a each poin of he cloud. There are wo ways o rea he glow energy. Wilhelms and Van Gelder [33] rea he glow energy as a physical propery of he cloud, whereas Williams and Max [38] consider he glow energy o be defined as g() = k()(), where he chromaiciy k() is considered o be a physical propery of each poin in he cloud. The HIAC sysem will generae images using eiher reamen of he glow energy, as chosen by he user. We assume he use of piecewise linear ransfer funcions for specifying he dependence of chromaiciy (or glow energy) and opical densiy on he scalar field being visualied. By he use of an inegraing facor and by applying boundary condiions a 1 and, we ge he following inegral equaion for he radiance using he Williams and Max reamen of glow energy, see [17], [38]: af - d - udu ch ch 1 afaf 1 1 I = I e + e k d. (5). This equaion is insaniaed once for each of he hree componen wavelenghs of ligh. The second erm represens he glow energy along he ray segmen, aenuaed by he opaciy in fron of i, and he firs erm represens he incoming illuminaion I( 1 ) a he far end of he ray, also aenuaed by he inervening opaciy. For Wilhelms and Van Gelder s [33] neon and smog reamen of glow energy, we ge he following inegral equaion, whose erms can be undersood in he same way: af 1 I = I e + e gd. (6) af af - d - udu ch ch 1 af 1 The inegral which is he second erm on he righ side of (5) and (6) canno be solved in closed form for general (). However, if he scalar field and he ransfer funcions vary piecewise linearly along a ray segmen wihin a cell, hen he equaions can be inegraed exacly over each piecewise linear region. This soluion is described by Williams and Max in [38] and discussed furher in he nex secion, ogeher wih is implemenaion. Le he second erm of he righ-hand side (of eiher equaion) be described as: a af e caf d. (7). 1 A general closed-form soluion for his inegral is no known when a() is cubic or higher order, regardless of he form of c(). Therefore, even hough he c() erm is lower order in he neon and smog reamen, he inegral sill canno be solved exacly when he scalar field is quadraic or higher order (even wih linear ransfer funcions). The neon and smog reamen does, however, permi he glow energy o be mapped independenly o a differen scalar field han he opical densiy, which is no possible wih he oher reamen. Neverheless, we have creaed successful visualiaions where k and each depended on separae scalar fields using he Williams and Max reamen of glow energy. The main advanage of he Williams and Max reamen is ha i makes he specificaion of he ransfer funcions somewha more inuiive. For example, increasing he exincion coefficien makes he surface color more dominan, raher han making he image darker and ulimaely black, as is he case wih he neon and smog model. More deails on his and on he relaive meris of he wo differen reamens of glow energy are given in an Appendix o [39]. The HIAC sysem allows he opical densiy o be mapped o a differen scalar field han he color, and he conour surfaces can be keyed o a hird scalar field. The HIAC sysem uses he cell projecion approach, herefore, a visibiliy ordering of he cells is required. The soring algorihm originally used in [16] was resriced o recilinear volumes or Delaunay riangulaions in 3D. The HIAC sysem uses a differen soring algorihm, a modified version of he one repored by Sein e al. in [31]. The revised algorihm, which works on an arbirary collecion of acyclic noninersecing convex polyhedra, is described in Secion 6. The nex secion describes how he HIAC rendering engine processes linear cells. Secion 5. describes he reamen of quadraic erahedra. Finally, Secion 5.3 describes he subpixel splaing procedure. 5.1 Linear Cells Linear cells, as discussed in Secion 3, are convex polyhedra where he scalar field is specified a he convenional verices and varies linearly along he edges of he cells. Afer visibiliy ordering he cells, each cell is checked o deermine if he range of he scalar field wihin i includes any ransfer funcion breakpoin values. If any are found, and he cell is erahedral, he cell is sliced a each breakpoin, resuling in slabs in which he color and opaciy are linear. Each slice is defined by a conour surface for he field value corresponding o a ransfer funcion breakpoin. Since he scalar field varies linearly wihin a erahedron, he slices are planar and parallel. An example slab is shown in Fig. 3. If an isosurface is o be separaely rendered, he erahedron mus also be sliced a hese conour values so he slabs and surface polygons can be composied individually in he correc order. Currenly, if he cell is nonerahedral, we firs subdivide he cell ino erahedra and hen slice he erahedra ino slabs. The visibiliy ordering of he erahedra wihin a hexahedron, or he slabs wihin a erahedron, is simple, and is done separaely from he global visibiliy ordering of all he cells. (Mehods like he marching cubes algorihm of Lorensen and Cline [11] can slice hexahedral cells direcly, bu he slices, which are curved surfaces, mus be divided ino riangles. If here are muliple conour levels inside a single

6 4 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 4, NO. 1, JANUARY-MARCH 1998 Fig. 3. Example of slab of linear erahedron. Wihin he slab, he color and opaciy are linear. The slicing planes defining he slab are conour surfaces for field values corresponding o ransfer funcion breakpoins or for user-specified isosurface values. cell, i is no clear ha hey will be noninersecing, or ha he volume pieces hey slice off will be convex, as required by our projecion algorihm.) We do no subdivide nonerahedral cells ino erahedra when he range of scalar funcion wihin he cell does no include a slicing value. Boh he sorer and he scan converer will handle he differen ypes of linear cells described in Secion 1. The rendering engine will correcly process convex cells or slabs wih any number of faces and verices. The slicing algorihm is robus, permiing up o hree verices of a erahedron o ake on he same slicing value. When all four verices have he same conour value, he se of poins aking on he conour value is no longer a surface, bu conains he whole erahedron. In ha case, he conribuion of such a erahedron o he isosurface is negleced, leaving a hole in he conour surface, bu he erahedron will sill conribue o he volume rendering. When a conour surface inersecs hree verices of a erahedron, he surface corresponds o one of he cell s faces, which may be shared wih anoher cell. In ha case, i is imporan o render he conour polygon only once. The boundary polygons for he slabs are found from a case-by-case analysis of he ways a slice plane can inersec a erahedron, and he ways in which he slab beween wo consecuive slice planes can inersec he erahedron s face riangles. For orhogonal projecion, he back-o-fron soring order of he slabs wihin a cell can be deermined from he - componen of he gradien of he scalar field S on he cell: S(x, y, ) = c 1 + c x + c 3 y + c 4. (Since he field is known a he four verices of he cell, he four consans can be deermined for he cell.) If he -componen of he gradien is greaer han ero, hen he back-o-fron order sars wih he slab having he larges scalar value. For perspecive projecion, consider he slicing planes defining he slabs o be infinie parallel planes in world coordinaes. If he viewpoin lies beween wo of hese infinie planes which define a slab, we call ha slab he eye slab. The slabs and conour surfaces are hen composied in wo groups: from one side up o, bu no including, he eye slab and, hen, from he oher side, up o and including he eye slab. If here is no eye slab, only one group is required, as in he parallel projecion case. Nex, he cells or slabs are sen in visibiliy order o he rendering engine for projecion and accumulaion. The firs sep in his process is o scan conver he cell or slab. The fron-facing polygons bounding he cell are scan convered ino a fron -buffer, wih values f, and he back-facing polygons ino a back -buffer, wih values b. The k and values are bilinearly inerpolaed along edges and across scan lines, as in Gouraud shading, and saved in he fron or back buffers, as k f and f, or k b and b, respecively. Then, for each pixel in he projecion of he cell, he lengh l of he ray segmen is compued as b - f, and he values of k and are assumed o vary linearly beween heir values in he fron and back buffers. In parallel projecion, his resuls in piecewise rilinear inerpolaion, where he subdivision ino rilinear pieces depends on he projecion of he polyhedron. Thus, as in piecewise bilinear Gouraud shading, he inerpolaion scheme is no roaionally invarian. However, for linear erahedra, or for slabs cu from hem on which k and are linear, his rilinear inerpolaion reduces o linear inerpolaion, which is roaionally invarian. Nex, he ray inegraion is performed for each pixel covered by he cell or slab. When he cells are linear erahedra and he ransfer funcions are piecewise linear, he inegral in (5) can be inegraed exacly as shown in [38] by compleing he square of he exponen and repeaed applicaion of inegraion by pars, yielding: a + b m + n1 I e 1 1 ch c - hc g + d + dh = - + d d cg + d h df F g + d I F 1 e erfi erfi HG KJ - g + d II hd d HG d KJ 1 1 ch - d d g + -g - HG 1 I e F HG I KJ KJ +, (8) where, a, b, d, m, n, g, and h are funcions of he four consans c i in he erahedral inerpolaion funcion S(x, y, ) = c 1 + c x + c 3 y + c 4, he consans describing he applicable linear pieces of he ransfer funcions, for example, (x, y, ) = a + bs(x, y, ), and he hree ray parameeriaion funcions, x = u 1 + u, ec., as described in [38]. The erfi() funcion will be discussed below. Key erms in (8) are: d, since i appears in he denominaor of several erms, and g + d, he numeraor in he argumen o erfi(). The erm d is equal o he slope of he perinen piece of he opical densiy ransfer funcion, i.e., b in he example above, imes he slope of he (linear) scalar field wihin he cell. The erm g + d is exacly (), as can be seen from he deailed derivaion given in [38]. The complex error funcion, erf(), is defined as: -u erf( ) = e du. p 0 The imaginary error funcion, erfi(), which appears in (8), is defined as erfi() = erf(i)/i. In (8), erfi s argumen is eiher real when d > 0, or pure imaginary when d < 0. When is argumen is real,

7 WILLIAMS ET AL.: A HIGH ACCURACY VOLUME RENDERER FOR UNSTRUCTURED DATA 43 x u erfi( x) = e du. p 0 When is argumen is pure imaginary, of he form = ib, wih real b, b -u erfi( ib) = i e du. p 0 When d < 0, he i in he laer expression cancels he i in he facor d -.5 in (8). When d = 0, he soluion o he inegral akes a differen and relaively simple form, as shown in [38], involving only he exponenial funcion, and no erfi(). (The formula for his case in [38] can be furher simplified by noing ha, when d = q = 0, q 5 is also ero, eliminaing half he erms.) Originally, in [38], we implemened he erfi() funcions in x - x erms of he indefinie inegrals e d and e d 0. We 0 precompued hese inegrals incremenally a equally spaced values of x using Simpson s rule, and sored he resuls in wo ables. To evaluae (8), we inerpolaed values from hese precompued ables. By inroducing Dawson s inegral, which is defined in x x [4] as D( x) = e - e d, he soluion o he inegral equaion can be simplified o eliminae he use of ables. Dawson s 0 inegral is relaed o he complex error funcion by: i Dx e x x af = - p erfaixf = e erfiaf - p - x. (9) An efficien and accurae numerical approximaion for Dawson s inegral, due o Rybicki [6] and described in [4], enables he calculaion of erfi() wihou he use of ables. The accuracy of Rybicki s approximaion increases exponenially as he sep sie, h, used in he approximaion, ges small. We use h = 0.4, which gives an accuracy of abou The funcion erfi(x) for imaginary x can be reduced, by a rivial change of variables, o he Error inegral, he inegral of a Gaussian normal disribuion, for which subrouines also exis, as described in [4]. Appendix B discusses deails of implemenaion and how o avoid overflow in he exponenials. For linear erahedra, he HIAC sysem uses he exac soluion from [38], uiliing subrouines for Dawson s inegral and he Error inegral as described above. When Wilhelm and Van Gelder s neon and smog reamen of he glow energy is used, he echniques given above sill apply, bu he erm k()() in he inegrand is replaced by g(), which is now linear, raher han quadraic, for linear scalar daa. Unforunaely, his does no permi any significan furher simplificaion in he calculus. When a perspecive view is specified, a bi of care is required o do mahemaically correc inerpolaion and inegraion, since he disance meric along an edge or ray is disored by he perspecive ransform. Afer performing he perspecive ransform, he scalar field no longer varies linearly along he edges of a cell nor on a ray hrough a cell. (For example, he midpoin of an edge in parallel projecion is no longer he midpoin of ha edge afer he perspecive ransform.) Therefore, inegraion echniques ha are suiable for linear funcions no longer perain. Our approach o his problem is o reverse he perspecive ransform and do he inerpolaion and inegraion in world coordinaes raher han screen coordinaes. When his is done, he lengh of he ray segmen mus be compued as a 3D (slaned) disance, raher han jus a difference in values. (For a perspecive ray from he origin hrough a pixel a ( x, y, 1 ), he difference of he world coordinae -values of he endpoins of he ray segmen mus be muliplied by x + y + 1.) The deails of his work are edious and he ineresed reader is referred o our code, which is in he public domain as indicaed in Secion 9. Near and far clipping planes parallel o he screen may be specified o achieve a volume slab of ineres. This - clipping is accomplished as follows. Cells enirely in fron of he near clipping plane are skipped, as are cells enirely behind he far clipping plane. Cells inersecing he slab are processed normally, bu, for each pixel, he viewingray/cell inersecion segmen is resriced o he region inside he slab. This could be done efficienly by 3D polyhedron clipping, bu he above per-pixel scissoring alernaive was easier o code. -clipping is also implemened for quadraic cells. 5. Quadraic Cells As discussed in Secion 3, quadraic cells are cells where he scalar field varies quadraically along he edges of he cell. In his secion, we deal wih quadraic erahedra, which have six inerior nodes, one per edge, as in Fig. 1. Oher higher-order cells, as well as linear bricks and prisms, are discussed in Appendix C. In a quadraic erahedron, he scalar field varies quadraically along any ray segmen hrough he cell because (3) conains only quadraic erms. Inside hese cells, conour surfaces will be curved, herefore, he slabs will be curved, and a viewing ray may inersec a single slab wice. Because of his, we do no acually pariion he cells ino slabs as we did for linear erahedra, bu, raher, process each ray hrough a cell in segmens. In each ray segmen, he color and opical densiy vary smoohly. The inerpolaion funcion for a quadraic erahedron has he form of (3), which we repea below: f(x, y, ) = c 1 + c x + c 3 y + c 4 + c 5 x + c 6 xy + c 7 y + c 8 y + c 9 + c 10 x. Subsiuing he coordinaes of he 10 nodes, along wih heir field values, ino his equaion, we ge 10 equaions for he 10 unknown polynomial coefficiens c i, which we solve wih he LINPACK linear algebra package. The segmen endpoins are found as follows: The ray equaions parameried by, are af c 0 h af c 0 h af c 0 h x = x -, y = y -, = -. (We assume he viewpoin is a he origin, he pixel is locaed a ( x, y, ), where is he disance from he viewpoin o he screen, and he ray is in he direcion of ligh flow, wih increasing oward he eye.) Subsiuing he ray equaions ino he quadraic inerpolaing funcion shown in (3), gives a quadraic polynomial in one variable: f() = a + b + c. When a is nonero, his polynomial will ake on a

8 44 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 4, NO. 1, JANUARY-MARCH 1998 Fig. 4. The variaion of he scalar field f() = a + b + c along a ray, oward he eye, hrough a quadraic erahedron as a funcion of he ray parameer. Example regions defined by conour surfaces corresponding o ransfer funcion breakpoins are also shown. The horional regions beween s i and s i+1, for 1 i 5 correspond o curved slabs in he erahedron. The ray eners he erahedron a f. maximum or minimum a a single value m = -b/(a). If he ray segmen does no conain m, he quadraic polynomial is monoonic in he segmen. Oherwise, i conains boh increasing and decreasing regions, as shown in Fig. 4. These regions, and heir monoonic direcion, can be deermined from he sign of a, which is he sign of he second derivaive of f, and he locaion of m relaive o n and f, he near and far endpoins of he ray segmen. For every relevan slicing value, s i, he corresponding breakpoins in can be found by using he quadraic formula for he roos of f() = s i. Beween every pair of consecuive breakpoins s i and s i+1, he color and opical densiy will each be represened by a smoohly varying polynomial in. The poins in he quadraic erahedron, which have scalar values in he inerval [s i, s i+1 ], define a curved slab bounded by he conour surfaces a he breakpoins and pars of he erahedron s surface faces. We will refer o his as he slab [s i, s i+1 ]. The ray/slab inersecion segmens and heir order along he ray can be deermined by comparing f( n ), f( f ), and f( m ) wih he slicing values s i. In he case illusraed in Fig. 4, s 1 < f( n ) < s < f( f ) < s 3 < s 4 < s 5 < f( m ) < s 6. Therefore, he ray eners he slab [s, s 3 ] hrough a erahedron face a he lef where f is increasing, passes hrough he slab [s 3, s 4 ], coninues hrough slab [s 4, s 5 ], hen eners he slab [s 5, s 6 ], coninues hrough [s 5, s 4 ] (wih f decreasing), and, hen, slabs [s 4, s 3 ], [s 3, s ], and [s 1, s ], and, finally, exis hrough a face of he erahedron. Though m is shown in Fig. 4, i is no one of he breakpoins; f reaches is maximum on he ray segmen inside he slab [s 5, s 6 ], bu coninues smoohly pas is maximum, as do g() or k(), and (), so here is no need o subdivide he inegraion here. Oher cases can be handled similarly: The code has wo loops over he increasing and he decreasing ranges of f(), bu one may no be needed. Assuming he Williams and Max reamen of glow energy, le k() and () be he chromaiciy and opical densiy, respecively, a posiion along he ray. Then, he oal opaciy from a ray segmen [p, q] is: q - af ds (10) p e and he oal radiance or color added by ha segmen is: q q - udu p af afaf. (11) e k d For k() and () quadraic in, (10) can be inegraed exacly, bu numerical inegraion is required for (11) because he a() in (7) is a cubic polynomial. We have used five-poin Gaussian inegraion [4], which gives exac answers for polynomials of up o degree nine, and very good approximaions for sufficienly smooh funcions ha are well approximaed by such polynomials, bu poor approximaions for funcions which are no smooh. This is he reason for breaking he range of inegraion up ino he subsegmens where k() and () are smooh polynomials. The color (11) and opaciy (10) on hese subsegmens are composied in he back-o-fron order described above. If a semiransparen conour surface is requesed a he near breakpoin of a subsegmen, i is composied afer he subsegmen. The surface normal is compued from he parial derivaives of (3), and used for Phong shading, as well as o make he surface appear more opaque when i is seen edge-on, as if i were a finie hickness of parially absorbing glass. This makes he conour surfaces appear appropriaely curved wihin a cell, even in he absence of refleced ligh. However, finie elemen simulaions rarely produce resuls which are C 1 across cell boundaries, so he conour surfaces may no be globally smooh. 5.3 Subpixel Splaing In curvilinear or irregular meshes designed o concenrae small cells near shocks, boundaries, or oher regions of rapid change or special ineres, projecions of iny bu imporan cells may fall beween he pixel ceners. This can also happen due o perspecive foreshorening. Any volume rendering algorihm which samples he image only a pixel ceners may, herefore, miss significan deails enirely, or include hem wih an inappropriae weighing. This is he case in boh ray racing and cell projecion mehods. The heoreically correc soluion o his problem is o deermine an analyic represenaion for he image as a funcion of he coninuous coordinaes on he image plane and, hen, convolve i wih a presampling filer kernel, before sampling i a he pixel ceners. Because of he geomeric and analyic complexiy of a volume rendered image, his is a formidable ask. We use an approximaion o his analyic anialiasing, suggesed by Wesover s splaing echnique [3]. If a cell s projecion overlaps oo few pixels (for example, less han wo), we assume ha is color and opaciy effecs on he image are concenraed a is cener of graviy. We, herefore, ake a dela funcion a he projecion of he cell s cener of graviy, and muliply i by he volume of he cell, he perspecive projecion area shrinkage facor, and he color and opaciy a he cell s cener of graviy. We hen convolve

9 WILLIAMS ET AL.: A HIGH ACCURACY VOLUME RENDERER FOR UNSTRUCTURED DATA 45 his weighed dela funcion wih a presampling filer kernel (described below), which is equivalen o aking a weighed ranslaed copy of he kernel. The resul is a spla o be composied ono he image. If subpixel splaing is urned on, when he rendering engine ges a cell, we firs do a rudimenary scan conversion o deermine he number of pixels covered. If he pixel coun is larger han a hreshold, we repea he scan conversion, doing he analyic inegraion for color and opaciy, and composie he resul ino he image. Oherwise, we composie a weighed ranslaed copy of he filer kernel. We use a piecewise biquadraic kernel, he produc of wo idenical 1D piecewise quadraic kernels in x and y, he B-spline kernel. This kernel is he wice ieraed convoluion of a pixel-sied box filer wih iself. In spaial frequency, his filer has he Fourier ransform sin 3 (px)/(px) 3, which grealy aenuaes frequencies greaer han he Nyquis limi and, so, gives good anialiasing. However, i does cause some minor blurring, since frequencies less han he Nyquis limi are also aenuaed, and he fooprin of each cell is a 3 3 square of pixels. A wider filer, such as he one we use, is superior o a pixel-sied box filer kernel when brigh objecs much smaller han a pixel move during animaion. Wih a box filer kernel (area averaging), he brigh objec would suddenly jump from one pixel o an adjacen one when i crossed he edge beween hem, bu, wih a wider kernel, he conribuions smoohly fade up and down. Raher han precompue and sore a high resoluion version of his spla, as Wesover did, we jus evaluae he simple quadraic polynomials each ime hey are needed. The polynomial variables are he fracional subpixel coordinaes of he projecion of he cell s cener of graviy. The original algorihm of Wesover used splas whose fooprin decreased as he projeced splas go closer ogeher, bu his mehod could also cause splas o be los beween pixels! Our soluion is o keep he spla sie o a hree-pixel square, and decrease he color/opaciy ampliude insead, as described above. Anoher approach (for regular grids only) is given by Mueller and Yagel in [18]. They use summed area ables o compue he inegral of he spla fooprin over he pixel area, so all splas will conribue heir effecs compleely o he image. We esed subpixel splaing by dividing a cube ino a large number of iny erahedra, each smaller han a pixel, and composiing heir splas. The resul was he same as he analyic inegraion over he projecion of he five larger erahedra represening he original cube, excep for sligh blurring. This splaing scheme is no a perfec soluion o he anialiasing problem. Suppose a iny cell is very brigh, bu i is oally occluded by anoher iny cell direcly in fron of i which happens o be dark and very opaque. Firs, he iny brigh cell conribues a proporion o a nearby pixel, hen he oally opaque cell conribues a proporion o he opaciy, bu, overall, he pixel will incorrecly reain some brighness. Varians of his problem wih per-pixel composiing occur wih any scheme ha does no represen he complee geomeric projecion of all cells overlapping he filer kernel. We are currenly working on an analyic anialiasing scheme which does ake ino accoun he complee geomery, bu we expec i o be very slow. 6 THE VISIBILITY ORDERING ALGORITHM The HIAC volume rendering sysem uses he cell projecion mehod which requires a visibiliy ordering of he cells. We use he accurae soring algorihm presened by Sein e al. [31], which is an O(n ) (wors case) mehod for visibiliy ordering n arbirary shaped, noninersecing convex polyhedra wih planar faces, whose visibiliy ordering does no conain cycles. The faces of adjacen cells need no be aligned, and he meshes may have disconneced porions. The algorihm is effecively a 3D generaliaion of he Newell e al. sor for polygons [3], [19], [0]. A -buffer is incorporaed in he rendering engine o serve as a winess o he correcness of he visibiliy ordering. A correcion o he original algorihm repored by Sein e al. is given in Secion 6.1; hen, in Secion 6., we describe a mehod ha, for large daa ses, increases he efficiency of he algorihm by up o wo orders of magniude. The original Sein visibiliy ordering algorihm, which oupus he cells in back-o-fron order, can be quickly described in he following seps: Firs, ransform all of he verices o screen coordinaes wih a perspecive correced. Nex, creae a roughly sored lis of he polyhedra by arranging he elemens in back-ofron order based on each polyhedron s rearmos coordinae. The algorihm QuickSor works well here. Las, fineune he sor by performing visibiliy ess for each relevan pair of polyhedra in he lis. This fine uning is described in more deail in he following paragraph. The goal of he fine-uning sage is o verify ha no polyhedron P obscures any oher polyhedron following i in he lis. If P does no obscure any polyhedron following i in he lis, hen P can be safely oupu. However, if P does obscure some elemen laer in he lis, hen a porion of he lis mus be rearranged. We deermine ha P does no obscure an elemen Q by finding wheher P lies behind a plane ha separaes he wo elemens. Because his can be difficul and ime-consuming, he algorihm has a predeermined lis of planes which i ries. This lis sars wih he planes ha are easies o calculae, such as he planes perpendicular o he X, Y, and Z axes, and ends wih he more compuaionally difficul possibiliies, such as he planes defined by he fron- and back-facing faces of P and Q. If a separaing plane canno be found from his lis, hen he explici screen-projecions of he wo polyhedra are examined by a subrouine ProjecsBehind(P, Q) o deermine wheher P obscures any par of Q. The fine-uning is described as follows: Given a roughly sored lis of elemens in back-o-fron order, P, he elemen a he head of he lis, can be oupu (i.e., i obsrucs no remaining cells) if, for all elemens Q whose -exen overlaps P s -exen, he following subrouine Obsrucs(P, Q) reurns false: Obsrucs(P, Q) if (P and Q have no overlapping X exens) reurn False else if (P and Q have no overlapping Y exens) reurn False

10 46 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 4, NO. 1, JANUARY-MARCH 1998 Fig. 6. Example of configuraion (b). Fig. 5. Two cases when ProjecsBehind(P, Q) canno find inersecions in he screen projecion of he edges of cells P and Q. else if (P is behind a back-face of Q) reurn False else if (Q is in fron of a fron-face of P) reurn False else if ProjecsBehind(P, Q) reurn False else reurn True If Obsrucs(P, Q) is false for all Q, hen P can be safely rendered. In his case, P is oupu, removed from he lis, and he nex elemen a he sar of he lis becomes he new P; he process repeas unil he lis is empy. If Obsrucs(P, Q) is rue for a pair of polyhedra P and Q, hen P obsrucs a leas a porion of Q. In his case, Q is moved o he head of he lis, hereby becoming he new P, and he process repeas for his new P. When a Q is moved o he head of he lis, i is agged as having been moved. If such a Q requiring moving has already been agged, hen he visibiliy ordering of he daa se conains a cycle, and his is repored. We have no ye implemened he polyhedron subdivision necessary for breaking cycles. ProjecsBehind(P, Q) examines he screen projecions of he wo polyhedra o deermine wheher P obscures any porion of Q. This subrouine searches for edge inersecions beween he screen projecions of all edges of cells P and Q. If an edge inersecion is found, he componens of he inersecion poin on P s and Q s acual edges are compared, enabling ProjecsBehind(P, Q) o reurn he appropriae value. If P s componen of he inersecion poin lies behind Q s componen of he inersecion poin, hen P lies behind Q. Oherwise, P obscures a leas a porion of Q. There are wo cases, however, when ProjecsBehind(P, Q) canno find edge inersecions in he screen projecion, as illusraed in Fig. 5. The original algorihm published by Sein e al. [31] did no deal wih his siuaion correcly. Boh he process of handling hese wo configuraions and a correcion o he original algorihm are given in he nex subsecion. In he even ha a face of a cell is nonplanar, we approximae a plane equaion o he verices defining he face, using Newell s mehod [5]. Then, for each verex (x, y, ) of he face, we calculae is deviaion, Ax + By + C + D, where A, B, C, D are he plane equaion coefficiens wih an ouward-poining normal, and reain he maximum deviaion along wih A, B, C, D for each face. To deermine if a face of cell P defines a separaing plane wih regard o cell Q, we subsiue he coordinaes of each verex of cell Q ino he plane equaion for he relevan face of cell P. If he plane equaion evaluaes o a posiive value for all verices, hen ha face is a separaing plane. To deal wih slighly nonplanar faces when evaluaing he plane equaion, he resul plus he deviaion mus be greaer han ero. 6.1 Correcion o he Soring Algorihm In configuraion (a) of Fig. 5, because he polyhedra are enirely disjoin, eiher one can be oupu wihou affecing he visibiliy of he oher. In configuraion (b) of Fig. 5, ha is no rue; elemen Q lies behind elemen P. The reason why Q lies behind P is as follows: Since ProjecsBehind(P, Q) has found no edge inersecions in he screen projecions of P and Q, and we are assuming configuraion (b) holds, hen eiher a fron face or a back face of P defines a separaing plane beween P and Q. Bu, he fourh es of Obsrucs(P, Q) has ruled ou he fron face of P as a separaing plane; herefore, Q mus be behind P. This issue was no addressed by Sein e al. in [31] because hey incorrecly saed ha configuraion (b) could no occur. We now describe how such a configuraion could occur. Suppose we have wo polyhedra a he beginning of our roughly sored lis (a back-o-fron soring based on he rearmos componen of each elemen), as illusraed in Fig. 6. Assuming ha he screen projecions of he edges of P and Q do no inersec and ha heir x and y exens are no disjoin, hen we have an insance of eiher configuraion (a) or (b) in Fig. 5 where he firs four ess of Obsrucs(P, Q) fail. The problem, hen, is o disinguish which of he wo configuraions exiss. To do his, we es wheher P and Q are enirely disjoin from each oher by deciding wheher he projecion of P is conained in he convex hull [8] of he projecion of Q, or wheher he projecion of Q is conained in he convex hull of he projecion of P. If boh of hose ess fail, indicaing configuraion (a), hen P and Q are enirely disjoin from each oher and ProjecsBehind(P, Q) reurns rue indicaing ha P can be oupu. Oherwise, if eiher of he convex-hull ess is rue, hen ProjecsBehind(P, Q) reurns false, indicaing we have configuraion (b) and ha he order mus be changed. 6. The Muliiled Sor The soring algorihm described above is O(n ) wors case, for n cells, because he firs -range overlap es may need o be performed for every pair of cells. Since, in general, a very large percenage of cells do no overlap each oher in x and y, we reasoned ha, by iling he view-plane window ino a recilinear grid of p iles, and, hen, for each ile, soring

11 WILLIAMS ET AL.: A HIGH ACCURACY VOLUME RENDERER FOR UNSTRUCTURED DATA 47 TABLE 1 RESULTS OF PRELIMINARY EXPERIMENT TO TEST EFFECT OF TILING ON THE SORTING ALGORITHM Time in minues o sor n cells using iling and a single CPU Tiles 13, , ,000 1,000, Shown are imes in minues o visibiliy order n cells by calling he original soring algorihm imes, each wih one of iles, using a single R10000 CPU of an SGI Power Onyx. only he cells ha projec ono i, we could gain a speed-up of up o p imes (since each ile would have approximaely n/p cells). Of course, many cells may overlap ile boundaries and, ypically, he algorihm does no require worscase quadraic ime. Bu, he sor was aking so long for large daa ses ha i was worh experimening o see wha improvemen could be achieved. We pariioned he screen ino p roughly load balanced iles and, hen, sored he cells in each ile sequenially, using a single CPU. The end resul was p sored liss of cells, one for each ile, which could hen be rendered on a ile-by-ile basis. The resuls of his experimen were very graifying; he imes repored in Table 1 are for an SGI Power Onyx sysem using a single R10000 processor. Using he mehod described above, if a cell projecs ono more han one ile, ha cell will appear in more han one sored lis, hus requiring muliple clipping and/or scan conversion passes. Therefore, we modified he soring algorihm o uilie iling inernally. The inernal iling process we describe nex produces a single sored lis of all he cells, for he enire view plane window. In he muliiled sor, in order o decide wheher polyhedron P can be safely oupu, we have o verify ha P lies behind all polyhedra Q ha overlap he same iles as P. The muliiled visibiliy soring algorihm can be described as follows: 1) Sor he elemens in back-o-fron order as before using he elemens rearmos coordinae as he soring crierion. We call his (roughly) sored lis he Global Soring Lis (GSL). ) Assign each elemen a unique ideniy (an elemen s iniial posiion in he GSL will suffice), and give each elemen a las_comparison variable o keep rack of he las polyhedron o which i was esed and found o lie in fron. In oher words, if elemen T s las_comparison holds elemen U s ideniy, hen T has been deermined o lie in fron of elemen U. 3) Divide he view-plane window ino disjoin recangular iles. By defaul, we assume a uniform disribuion of cells in he volume, and pariion he window ino n 1/6 by n 1/6 equal sied iles, where n is he oal number of cells. While n 1/3 by n 1/3 equal sied iles would ideally resul in he fewes number of cells per ile, n 1/3, large values of n would lead o a large number of iles and, in urn, a large amoun of sorage. Thus, he somewha arbirary 1 exponen was chosen 6 o keep he number of windows down o roughly n 1/3. The sor can opionally be called wih an array of alernaive ile dimensions o allow he use of load balanced iles. The HIAC sysem uses a load-balancing scheme designed for unsrucured meshes which will be described in a subsequen publicaion. 4) For each ile, creae a Tile Soring Lis (TSL) which is a linked lis of poiners o he polyhedra overlapping he given ile. In he TSL, he polyhedra are sored in back-o-fron order based on heir rearmos coordinae. For each polyhedron, creae a lis of poiners o iles ha he polyhedron overlaps. The following seps are essenially a merge of he separae TSLs. 5) Begin he merge by selecing he head polyhedron in he GSL and calling i P. For each ile ha P belongs o, deermine wheher P lies behind all of he polyhedra Q ha occupy P s iles and whose rearmos coordinae lies behind P s fronmos coordinae, by using he subrouine Obsrucs(P, Q). However, before acually esing each Q, examine Q s las_comparison variable o see wheher Q has already been esed wih elemen P in a differen ile (each elemen can only occur once in any TSL). If he variable conains P s ideniy, hen P has already been found o lie behind Q and he ess can be skipped. Proceed o he nex elemen following Q in he TSL. Oherwise, deermine wheher P lies behind Q by calling Obsrucs(P, Q). Record P s ideniy in Q if P can be safely oupu before Q and, hen, proceed o he nex elemen following Q in he TSL. 6) When an elemen P fails a es agains a paricular Q, ag Q and move i o he head of he GSL (Q does no move in any of he TSLs and, herefore, does no affec he early erminaion condiion menioned in he following sep. If Q has already been agged as moved, hen he daa se conains a cycle.) This Q now becomes he new P, and he whole process is repeaed. 7) Once we find an elemen Q whose rearmos coordinae lies in fron of P s fronmos coordinae, we may erminae any furher ess in ha paricular TSL because he remaining elemens in he TSL lie fully in fron of P. Proceed o he nex TSL ha P occupies and repea he ess for all he applicable Q. 8) When an elemen P passes he ess for all of he applicable Q in each of he TSLs i occupies, P is oupu and removed from he GSL as well as from all of he TSLs o which i belongs. The nex elemen a he head of he GSL becomes he new P and he whole process repeas. Comparaive imings for he above muliiled sor versus he original soring algorihm repored by Sein e al. [31] (wih he correced algorihm) are given in Table. Load balanced ile dimensions were provided o he sorer for

12 48 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 4, NO. 1, JANUARY-MARCH 1998 TABLE COMPARATIVE TIMINGS IN MINUTES TO VISIBILITY ORDER n CELLS USING THE ORIGINAL SORT AS REPORTED BY STEIN ET AL. IN [31] VERSUS THE REVISED ALGORITHM USING TILES DESCRIBED HEREIN, USING A SINGLE R10000 CPU OF AN SGI POWER ONYX No. Cells Original Sor Muliiled Sor 13, min. 0.1 min. 190, min..7 min. 600, min. 9.5 min. 1,000, min min. hese ess. For 1,000,000 cells, he iling of he sor resuled in a 60 fold speed-up in soring ime. 7 HARDWARE BASED POLYHEDRON PROJECTION The volume rendering sysem described o his poin is no ineracive, because i uses a precisely correc soring, and a slow, accurae, analyic, or numerical inegraion along each ray segmen on which he ransfer funcions are linear. For rapid preview, polygon-based rendering hardware can be used insead. Shirley and Tuchman [9] divided he projecion of a erahedron ino from one o four riangles, and used hardware scan conversion, ransparency, and back-ofron composiing o produce an image. Sein e al. [31] poin ou ha he linear ransparency inerpolaion beween he riangle verices replaces wha should be an exponenial compuaion per pixel in (5), and can produce Mach bands. They sugges a more accurae mehod using hardware exure mapping, which we have now generalied from erahedra o arbirary convex polyhedra. The bilinear inerpolaion of,, and k across faces, described in Secion 5.1, could be performed in a sandard rendering pipeline, bu sandard hardware does no permi he inerpolaed values o be sored in separae fron and back buffers and combined laer. Therefore, we need smaller homogeneous regions where a single value for each of hese parameers will suffice and can be inerpolaed linearly or bilinearly. Consider he se S of polygonal regions ino which he projecion of a convex polyhedral cell P is divided by he projecions of all is edges. Since no projeced edges of P cross he inerior of any polygonal elemen R of S, R lies wihin he projecion of a single fron facing face of P, and of a single back facing face of P. Since each face is planar, he hickness l = b - f of R varies linearly over R, and can be bilinearly inerpolaed in hardware from is values a he verices of R. Since f and b are linear (or bilinear) over R, heir average avg = ( f + b )/ is also, and i oo can be inerpolaed in hardware. Consider a ray segmen s of lengh l = - 1 = b - f, on which varies linearly beween b a 1 and f a. Then, d he ransparency T = e - () 1 on s reduces o T e l avg = -. The sandard composiing hardware produces a linear combinaion: I new = (1 - a)i old + a I add (1) (- g + d ) d where I old is he curren color in he frame buffer, I add is he inerpolaed color for he curren polygon, and I new is he new color o be placed in he frame buffer. The opaciy a = 1 - T can be inerpolaed from is verex values, as in [9] or [33], or aken from a exure map, as in [31]. The D exure map M(u, v) is preloaded wih he funcion 1 - e -uv, and he exure parameers are se a he verices of R wih u = l = b - f, he lengh of he ray segmen s, and v = avg. When u and v are bilinearly inerpolaed by he hardware, and M(u, v) is used for a, he hardware finds, per pixel, he exponenial needed for correc ransparency. The access o he SGI graphics hardware pipeline is hrough OpenGL, so he code should be fairly porable. The chromaiciy k is bilinearly inerpolaed by he shading hardware, and used as I add in he composiing equaion (1). We have hree mehods for calculaing k a he verices of R. They are, in increasing order of accuracy and compuaion ime: (M1) he average chromaiciy (k f + k b )/, (M) he able-based evaluaion of (8) described in [38], and (M3) he subrouine-based evaluaion of (8) described in Secion 5.1 and Appendix B. In mehods (M) and (M3), we divide he radiance from (8) by he opaciy a, o ge an effecive chromaiciy a he verex. When his quaniy is inerpolaed, and used as I add in he composiing equaion (1), he resul gives some of he effecs one would expec, such as a closer color k f parially obscuring a farher color k b along he same ray, alhough i is no as accurae as evaluaing (8) a each pixel. We also offer a mehod (M0) which jus uses he hardware bilinear inerpolaion of a from is values a he verices of R, insead of he exure mapping. I is even faser and less accurae han mehod (M1). Mehod (M0) is a direc generaliaion of he mehod of [9] and may be used if exure mapping hardware is no available. If he chromaiciy k is consan on a ray segmen, he inegral in (7) reduces o (1 - T)k = ak, as explained in [17], so mehod (M1) is appropriae for cells of consan chromaiciy. Mehod (M) may be sufficienly accurae for 8-bi-per-color images, if he precompued ables for he inegrals of e and e - are large enough in range and resoluion. However, since each inegral is looked up individually, he separae exponenial facor e may cause exponen overflow or underflow. If his happens (we es for overflow in advance) or if he range of he precompued ables is exceeded, we rever o mehod (M1). As explained in Appendix B, mehod (M3) handles all possible inpus correcly and gracefully. Now, consider he problem of subdividing he projecion of P ino homogeneous regions R. Shirley and Tuchman [9] used a caalogue of four possible projecion opologies for a erahedron, and Wilhelms and Van Gelder [33] used a line sweep algorihm for he case of a hexahedron. For a general convex polyhedral cell, we have used an incremenal approach o build up a winged-edge daa srucure [] for he subdivision. We add he projeced edges one a a ime, saring wih an empy subdivision wih a single unbounded face. The new projeced edge is exended from is

13 WILLIAMS ET AL.: A HIGH ACCURACY VOLUME RENDERER FOR UNSTRUCTURED DATA 49 TABLE 3 TYPICAL TIMINGS FOR THE VOLUME RENDERING ENGINE FOR THE DIFFERENT INTEGRATION METHODS FOR BOTH 100,000 AND FOR 1,000,000 PIXEL IMAGES TYPE NUMBER RENDERING ENGINE OF OF TIME INTEGRATION PIXELS 13,000 cells 600,000 cells approximae mehod 100, min..0 min. 1,000, min. 7.6 min. exac linear 100, min..9 min. 1,000, min min. quadraic 100, min. 3.5 min. 1,000,000.0 min..8 min. The oal ime for he HIAC sysem is he sum of he rendering engine ime shown here, plus he muliiled soring ime given in Table (0.1 minues for 13,000 cells, and 9.5 minues for 600,000 cells). All imes are from an SGI Power Onyx using one R10000 CPU. saring verex, slicing one-by-one hrough he exising polygonal regions, and he winged-edge daa srucure is adjused accordingly. When all edges have been added, he bounded polygons in he subdivision are he desired homogeneous regions. Our principal curren use of his hardware composiing of general polyhedra is for he slabs of Fig. 3, ino which a linear erahedron is divided by breakpoins in he piecewise linear ransfer funcions. In his case, he HIAC sysem sill uses he slow bu precise back-o-fron sor of Secion 6. However, i also comes wih a version of he much quicker approximae sor of Williams [36], [37], which is more useful for ineracive applicaions. 8 RESULTS Timings for he HIAC rendering engine are given in Table 3. Times are shown for he exac linear inegraion using he Dawson and Error inegral, and for quadraic inegraion using Gaussian quadraure. For comparison, he imes for he approximae mehod, as described in Secion 4, are also shown. Times are given for images wih 100,000 pixels and wih 1,000,000 pixels. There is no significan difference in rendering ime when several semiransparen illuminaed isosurfaces are embedded in he image. Toal volume rendering ime is he sum of he ime shown here plus he appropriae muliiled soring ime shown in Table. Figs. 7, 8, 9, 10, 11, 1, 13, and 14 show volume rendered images of coolan velociy magniude from a finie elemen simulaion of coolan flow inside a componen of he French Super Phoenix nuclear reacor. The daa is defined on a mesh of 13,000 quadraic erahedra. Fig. 7 is an image creaed using he inegraion mehod for quadraic erahedra described in Secion 5.. This image is o be compared wih he nex four images, which were creaed using he same inpu specificaions as used for Fig. 7, bu differen volume rendering mehods. Fig. 8 was generaed using he exac inegraion mehod for linear erahedra described in Secion 5.1, by neglecing he daa a he inerior nodes. Fig. 9 was creaed using he approximae mehod described in Secion 4, which assumes k is a consan on each ray segmen, wih he value (k f + k b )/. Fig. 10 was generaed using he approximae mehod, bu wihou slicing he cells ino slabs. Fig. 11 was creaed using he hardware based polyhedron projecion mehod, (M3), for sliced linear erahedra, described in Secion 7. Differences beween hese images are clearly visible in he original images which are available for downloading a heir full sie and resoluion in SGI RGB forma a: hp:// Figs. 1 and 13 show volume rendered images wih embedded semiransparen illuminaed isosurfaces; boh were generaed using he inegraion mehod for quadraic erahedra. Fig. 14 shows he same view as Fig. 1, bu was creaed using he inegraion mehod for linear erahedra. Figs. 15 and 16 show volume rendered images of he densiy field from a finie elemen mehod simulaion of air flow pas an F117a je aircraf flying a a 0 degree angle of aack. There is a vorex generaed ha breaks a he wing railing edge. This daa se is composed of 50,000 linear erahedra in a highly adapively refined mesh. Fig. 15 was creaed using he exac inegraion mehod for linear erahedra. Subpixel splaing was urned on for he generaion of his image; here were 9,00 projeced cells covering less han wo pixels, which were splaed. Fig. 16 was creaed using he approximae inegraion mehod, wih splaing urned off. 9 FUTURE WORK AND CONCLUSION The HIAC volume rendering sysem described in his paper creaes highly accurae images of unsrucured daa ses whose cells are eiher linear or quadraic erahedra and whose faces are planar or nearly planar. The sysem was specifically designed o deal wih daa ses from he finie elemen mehod bu i is no limied o his ype of daa. Currenly, he HIAC visibiliy ordering algorihm and he rendering engine will handle erahedra, bricks, prisms, pyramids, and wedges, or any combinaion hereof (oo meshes); bu will only use high accuracy inegraion for linear and quadraic erahedra. We plan o implemen he procedure o perform accurae inegraion for linear and quadraic bricks and prisms, and cubic erahedra, which, along wih he linear and quadraic erahedron, are he mos widely used finie elemens. The accurae inegraion procedure for hese oher elemens is very similar o ha for quadraic erahedra, and is discussed in Appendix C.

14 50 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 4, NO. 1, JANUARY-MARCH 1998 Fig. 7. Image creaed by inegraion mehod for quadraic erahedra. Fig. 10. Image creaed using he approximae mehod, wihou slicing cells. Fig. 8. Image creaed using exac inegraion mehod for linear erahedra. Fig. 11. Image creaed using he hardware based projecion mehod (M3), wih cell slicing. Fig. 9. Image creaed using he approximae mehod. Fig. 1. Image creaed by inegraion mehod for quadraic erahedra, wih embedded isosurfaces.

15 WILLIAMS ET AL.: A HIGH ACCURACY VOLUME RENDERER FOR UNSTRUCTURED DATA 51 Fig. 15. Image creaed using exac inegraion mehod for linear erahedra, wih subpixel splaing. Fig. 16. Image creaed using approximae mehod, wihou subpixel splaing. parameers and he creaion of he image specificaion file.) To improve he efficiency of he HIAC sysem, we have parallelied i for an MPP, he IBM SP-. This parallel work, which includes load balancing echniques for unsrucured meshes and mehods for dealing wih disribued finie elemen daa ses has been implemened and will be described in a subsequen publicaion. I may be possible o increase he efficiency of he sysem somewha by he use of fron-o-back composiing wih early erminaion. This can be done on a pixel by pixel basis wihin cells, and will save unnecessary calls o he numerical rouines for he complex error funcion for hose pixels. The exac visibiliy ordering algorihm described by Fig. 13. Differen view of Fig. 1. Sein e al. [31], and used in he HIAC sysem, is O(n ) wors case. Therefore, i is no suiable for ineracive use wih he previewer. A faser exac deph ordering algorihm 4/3+ Fig. 14. Same view as Fig. 1, bu using exac inegraion mehod for linear erahedra. The sysem is no inended o be highly ineracive, bu, raher, o operae in bach mode o creae high qualiy/accuracy images for publicaion or in-deph sudy, or for animaions. (I would be nice if a graphical user inerface were developed o faciliae he selecion of he inpu is described by de Berg e al. [] which runs in ime O(n ) for any fixed > 0. However, his algorihm, which is based on a general framework for compuing and verifying linear orders exending implicily defined binary relaions, is quie heoreical and is no readily implemened. A presen, Williams [36] MPVO visibiliy ordering algorihm, which is a heurisic for nonconvex meshes, is used for he hardware assised previewer described herein. However, he MPVO algorihm has a large sorage requiremen for is preprocessed daa srucures; herefore, i would be useful o invesigae replacing his algorihm wih a differen soring heurisic, such as one ha sors he cells by heir ceners of graviy. For daa defined on 3D Delaunay riangulaions, Karasick e al. [9] describe an efficien exac soring algorihm based on soring he erahedral cells by heir powers. An especially ineresing and challenging projec for he fuure is how o accuraely volume render finie elemen daa whose cells have been deformed by a parameric mapping funcion resuling in cells wih highly curved faces. In his case, a ray hrough he volume may ener and exi he same cell more han once. Thus, a global visibiliy ordering may be impossible. In addiion, he parameric mapping mus be invered before he scalar funcion can be evaluaed.

16 5 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 4, NO. 1, JANUARY-MARCH 1998 As menioned a he end of Secion 5.3, our curren splaing mehod is no always correc, and we are also working on analyic anialiasing using an exac geomeric subdivision of he image plane by he projeced edges of all cells. The source code for he HIAC volume rendering sysem including he visibiliy ordering algorihm by Sein e al. [31], a previewer using Williams ineracive splaing sysem [37] (based on he projeced erahedron algorihm of Shirley and Tuchman) and his MPVO visibiliy ordering algorihms [36], and he hardware-assised projecion and composiing sysem uiliing exure mapping hardware described in [31] and exended o deal wih arbirary convex polyhedra as described herein, is available a: hp:// This research code is wrien in FORTRAN, C, and C++, and uilies OpenGL for he hardware rendering. APPENDIX A This appendix describes he HIAC daa srucures for use wih daa ses having mixed cell ypes (oo meshes). We allow he use of five differen cells ypes in he inpu daa se: erahedra, pyramids, prisms, wedges (cells wih seven nodes, also called anvils), and bricks, each of which may be linear, quadraic, or cubic. For he higher order elemens, we assume he exra nodes are locaed on he edges, wih he excepion of he cubic erahedron, which also has a node in he cener of each face. (More han five cell ypes could be used as long as each cell ype has a unique oal number of nodes per cell.) Each scalar field in he daa se is sored in a floaing poin array wih one elemen per node, in he same ordering as is used for he xy array, described below. In addiion o he daa arrays, hree oher basic arrays are used: elems, nodes, and xy. The elemens of he elems and he nodes arrays are ineger values and he elemens of he xy array are hree-uples of floaing poin values. Each cell has one enry in he elems array, is oal number of nodes nn. However, raher han encoding nn direcly, he elems array sores he cumulaive oal of nn. So cell i will have nn = elems[i] - elems[i - 1] nodes. The nodes for cell i are in he nn enries in he nodes array, saring wih nodes[elems[i]]. The nodes sored in he nodes array are poiners o he coordinaes of he nodes which are sored in he xy array. The convenional nodes are specified firs in he nodes array in a sandard order, followed by he inerior nodes, if any. We also keep a flag indicaing wheher he daa se is linear, quadraic, or cubic. This flag disambiguaes differen ypes of cells wih he same number of nodes, e.g., a quadraic brick has he same number of nodes as a cubic erahedron. We assume elemens of differen orders (linear, quadraic, cubic) will no be combined in one mesh. APPENDIX B This appendix gives implemenaion deails for he evaluaion of (8) of Secion 5.1. In he noaion of ha secion, le f 1 g = + d d 1 and g + d f =. d Since he numeraors represen he exincion coefficiens ( 1 ) and ( ) a he wo endpoins of he ray segmen, f 1 and f are nonnegaive real numbers. (If numerical inaccuracy resuls in a slighly negaive value, i is replaced by ero.) Noe ha he erfi erms in (8) are muliplied by a facor of e - f if d > 0. When erfi(f ) is replaced by he Dawson inegral from (9), he above facor e - f cancels he facor e f in f erfi( f ) = p e D( f ). Thus, he produc e ( erfi( f )- erfi( f )) f1 - f - f 1 can be evaluaed as ( Df ( p ) - e Df ( 1 )). If d > 0, f > f 1, so f1 - f is negaive, and if i becomes so negaive ha i leaves he valid domain of he exponenial (causes underflow), hen i is safe o replace e f 1 - f by ero. If d < 0, hen he corresponding produc is 1 f F HG F HG i e erfi fi I KJ - 1 erfi F H G f i II KJ ch ch f d 1 i. e erf f erf f KJ = - (The 1 in fron comes from he d -.5 facor in (8).) We use an i approximaion o erf(x) for x > 0, given in [4] under he guise of he complemenary error funcion 1 - erf(x), of he form erf x ue x (- + ( ) p ( u )) 1 -, where u = +, and p(u) is a x ninh degree Chebyshev polynomial seleced o give an accurae fi o he ail of he error funcion. Thus, we ge f p( u ) f - f p( u ) e erf f erf f u e u e 1 1 ( ( ) - ( - 1 e, which avoids loss of accuracy when f 1 and f are large, since he 1s cancel. In his case, f 1 > f, and we can again se e f - f 1 o ero if f - f is oo negaive. 1 APPENDIX C Secion 3 discussed inerpolaion funcions for he linear erahedron and he quadraic erahedron. This appendix discusses inerpolaion funcions for he oher common cells used in he finie elemen mehod he prism, he cubic erahedron, and he quadraic brick and oulines how he volume rendering mehods described in his paper may be exended o include hese oher cells. The inerpolaion funcion for he six-node penahedron or prism is: f(x, y, ) = c 1 + c x + c 3 y + c 4 + c 5 x + c 6 y. (13) The prism is mainly used as a ransiion elemen o glue ogeher erahedra and bricks in meshes ha use a combinaion of differen elemen ypes. FEM simulaion code ha uses such a mesh of mixed elemen ypes is ofen called a oo code. The inerpolaion funcion for he cubic erahedron, which has 0 nodes, is a cubic polynomial, and is complee hrough he cubic erms of he 3D power series. Therefore,

17 WILLIAMS ET AL.: A HIGH ACCURACY VOLUME RENDERER FOR UNSTRUCTURED DATA 53 is inerpolaion funcion has all he erms shown in (1). This cell has wo inerior nodes per edge, usually a 1 3 and 3 of he edge, as well as a node a he cener of graviy of each face of he cell. Here, he field varies cubically along any ray hrough he cell. The quadraic brick, wih 0 nodes, has he following inerpolaion funcion: f(x, y, ) = c 1 + c x + c 3 y + c 4 + c 5 x + c 6 y + c 7 + c 8 xy + c 9 x + c 10 y + c 11 x y + c 1 x + c 13 xy + c 14 x + c 15 y + c 16 y + c 17 xy + c 18 x y + c 19 xy + c 0 xy.(14) The brick s inerior nodes are locaed a he ceners of is edges. A presen, he erahedron and brick, in heir linear and quadraic forms, dominae pracical applicaions [14]. Therefore, in his paper, we limi our coverage of accurae volume rendering mehods o hese cells and he prism. Similar echniques o hose given in his paper can be applied o oher ypes of cells, provided he inerpolaion funcion is fourh degree or lower. We have implemened he high accuracy volume rendering mehods for he fournode erahedron and 10-node erahedron, and we describe below how one migh exend he sysem o deal wih he linear brick and prism, and he quadraic brick. A presen, he HIAC sysem will also handle bricks, prisms, pyramids, and wedges, alhough no wih he highes precision. When we implemen he accurae rendering scheme for linear bricks and prisms described here, hese elemens will be no be subdivided ino erahedra prior o rendering. The same basic procedure described in Secion 5. for quadraic erahedra can be used for high accuracy rendering of he quadraic brick as well as he linear brick and prism. The inerpolaion equaion for he quadraic brick is given in (14); i is a fourh order polynomial in x, y, and, so is evaluaion along a linear ray gives a fourh order polynomial f() in he ray parameer. The poins on a ray where he polynomial akes on a conour value s i can be found analyically by he closed form nonieraive soluion of he quaric equaion firs published by Ferarro, see [40]. Here, a case analysis, similar o, bu more complex han, he one described above for quadraic erahedra, is required o find he regions where he polynomials () and g() are smoohly varying, i.e., include no breakpoins. A diagram of he case analysis for finding he ray segmens is given in Fig. 17. The poins 1,, and 3, where f () = 0 separae he monoone ranges of f(), can be found as roos of he cubic polynomial f (). Eiher Gaussian quadraure or he power series mehod of Novins and Arvo [1] can be used o do he inegraion. The inerpolaion funcion for he linear brick is given in (). I is rilinear, herefore, conours wihin he cells are curved and, so, he mehods described above perain. The funcion f() is cubic because of he xy erm in (). To find he roos of he cubic polynomial, f() - s i = 0, we can use he closed form soluion given in [4]. The linear prism has an inerpolaion funcion given by (13). This has he bilinear erms x and y, and, so, conour surfaces in he inerior of he linear prism will be curved. Subsiuing he ray parameeriaion ino he inerpolaion Fig. 17. Analysis of ray segmen hrough a quadraic brick elemen. equaion, we ge a quadraic polynomial which can be analyed on a case by case basis similarly o ha used for he quadraic erahedron. The inerpolaion equaion for he cubic erahedron is: f(x, y, ) = c 1 + c x + c 3 y + c 4 + c 5 x + c 6 y + c 7 + c 8 xy + c 9 x + c 10 y + c 11 x 3 + c 1 y 3 + c c 14 x y + c 15 x + c 16 xy + c 17 x + c 18 y + c 19 y + c 0 xy. (15) Since his equaion is of degree hree, i can be deal wih in a similar way o ha described above for he oher higher order elemens. ACKNOWLEDGMENTS We are graeful o Roger Crawfis of Ohio Sae Universiy for his conribuion o he scanvol code which was modified for use in he sysem described herein. We also appreciae echnical assisance from Barry Becker of Silicon Graphics, Inc., Kwan-Liu Ma a ICASE, and Mark Duchaineau a LLNL. Peer L. Williams is grealy indebed o Sam Uselon and Tom Lasinski a NAS, NASA Ames Research Cener, for heir generous summer suppor for hree years and for equipmen loans wihou which his work would no be possible. He is also graeful for he use of he Large-Scale Ineracive Visualiaion Environmen (LIVE) a NAS, NASA Ames, which was used o generae he images in his paper. Peer L. Williams and Nelson L. Max received summer suppor for wo years, arranged by Becky Springmeyer, from he Acceleraed Sraegic Compuing Iniiaive (ASCI). Rober Haimes of MIT graciously provided he F117a daa se and Bruno Nirosso a Elecricié de France provided he Super Phoenix daa se. This work was parially performed under he auspices of he U.S. Deparmen of Energy by Lawrence Livermore Naional Laboraory under conrac number W-7405-ENG-48. Peer L. Williams performed he majoriy of his conribuion o his research while a summer visior a Lawrence Livermore Naional Laboraory, and a NAS, NASA Ames Research Cener.

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Visualiaion 9, pp , Boson, Oc [38] P.L. Williams and N.L. Max, A Volume Densiy Opical Model, Proc. 199 Workshop Volume Visualiaion, pp , Boson, Oc [39] P.L. Williams and S.A. Uselon, Merics and Generaion Specificaions for Comparing Volume Rendered Images, NASA-Ames TR NAS-96-01, Dec. 1996, hp://science.nas.nasa.gov/pubs/techrepors/ NASrepors/NAS-96-01/NAS hml. [40] P.H. Winson and B.K.P. Horn, Lisp, second ediion. Reading, Mass.: Addison Wesley, [41] R. Yagel, D. Reed, A. Law, P-W. Shih, and N. Shareef, Hardware Assised Volume Rendering of Unsrucured Grids by Incremenal Slicing, Proc Symp. Volume Visualiaion, pp. 55-6, Nov [4] Handbook of Chemisry and Physics. Cleveland, Ohio: Chemical Rubber Publishing. Peer L. Williams received he PhD in compuer science from he Universiy of Illinois a Urbana- Champaign, and he BS in engineering-physics from he Universiy of California a Berkeley. He has augh compuer science a Vassar College, he Universiy of Connecicu a Sorrs, and Harvey Mudd College. He is currenly a visiing scienis a he IBM T.J. Wason Research Cener, where he is a member of he IBM Daa Explorer research group. A IBM, Dr. Williams is developing a disribued visualiaion sysem for use on he IBM SP- for he Naional Laboraories erascale compuing effor. His research ineress include graphics, scienific visualiaion, volume rendering (especially unsrucured daa), and high performance parallel and disribued compuing. Nelson L. Max has research ineress in he areas of scienific visualiaion, volume and flow rendering, compuer animaion, molecular graphics, and realisic compuer rendering, including shadow and radiosiy effecs. Since 1977, he has been a compuer scienis a Lawrence Livermore Naional Laboraory, and has been eaching par ime a he Universiy of California, Davis, currenly as a 50 percen professor of applied Science. Dr. Max has augh mahemaics and compuer science a he Universiy of California a Berkeley, he Universiy of Georgia, Carnegie Mellon Universiy, and Case Wesern Reserve Universiy. He was direcor of he U.S. Naional Science Foundaion suppored Topology Films Projec in he early 1970s, which produced compuer animaed educaional films on mahemaics. He has worked in Japan for hree and a half years as codirecor of wo Omnimax (hemisphere screen) sereo films for inernaional exposiions, showing he molecular basis of life. For he pas eigh years, he has concenraed on volume, vecor field, and flow visualiaion for 3D simulaions. Clifford M. Sein graduaed from Harvey Mudd College and received his PhD in compuer science from he Universiy of California a Davis in He was employed a Lawrence Livermore Naional Laboraory from 199 o His curren research ineress include animaion, rendering, and physically-based modeling.