High-Quality Unstructured Volume Rendering on the PC Platform

Transcription

1 Graphics Hardware (2002), pp. 1 8 Thomas Ert, Wofgang Heidrich, and Michae Doggett (Editors) High-Quaity Unstructured Voume Rendering on the PC Patform Stefan Guthe Stefan Roettger Andreas Schieber Wofgang Strasser Thomas Ert WSI/GRIS University of Tuebingen IfI/VIS University of Stuttgart Abstract For the visuaization of voume data the appication of transfer functions is used widey. In this area the preintegration technique aows high quaity visuaizations and the appication of arbitrary transfer functions. For reguar grids, this approach eads to a two-dimensiona pre-integration tabe which easiy fits into texture memory. In contrast to this, unstructured meshes require a three-dimensiona pre-integration tabe. As a consequence, the avaiabe texture memory imits the resoution of the pre-integration tabe and the maximum oca derivative of the transfer function. Discontinuity artifacts arise if the resoution of the pre-integration tabe is too ow. This paper presents a nove approach for accurate rendering of unstructured grids using the muti-texturing capabiities of commodity PC graphics hardware. Our approach achieves high quaity by reconstructing the coors and opacities of the pre-integration tabe using the high interna precision of the pixe shader. Since we are using standard 2D muti-texturing we are not imited in the size of the pre-integration tabe. By combining this approach with a hardware-acceerated cacuation of the pre-integration tabe, we achieve both high quaity visuaizations and interactive cassification updates. Categories and Subject Descriptors (according to ACM CCS): I.3.1 [Computer Graphics]: Picture and Image GenerationGraphics processors; I.3.3 [Computer Graphics]: Picture and Image GenerationViewing agorithms. Keywords: Voume Rendering, Unstructured Meshes, Ce Projection, Graphics Hardware, Texture Mapping. 1. Introduction Due to the increasing fexibiity of commodity graphics hardware the pre-integration technique has become widey avaiabe for the visuaization of voume data on reguar grids. Previous approaches for unstructured meshes, that is irreguar tetrahedra grids, empoyed a 3D texture to effectivey appy pre-integration. Athough the resuting images are of high quaity, there are severa restrictions due to the imited amount of avaiabe texture memory. Transfer functions with high gradients require a high resoution preintegration tabe, which does not fit easiy into the dedicated texture memory. To circumvent this restriction we propose to impicity store the 3D texture by means of mutipe 2D textures. Then the coors and opacities of the three-dimensiona pre-integration tabe can be reconstructed accuratey with the high interna precision of the pixe shader Reated Work Direct voume rendering of unstructured meshes was dramaticay acceerated by the Projected Tetrahedra (PT) agorithm of Shirey and Tuchman In order to compensate for the imitations of the PT agorithm there exist numerous competing approaches, such as ray tracing 3, ray casting 14, sicing 23, and sweep-pane agorithms 17. The origina PT agorithm interpoates the opacities and coors ineary between the vertices, resuting in Mach bands as reported by Max et a. 6. Extending the origina agorithm, Stein et a. 16 presented a soution for the correct interpoation of opacities by utiizing 2D texture mapping. However, this method is restricted to inear transfer functions for the opacity and sti interpoates coors ineary ignoring the transfer function inside the tetrahedra. Roettger et a. 11 presented an improvement to this agorithm using pre-integrated 2D textures and a inear interpoation of the opacity. They aso introduced a new tech-

2 Guthe and Roettger / High-Quaity Unstructured Voume Rendering nique which is based on 3D texture mapping. This technique aows the correct interpoation of both the coors and the opacities empoying the pre-integration method first introduced by Max et a. 7. Athough 3D texture mapping is avaiabe on recent PC graphics hardware, the imited amount of texture memory restricts the accuracy of the cassification. Therefore cassifications containing high gradients do not render acceptabe resuts. Modern PC graphics hardware, for instance the ATI Radeon 8500 and the NVIDIA GeForce4, aow more sophisticated approaches using dependent textures, muti-texturing, per-pixe shading, and hardware acceerated pre-integration. This enabes us to overcome the imited size of the three-dimensiona pre-integration tabe Paper Overview The remainder of this paper is organized as foows: In Section 2 the basic ce projection agorithm and the optica mode used throughout the paper are described. A new approach for approximating the three-dimensiona ray integra by means of mutipe 2D textures is discussed in Section 3. An efficient hardware acceerated cacuation of the ray integra is given in Section 4. Empoying the new approximations, the experimenta resuts obtained on different graphics hardware are presented in Section 5. Finay, we concude our paper and outine future work. 2. Basic Agorithm The Projected Tetrahedra agorithm can be outined as foows: A tetrahedra are sorted according to their visibiity (see aso ), cassified to their projective profie and spit into trianges as seen in Figure 1. In a more genera approach each ce is spit into severa tetrahedra during a precomputationa step. However, it is aso possibe to render hexahedra ces directy using the extension of Schussman and Max 12. Athough our approach is compatibe with this extension, we restrict ourseves to tetrahedra throughout this paper. Whie the origina PT agorithm uses a inear interpoation of coors and opacities between the triange vertices, Stein et a. suggest to use a 2D texture map for the exponentia interpoation of the opacities. The texture coordinates assigned to each vertex correspond to the average extinction coefficient τ and the thickness of the projected ce, whereas the texture map contains the α-component set to α 1 e τ. Since the texture coordinates are interpoated ineary, this approach is restricted to inear transfer functions τ τ f x y z with f being the scaar function. In their high accuracy voume renderer (HIAC) Wiiams et a. 21 extended this approach to piecewise inear transfer functions. Assuming a inear interpoation inside the tetrahedra, it can easiy be seen that f varies ineary aong each viewing ray. Therefore the integrated chromaticity C and opacity α!"##%$ Figure 1: Cassification of non-degenerated projected tetrahedra (top row) and the corresponding decomposition (bottom row) according to 13. of the ray segment can be defined by a three-dimensiona function depending on S f, S b and (see Figure 2). Using the optica mode of Wiiams and Max with the chromaticity vector κ κ f x y z& and the scaar optica density ρ ρ f x y z&, the ray integra is given as foows: S x' S f ( C S f S b ) α S f S b ) * x S b S f e,+ t ρ- 0 S - u./. du κ S t& ρ S t& dt 0 1 e,+ ρ- 0 S - t./. dt Sf Sb Figure 2: Intersection of a tetrahedra ce with a viewing ray. S f and S b are the scaar vaues of the front and back face, respectivey; denotes the ength of the ray segment. These integras are computed using the numerica integration described by Enge et a. 2. For each compositing step we cacuate the coor I0 of a pixe from the previous coor I by using the foowing bend operation: I0 C ( 1 α21 I3

3 5 Empoying the 3D texturing approach as proposed by Roettger et a. 11 both κ and ρ need not be inear, thus arbitrary transfer functions can be appied. However, 3D textures require a huge amount of texture memory to achieve high quaity images for arbitrary transfer functions. Beside the memory imitations of consumer graphics hardware 3D texture mapping aso resuts in a decreased rendering performance. Guthe and Roettger / High-Quaity Unstructured Voume Rendering 3. High Resoution Ray Integra Since high resoution 3D textures require huge amounts of texture memory, we separate the three-dimensiona function of the voume density optica mode. Unfortunatey, ony the opacity can be separated easiy. The chromaticity needs to be approximated by means of a inear combination of twodimensiona functions. inear α wireframe 3.1. Opacity Reconstruction Since the opacity depends on the average density aong the viewing ray and the ength of the ray segment, it can be separated as foows: * 1 ˆρ S f S b ' α 1D x' 0 1 e x ρ S f ( t S b S f & dt (1) For each rendered pixe we derive the average density ˆρ from a 2D texture map (Equation 1) and compute the fina opacity α 1D by means of a 1D dependent texture ookup (Equation 2). In order to further increase the accuracy of the reconstructed α vaues, the dependent texture is extended to hod the higher 8 bits of a 16 bit α vaue in the apha channe A and the ower 8 bits in the additiona uminance channe L. In order to map the maximum 16 bit α vaue to 1, it is scaed by the factor Since the resuting equation α 1D 256A L 257 ( 257 is inear, the texture interpoation deivers a true 16 bit α ookup. Compared to the inear approximation of the opacity using the 2D texturing approach of Roettger et a. 11 the resuting images are significanty improved, as iustrated in Figure 3. (2) inear α exponentia α Figure 3: Comparison between inear approximation and correct exponentia α. The corresponding transfer function which is spit into normaized emission and opacity can be seen beow each image. 1 Ĉ 0 0 S f S b ' 65 κ S 1 t& ρ S 1 t 1 ρ 0 S 1 t& C S Ĉ max S f S b ' f S b max α S f S b max C 1 S f S b ' Ĉ max S f S b 7 Ĉ 0 S f S b Using the texture setup of Tabe 1, we impement the foowing approximation of the voume optica density mode by utiizing dependent textures and the pixe shader on the NVIDIA GeForce4 24 and the ATI Radeon graphics adapter: C S f S b )8 Ĉ in S f S b α S f S b Ĉ in S f S b ) Ĉ 0 S f S b ( C max 1 S f S b α S f S b ) α 1D ρ S f S b & 3.2. Chromaticity Reconstruction In order to achieve a high-quaity approximation of the chromaticity, we pre-integrate the normaized chromaticities C Ĉ α for 4 0 and max with max being the maximum ength of the ray segments. The normaized emission Ĉ 0 and the difference C 1 of the normaized emissions Ĉ 0 and Ĉ max are stored in two high resoution 2D textures. The atter emissions are defined as foows: unit coordinates RGB A 0 S f, S b Ĉ 0 S f S b ρ S f S b 1 S f, S b C 1 S f S b - 2 ρ S f S b - α 1D ρ S f S b & Tabe 1: Texture setup for dependent texture mapping.

4 Guthe and Roettger / High-Quaity Unstructured Voume Rendering This is a inear approximation in for every pair of S f and S b. As seen in Figure 4, the inear approximation is not accurate for transfer functions that contain high gradients. For an improved reconstruction we approximate the chromaticity by a poynomia of degree n 9 1 in with the coefficients C i, i 03&33 n. This is simiar to the poynomia texture mapping approach of Mazbender et a. 5, which reconstructs the coors of a surface by a biquadratic poynomia. In our case the approximated chromaticity is given by the poynomia C S f S b )8 n α S f S b i: 0 i i max C i S f S b ;3 inear chromaticity wireframe To compute the poynomia coefficients C i we preintegrate the chromaticity at for i 03&3&3 n and i construct a poynomia through each of these points for every pair of S f and S b. This corresponds to the computation of n( 1 sices with const of the pre-integration tabe. Since the number of texture units is imited, we can ony use a poynomia approximation with a degree of up to 2 on the GeForce4 and of up to 4 on the Radeon In the atter case the rasterization performance drops by amost 50%, but the quaity of the approximation is ony improved sighty. Therefore a poynomia degree of 2 shoud be preferred (see Figure 4). The corresponding texture setups are depicted in Tabe 2. The poynomia coefficients are scaed to the maximum possibe texe range < 1 3&3&3 1= to improve the precision of the approximation. Additionay, the α vaues are reconstructed with 16 bits of accuracy. max unit coordinates RGB A 0 S f, S b C 0 S f S b ρ S f S b 1 S f, S b C 1 S f S b n S f, S b C n S f S b - n+1 ρ S f S b - α 1D ρ S f S b & Tabe 2: Texture setup for poynomia coor approximation of the three-dimensiona ray integra (with a maximum poynomia degree of n 2 on the GeForce4 and of n 4 on the Radeon 8500). 4. Hardware Acceerated Pre-Integration In order to visuaize voume data comfortaby one needs to change the transfer function interactivey. Whenever the transfer function is modified the pre-integration tabe has to be recomputed. For a resoution of and a poynomia degree of 4, for instance, this requires approximatey 11 seconds on a Pentium 4 running at 2 GHz which is far too sow for interactive updates of the transfer function. In order to inear chromaticity quadratic chromaticity Figure 4: Comparison between inear and quadratic coor approximation combined with 16 bit α, for the transfer function seen beow each image. speed up the cacuation of the pre-integration tabe we utiize graphics hardware for the purpose of numerica integration. We maintain a high eve of accuracy by using the high interna precision of the pixe shader. The numerica integration of the ray segments is performed by samping the integra m times. At each samping step, the integrated chromaticity κ and the integrated opacity α are bended with the corresponding entries of the transfer function. As described by Enge et a. 2 the integrated opacity can be cacuated quicky by the difference of two definite integras. If sef-attenuation is assumed to be negigibe the same approach can be used to efficienty cacuate the integrated chromaticities. This assumption is vaid for voume sicing, since the ray segment engths are usuay sma. In the case of unstructured voume rendering, however, this assumption does not hod, thus sef-attenuation cannot be negected. As a consequence, the numerica integration of the chromaticities is not fast enough to achieve interactive updates of the transfer function. But according to Roettger et a 10, the chromaticities of one sice of the pre-integration tabe can be integrated in parae by using a hardware-acceerated approach. For each sice with a constant ray segment ength this is accompished by bending m quadriateras containing the samped transfer function for every pair of S f and S b into the frame buffer. The samped transfer function is reconstructed from a 1D texture (see Tabe 3). For this purpose, the texture coordinate s of each vertex of the quadriateras is assigned as shown in Figure 5.

5 Guthe and Roettger / High-Quaity Unstructured Voume Rendering s=1-i/(m-1) Sb i=0,...,m-1 s=1 (ow 4 bits). First, the origina 12 bit chromaticity is reconstructed in the pixe shader by mutipying the ow bits with and adding the resut to the high bits. Note that a texture entry of 255 in the high bits aready represents a vaue of Next, the chromaticity and opacity of the transfer function are reconstructed from the 1D texture in the same fashion. Then the chromaticity is mutipied by the opacity, the resut of the previous iteration is mutipied by one minus the opacity, and the sum of both yieds the new integrated chromaticity. Finay, the integrated chromaticity is spit into 8 high and 4 ow bits and is written back into the corresponding ping pong texture. Sf s=0 s=i/(m-1) Figure 5: Texture coordinate setup for the hardwareacceerated pre-integration of one sice of the threedimensiona pre-integration tabe. As the 8 bit frame buffer depth of current PC graphics hardware imits the accuracy of the numerica integration, we integrate the chromaticity with the higher interna accuracy of the pixe shader. Combining two channes of the frame buffer for each integrated coor component of the chromaticity, a tota accuracy of 16 bit can be achieved. In practice however, a bit depth of 12 has turned out to be sufficient. We store the chromaticity and opacity of the transfer function for a given ength and the number of integration steps m in a 1D texture as defined in Tabe 3. To effectivey represent high gradients in the transfer function, we construct the 1D texture with the highest possibe resoution instead of using a inear interpoation of the 1D texture. The Radeon 8500 masks out a bits representing vaues higher than 13 0 or ower than Therefore the high 8 bits are extracted automaticay, whereas the ow 4 bits are extracted by simpy mutipying the 12 bit chromaticity with 256. In contrast to this, the GeForce4 aways uses saturation ogic instead of bit masking. Therefore the ow 4 bits can ony be extracted on the Radeon A speedup of neary two is achieved by performing four subsequent integration steps at once in the pixe shader. Since each RGB coor component has to be computed separatey, the hardware-acceerated pre-integration needs to be performed three times for every required sice of the preintegration tabe. Each component of a pre-integrated sice is transferred back into main memory and recombined with the other coor channes. This resuts in 9 pre-integration cyces for a poynomia approximation of a degree of 2, for exampe. In contrast to software numerica integration, this hardware-acceerated approach aows to update the preintegration tabe interactivey. With respect to integration accuracy the hardware-acceerated method exhibits a higher integration error which is due to the 12 bit quantization. An exampe of these quantization artifacts is given in Figure 6. channe meaning vaue red high 8 bit (chromaticity) κ s green ow 4 bit (chromaticity) κ s bue high 8 bit (opacity) 1 e apha ow 4 bit (opacity) 1 e m m ρ- s. ρ- s. Tabe 3: 1D texture used for hardware-acceerated preintegration. On the Radeon 8500 the numerica integration is impemented using a method caed ping pong fitering 9. For each bending step an RGBA texture contains the previousy integrated chromaticity in the red (high 8 bits) and apha channe Figure 6: Comparison between hardware (eft) and software (midde) pre-integration, incuding the error (right, scaed by a factor of 8 and inverted) for m 128 samping steps.

6 Guthe and Roettger / High-Quaity Unstructured Voume Rendering 5. Resuts In the previous chapters we have demonstrated that the muti-texturing capabiities of modern PC graphics acceerators can be utiized to bring high-quaity pre-integrated voume rendering of unstructured grids to the PC patform. A comparison of the visua quaity of the proposed methods is given in Figure 7. The best approximation of the preintegration tabe is achieved by using 16 bits for the representation of the opacities and a poynomia of degree 4 for the reconstruction of the chromaticities. A poynomia of degree 2 is ony sighty ess accurate, but performs significanty faster due to reduced rasterization requirements. Because of the high interna precision of the pixe shader and the representation of the opacities with 16 bits the resuts are even better than those obtained with a 3D texturing setup. Using our hardware-acceerated pre-integration approach we are abe to maintain high update rates of the pre-integration tabe. In comparison to software integration the achieved speedup is about 700% on a PC equipped with a Pentium 4 running at 2 GHz and an ATI Radeon 8500 (compare Tabe 4). software setup of textures inear coor (n 1) 4.4s poynomia n 2 6.6s poynomia n s Radeon 8500 setup of textures inear coor (n 1) 0.6s poynomia n 2 1.0s poynomia n 4 1.7s Tabe 4: Preprocessing times for 2D muti-texturing with a texture resoution of The tota rendering time is amost independent of the chosen reconstruction method (except for n 4). It depends mainy on the sorting agorithm and the transfer speed between the CPU and the graphics adapter (see Tabe 5). For comparison purposes the experimenta resuts are given for a poynomia degree of 2. We achieve up to 600,000 tetrahedra per second depending on the sorting agorithm. Approximatey haf of the time is spent by sorting, whie the other haf is spent by rendering. The ower performance for rendering the Bucky Ba data set is due to a arger variation of the scaar vaues which ead to a reduced texture cache coherence. 6. Concusion and Future Work We presented a new approach for pre-integrated rendering of projected tetrahedra on commodity PC graphics hardware. We empoyed 2D muti-texturing and pixe shading to reconstruct the three-dimensiona pre-integration tabe. Because GeForce4 #tetra numeric MPVO XMPVO Bunt Fin 187k 3.18 fps 2.64 fps 2.35 fps Bucky Ba 177k 2.46 fps 2.19 fps 2.05 fps Radeon 8500 #tetra numeric MPVO XMPVO Bunt Fin 187k 2.51 fps 2.20 fps 1.99 fps Bucky Ba 177k 2.09 fps 1.98 fps 1.87 fps Tabe 5: Dispay times incuding visibiity sorting on a Pentium 4 running at 2 GHz using a poynomia approximation of degree 2 and a view port. The appied sorting agorithms are numerica sorting 22, MPVO 19, and XMPVO 15. of the reduced memory requirements of the empoyed 2D textures, our method is capabe of appying high resoution transfer functions. We further presented a high quaity numerica pre-integration method which utiizes the graphics hardware to decrease the cassification update time. Since our approach uses the high interna precision of the pixe shader, the resuting images are of a much higher quaity in comparison to the previousy appied 3D texture mapping approach. 7. Acknowedgements The authors woud ike to thank Martin Kraus for his ideas and discussions and Michae Dogget from ATI for providing a Radeon Part of this work has been funded by the SFB grant 382 of the German Research Counci (DFG). References 1. João Comba, James T. Kosowski, Neson Max, Joseph S. B. Mitche, Caudio T. Siva, and Peter L. Wiiams. Fast Poyhedra Ce Sorting for Interactive Rendering of Unstructured Grids. In Computer Graphics Forum (Eurographics 99), voume 18(3), pages , K. Enge, M. Kraus, and Th. Ert. High-Quaity Pre-Integrated Voume Rendering using Hardware- Acceerated Pixe Shading. In Proc. Eurographics / SIGGRAPH Workshop on Graphics Hardware 01, Annua Conference Series, pages 9 16, , 4 3. James T. Kajiya. Ray Tracing Voume Densities. In Proc. SIGGRAPH 84, pages ACM, A. Kanitsar. Christmas Tree Data Set. XMasTree.htm, Tom Mazbender, Dan Geb, and Hans Woters. Poynomia Texture Maps. In SIGGRAPH 2001, Com-

7 Guthe and Roettger / High-Quaity Unstructured Voume Rendering puter Graphics Proceedings, Annua Conference Series, pages , N. Max, B. Becker, and R. Crawfis. Fow Voumes for Interactive Vector Fied Visuaization. In Proc. Visuaization 93, pages IEEE Computer Society Press, N. L. Max, P. Hanrahan, and R. Crawfis. Area and Voume Coherence for Efficient Visuaization of 3D Scaar Functions. Computer Graphics (San Diego Workshop on Voume Visuaization), 24(5):27 33, Neson Max. Optica Modes for Direct Voume Rendering. IEEE Transactions on Visuaization and Computer Graphics, 1(2):99 108, Jason L. Mitche. 1.4 Pixe Shaders. Metdown, , Stefan Roettger and Thomas Ert. A Two-Step Approach for Interactive Pre-Integrated Voume Rendering of Unstructured Grids. In Proceedings of the 2002 Symposium on Voume Visuaization (VOLVIS- 02). ACM Press, 2002 (to appear) Stefan Roettger, Martin Kraus, and Thomas Ert. Hardware-Acceerated Voume and Isosurface Rendering Based on Ce-Projection. In Proc. Visuaization 2000, pages IEEE Computer Society Technica Committee on Computer Graphics, , Greg Schussman and Neson Max. Hierarchicha Perspective Voume Rendering using Triange Fans. In Voume Graphics, Proceedings of the Internationa Workshop on Voume Graphics, pages , Peter Shirey and Aan Tuchman. A Poygona Approximation for Direct Scaar Voume Rendering. In Proc. San Diego Workshop on Voume Visuaization (SIGGRAPH), pages 63 70, , Caudio T. Siva and Joseph S. B. Mitche. The Lazy Sweep Ray Casting Agorithm for Rendering Irreguar Grids. IEEE Transactions on Visuaization and Computer Graphics, 3(2): , , Caudio T. Siva, Joseph S. B. Mitche, and Peter L. Wiiams. An Exact Interactive Time Visibiity Ordering Agorithm for Poyhedra Ce Compexes. In Proceedings of the 1998 Symposium on Voume Visuaization (VOLVIS-98), pages ACM Press, , Cifford Stein, Barry Becker, and Neson Max. Sorting and Hardware Assisted Rendering for Voume Visuaization. In Proc Symposium on Voume Visuaization, pages ACM SIGGRAPH, Rüdiger Westermann and Thomas Ert. The VS- BUFFER: Visibiity Ordering of Unstructured Voume Primitives by Poygon Drawing. In Proc. IEEE Visuaization 1997, pages 35 42, J. Wihems and A. van Geder. A Coherent Projection Approach for Direct Voume Rendering. Computer Graphics, 25(4): , Peter L. Wiiams. Visibiity Ordering Meshed Poyhedra. ACM Transactions on Graphics, 11(2): , , Peter L. Wiiams and Neson Max. A Voume Density Optica Mode Workshop on Voume Visuaization, pages 61 68, Peter L. Wiiams, Neson L. Max, and Cifford M. Stein. A High Accuracy Voume Renderer for Unstructured Data. IEEE Transactions on Visuaization and Computer Graphics, 4(1):37 54, C. M. Wittenbrink. CeFast: Interactive Unstructured Voume Rendering. In IEEE Visuaization 99 Late Breaking Hot Topics, pages 21 24, Roni Yage, David M. Reed, Asish Law, Po-Wen Shih, and Naeem Shareef. Hardware Assisted Voume Rendering of Unstructured Grids by Incrementa Sicing. In Proc. IEEE 1996 Voume Visuaization Symposium, pages 55 62, Hansong Zhang. Vertex Program 1.1 and Texture Shader 3. Game Deveopers Conference,

8 Guthe and Roettger / High-Quaity Unstructured Voume Rendering inear approximation of α exponentia interpoation of α inear coor reconstruction quadratic coor reconstruction Figure 7: Comparison between different approximations of the three-dimensiona ray integra. The appied transfer functions are depicted beow each image. Figure 8: Bucky Ba with per-vertex ighting of origina data set and part of the Christmas Tree data set 4, both with quadratic poynomia approximation of chromaticity and accurate 16 bit α. Figure 9: Bunt Fin dataset using quadratic poynomia approximation of chromaticity and 16 bit α. Due to the high reconstruction quaity of the pre-integration tabe, the undersamping within the origina data set can easiy be seen.