Implementing Ray Casting in Tetrahedral Meshes with Programmable Graphics Hardware (Technical Report)

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1 Implemening Ray Casing in Terahedral Meshes wih Programmable Graphics Hardware (Technical Repor) Marin Kraus, Thomas Erl March 28, Inroducion Alhough cell-projecion, e.g., [3, 2], and resampling, e.g., [4, 5] are he mos popular approaches o hardware-assised volume rendering of erahedral meshes, hey are no he only ones; for example, Wesermann and Erl [6] suggesed a hardware-assised echnique for ray casing in erahedral meshes. In his echnical repor anoher hardware-assised implemenaion of he ray casing algorihm is skeched. However, i employs programmable pixel shading o perform basically all compuaions in he graphics hardware and hereby avoids any significan daa ransfer beween he graphics subsysem and he main memory. Alhough his approach was neiher implemened ye, nor is i likely o perform as well as he menioned mehods on oday s graphics hardware, i has he poenial of overcoming bandwih limiaions on fuure graphics hardware. 2 Hardware-Based Ray Casing in Terahedral Meshes The basic principle of a ray casing implemenaion based on programmable pixel shading is o race all viewing rays from fron o back a he same ime as illusraed in Figure 1a. Each of hese viewing rays corresponds o one pixel of he frame buffer. In an iniializaion sep, he firs inersecion of each ray wih he mesh is compued. Afer his, he rays are propagaed one cell in each of he following passes. Noe ha one of hese passes will usually ake several acual rendering passes because of limiaions of he pixel shading hardware. As pixel shading programs may usually no access he frame buffer, several wo-dimensional exure maps of he dimensions of he frame buffer are necessary o sore he inermediae informaion abou inersecions of rays wih cells of he mesh. Each exel of hese exure maps corresponds o exacly one Visualizaion and Ineracive Sysems Group, IfI, Universiy of Sugar, Breiwiesensr , Sugar, Germany. Marin.Kraus 1

2 v,2 n,0 pixels in view plane (a) erahedral mesh e n,2 Λ 2 r Λ 1 r v,0 n v,1,1 (b) Λ 0 r Figure 1: Hardware-assised ray casing: (a) For each pixel one viewing ray is raced, which sops a all inerseced cell boundaries. The iniial inersecions are marked wih dos ( ), furher inersecions wih circles ( ), squares ( ), and diamonds ( ). (b) Inersecions of a viewing ray wih a cell s boundary. pixel of he frame buffer and, herefore, o exacly one of he viewing rays. Thus, he algorihm will work as follows: 1. Clear he frame buffer and iniialize he exure maps ha conain he informaion abou inersecions. The iniial values are deermined by he firs inersecion of he viewing ray associaed wih each exel and he cells of he mesh. 2. Updae each of he menioned exure maps by rendering one recangle of he same dimensions ino he exure memory. In he raserizaion of each exel, compue he nex inersecion poin of he ray corresponding o he exel wih a cell s boundary. Duplicae he previous inersecion poin if here are no furher inersecions. 3. Render anoher recangle of he same dimensions ino he frame buffer. This ime, compue he volume rendering inegral beween he previous inersecion poin and he poin compued in sep 2 and blend he resuling color ino he frame buffer. 4. Coninue wih sep 2 unless a specified ime limi has been reached. Sep 1 may be implemened using a raserizaion of he visible boundary faces of he mesh. As here are usually far less boundary faces han cells in a mesh, his sep is no ime criical and may also be performed in sofware. The compuaion of he volume rendering inegral in sep 3 may be performed as discussed in [2]. However, he blending has o be adaped from back-o-fron o frono-back blending. In he noaion of [1] he accumulaed associaed color C i and opaciy α i afer i passes wih fron-o-back blending is given by C i C i 1 1 α i 1 C i and α i 1 1 α i 1 1 α i where C i and α i denoe he associaed color and opaciy of he segmen of he volume rendering inegral in he i-h pass. 2

3 The sop condiion in sep 4 guaranees an almos consan frame rae and is easy o evaluae; however, he algorihm migh no render he whole mesh. In order o guaranee he rendering of he complee mesh, one has o es wheher any new inersecion poins have been compued in sep 2. v,2 n,0 v,0 v,1 n,2 n,1 a,0 a,2 a,1 f,0 f,2 f,1 (a) (b) (c) Figure 2: Nomenclaure in his paper: (a) The verex v i is in and he face normal n i is perpendicular o he i-h face of cell. For erahedral cells i is 0, 1, 2, or 3. (b) The neighbor a i of cell shares he i-h face. (c) Face indices f i of s neighbors: The i-h face of corresponds o he f i-h face of s neighbor a i. In order o describe he compuaions of sep 2, a few noaions have o be inroduced; see also Figure 2. Terahedral cells of a mesh consising of n cells will be idenified by an ineger index from 0 o n 1; ofen his index will be called. Each erahedron has four faces, he normal vecors of which are denoed by n i where i specifies he face and is 0, 1, 2, or 3. Noe ha normal vecors will always poin o he ouside of heir cell. Each erahedron also defines four verices v i. (See Figure 2a.) In conras o he sandard convenion verex v i is par of he i-h face. As indicaed in Figure 2b, he neighbor of a erahedron ha shares he i-h face is denoed by a i. The index of he face of a i ha corresponds o he i-h face of is denoed by f i; see Figure 2c. The scalar field value s x a a poin x is linearly inerpolaed from he scalar values a he verices of he mesh; hus, s x wihin one erahedral cell is a linear funcion and can be compued as s x g x x 0 s x 0 g x g x 0 s x 0 where g is he gradien of s x and x 0 is any poin, e.g. one of he verices. Therefore, s x may be specified by he vecor g and he scalar ġ g x 0 s x 0. As indicaed in Figure 1b, e denoes he eye poin and he direcion of a viewing ray is specified by a normalized vecor r. Any poin x on a viewing ray will be idenified by he facor λ such ha x e λr. All daa have o be sored in exure maps in order o access hem in a hardwarebased pixel shading program. Table 1 summarizes one possibiliy o organize hese exure maps. Noe ha he cell indices are encoded in wo exure coordinaes and wo color componens as heir range will usually exceed he range of a single exure coordinae or color componen. Apar from he five consan exure maps for verices, face normals, neighbor daa, linear funcions, and normalized direcions of he viewing rays 3

4 n n daa in exure coordinaes exure daa exure map u v w r g b α verices i v i face normals i n i neighbor daa i a i f i s x i g ġ ray direcions raser pos. r for his ray enered cells raser pos. i inersecion poins raser pos. ray facor λ s e λr Table 1: Summary of he exure maps described in he main ex. associaed wih he pixels of he frame buffer, here are wo exure maps describing he inersecions of viewing rays wih he boundaries of cells: One exure map specifies he mos recenly enered cell for each viewing ray indexed by he raser posiion of he corresponding pixel, and similarly anoher exure map specifies he posiion of and he scalar value a he las inersecion poin. This informaion is necessary in order o compue he segmen of he volume rendering inegral beween wo successive inersecion poins in sep 3 of he algorihm. In fac, he laer exure map has o be duplicaed as he algorihm needs o access he daa of he new and he previous inersecion poin. In sep 2 of he algorihm he nex inersecion of each viewing ray has o be deermined; hus, he wo exure maps specifying he enered cells and he inersecion poins have o be recompued. The nex inersecion poin is he exi poin of he cell enered a he previous inersecion poin. This poin may be deermined by compuing all inersecion poins of he ray wih he four faces of he enered cell and choosing he inersecion poin ha is closes o he eye poin bu no on a face ha is visible from he eye poin. Wih he eye poin e and he normalized direcion r of he viewing ray (see Figure 1b), he four inersecion poins wih he faces of cell are e λ i r wih 0 i 4 and λ i v i e r i i This equaion is easily implemened wih pixel shading operaions as hree-dimensional vecor operaions are usually well suppored. A face is visible, if he denominaor in he previous equaion is negaive; hus, his es comes almos for free. If λ i is se o an appropriaely large number for all visible faces, min λ i 0 i 4 will idenify he exi poin. Deermining he minimum of four numbers is usually less well suppored by pixel shading hardware, bu i may be implemened wih he help of a sequence of comparisons. Once he minimum λ i and is face i are idenified, he inersecion poin x may be compued as x e λ i r and he value of he scalar field a ha poin is s x g x ġ. For each inersecion poin, λ i and s x is sored as his informaion is required in sep 3 of he algorihm o compue he volume rendering inegral beween wo successive 4

5 inersecion poins. The cell enered a x is given by a i and he new face index in his cell is f i. The values of a i and f i will also indicae wheher x is on he boundary of he mesh, as here is no neighbor in his case. Once a viewing ray leaves he mesh, he inersecion poin should be kep consan because consan inersecion poins will resul in segmens of lengh zero, which will no conribue o he volume rendering inegral. However, his procedure is only useful for convex meshes as here may be re-enries if he mesh is non-convex. 3 Discussion and Conclusions As each pass of he algorihm requires he updae of wo exure maps and one updae of he frame buffer, here are a leas hree rendering passes per ieraion of he algorihm. On oday s graphics hardware furher rendering passes and addiional exure maps for inermediae resuls are necessary because he maximum number of operaions per pixel shading program is raher small. Anoher disadvanage of his mehod is he need o raserize exels and pixels, alhough heir viewing rays have already lef he mesh. However, i migh be possible o use early deph-ess in order o minimize he associaed performance penaly. The main advanage of his approach is ha he complee mesh daa is sored in he exure memory. Therefore, here is almos no daa ransfer beween he main memory and he graphics subsysem provided ha here is enough exure memory. This algorihm may also serve as a (benchmark) es for programmable pixel shading hardware and pixel shading languages, as i is an applicaion of programmable pixel shading o an acual problem of compuer visualizaion and i relies on several aspecs of programmable pixel shading (including dependen exures, hree-dimensional exure maps, vecor operaions, and rendering o exure memory). References [1] K. Engel, M. Kraus, and T. Erl. High-Qualiy Pre-Inegraed Volume Rendering Using Hardware-Acceleraed Pixel Shading. In Proceedings Eurographics/SIGGRAPH Workshop on Graphics Hardware 01, pages 9-16, [2] S. Röger, M. Kraus, and T. Erl. Hardware-Acceleraed Volume And Isosurface Rendering. In Proceedings IEEE Visualizaion 00, pages , [3] P. Shirley and A. Tuchman. A Polygonal Approximaion To Direc Scalar Volume Rendering. ACM Compuer Graphics (San Diego Workshop on Volume Visualizaion), 24(5):63-70, [4] M. Weiler and T. Erl. Hardware-Sofware-Balanced Resampling for he Ineracive Visualizaion of Unsrucured Grids. In Proceedings IEEE Visualizaion 2001, pages ,

6 [5] R. Wesermann. The Rendering of Unsrucured Meshes Revisied. In Proceedings EUROGRAPHICS/IEEE Symposium on Visualizaion 2001, [6] R. Wesermann and T. Erl. The VSBUFFER: Visibiliy Ordering Unsrucured Volume Primiives By Polygon Drawing. In Proceedings IEEE Visualizaion 97, pages 35-43, [7] R. Wesermann and T. Erl. Efficienly Using Graphics Hardware In Volume Rendering Applicaions. ACM Compuer Graphics (Proceedings SIGGRAPH 98), pages ,