The Community for Technology Leaders
RSS Icon
Issue No.06 - November/December (2009 vol.15)
pp: 1587-1594
Alan Chu , Chinese University of Hong Kong
Chi-Wing Fu , Nanyang Technological University, Singapore
Andrew Hanson , Indiana University, Bloomington
Pheng-Ann Heng , Chinese University of Hong Kong
This paper describes GL4D, an interactive system for visualizing 2-manifolds and 3-manifolds embedded in four Euclidean dimensions and illuminated by 4D light sources. It is a tetrahedron-based rendering pipeline that projects geometry into volume images, an exact parallel to the conventional triangle-based rendering pipeline for 3D graphics. Novel features include GPU-based algorithms for real-time 4D occlusion handling and transparency compositing; we thus enable a previously impossible level of quality and interactivity for exploring lit 4D objects. The 4D tetrahedrons are stored in GPU memory as vertex buffer objects, and the vertex shader is used to perform per-vertex 4D modelview transformations and 4D-to-3D projection. The geometry shader extension is utilized to slice the projected tetrahedrons and rasterize the slices into individual 2D layers of voxel fragments. Finally, the fragment shader performs per-voxel operations such as lighting and alpha blending with previously computed layers. We account for 4D voxel occlusion along the 4D-to-3D projection ray by supporting a multi-pass back-to-front fragment composition along the projection ray; to accomplish this, we exploit a new adaptation of the dual depth peeling technique to produce correct volume image data and to simultaneously render the resulting volume data using 3D transfer functions into the final 2D image. Previous CPU implementations of the rendering of 4D-embedded 3-manifolds could not perform either the 4D depth-buffered projection or manipulation of the volume-rendered image in real-time; in particular, the dual depth peeling algorithm is a novel GPU-based solution to the real-time 4D depth-buffering problem. GL4D is implemented as an integrated OpenGL-style API library, so that the underlying shader operations are as transparent as possible to the user.
Mathematical visualization, four-dimensional visualization, graphics hardware, interactive illumination
Alan Chu, Chi-Wing Fu, Andrew Hanson, Pheng-Ann Heng, "GL4D: A GPU-based Architecture for Interactive 4D Visualization", IEEE Transactions on Visualization & Computer Graphics, vol.15, no. 6, pp. 1587-1594, November/December 2009, doi:10.1109/TVCG.2009.147
[1] E. A. Abbott, Flatland. Dover Publications, Inc., 1952.
[2] T. F. Banchoff, Visualizing two-dimensional phenomena in four-dimensional space: A computer graphics approach. In E. Wegman, and D. Priest editors, , Statistical Image Processing and Computer Graphics, pages 187–202. Marcel Dekker, Inc., New York, 1986.
[3] T. F. Banchoff, Beyond the third dimension: Geometry, computer graphics, and higher dimensions. Scientific American Library, 1990.
[4] D. C. Banks, Interactive display and manipulation of two-dimensional surfaces in four dimensional space. In Symposium on Interactive 3D Graphics, pages 197–207, New York, 1992. ACM.
[5] D. C. Banks, Illumination in diverse codimensions. In Computer Graphics, pages 327–334, 1994. SIGGRAPH 1994.
[6] D. C. Banks, Screen-parallel determination of intersection curves. Parallel Computing, 23 (7): 953–960, 1997.
[7] L. Bavoil and K. Myers, Order independent transparency with dual depth peeling, 2008. White paper, NVidia, 10/opengl/src/dual depth peeling/docDualDepthPeeling.pdf.
[8] P. Bhaniramka, R. Wenger, and R. Crawfis, Isosurfacing in higher dimensions. In Proc. of IEEE Visualization 2000, pages 267–273, 2000.
[9] P. Brown and B. Lichtenbelt, Ext geometry shader4 extension specification, 2007. spe csGL EXT geometry shader4.txt (last modified: May 2007).
[10] S. A. Carey, R. P. Burton, and D. M. Campbell, Shades of a higher dimension. Computer Graphics World, pages 93–94, October 1987.
[11] R. A. Cross and A. J. Hanson, Virtual reality performance for virtual geometry. In Proc. of IEEE Visualization 1994, pages 156–163, 1994.
[12] K. L. Duffin and W. A. Barrett, Spiders: a new user interface for rotation and visualization of n-dimensional point sets. In Proc. of IEEE Visualization 1994, pages 205–211, 1994.
[13] R. Egli, C. Petit, and N. F. Stewart, Moving coordinate frames for representation and visualization in four dimensions. Computers and Graphics, 20 (6): 905–919, 1996.
[14] E. Eisemann and X. Décoret, Fast scene voxelization and applications. In Proc. of the 2006 symposium on Interactive 3D graphics and games, pages 71–78, 2006.
[15] C. Everitt, Interactive order-independent transparency, 2001. White paper, NVidia, Order Transparency.html.
[16] S. Fang and H. Chen, Hardware accelerated voxelization. Computers & Graphics, 24 (3): 433–442, 2000.
[17] S. Feiner and C. Beshers, Visualizing N-dimensional virtual worlds with N-vision. In SIGGRAPH 1990, pages 37–38, 1990.
[18] A. R. Forsyth, Geometry of Four Dimensions. Cambridge U. Press, 1930.
[19] G. K. Francis, A Topological Picturebook. Springer Verlag, 1987.
[20] A. J. Hanson, A construction for computer visualization of certain complex curves. Notices of the Amer. Math. Soc., 41 (9): 1156–1163, 1994.
[21] A. J. Hanson and R. A. Cross, Interactive visualization methods for four dimensions. In Proc. of IEEE Visualization 1993, pages 196–203, 1993.
[22] A. J. Hanson and P. A. Heng, Four-dimensional views of 3D scalar fields. In Proc. of IEEE Visualization '92, pages 84–91, 1992.
[23] A. J. Hanson and P. A. Heng, Illuminating the fourth dimension. IEEE Computer Graphics and Applications, 12 (4): 54–62, July 1992.
[24] A. J. Hanson and H. Zhang, Multimodal exploration of the fourth dimension. In Proc. of IEEE Visualization 2005, pages 263–270, 2005.
[25] D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination. Chelsea, New York, 1952.
[26] C. M. Hoffmann and J. Zhou, Some techniques for visualizing surfaces in four-dimensional space. Computer Aided Design, 23 (1): 83–91, 1991.
[27] S. Hollasch, Four-space visualization of 4D objects, 1991. Master thesis, Arizona State University.
[28] E. Lindholm, M. J. Kligard, and H. Moreton, A user-programmable vertex engine. In SIGGRAPH 2001, pages 149–158, 2001.
[29] I. Llamas, Real-time voxelization of triangle meshes on the GP U. In, ACM SIGGRAPH 2007 sketches, page 18, 2007.
[30] Miller and Gavosto The immersive visualization probe for exploring n-dimensional spaces. , IEEE Comp. Graph. and App., 24 (1): 76–85, 2004.
[31] N. Neophytou and K. Mueller, Space-time points: 4D splatting on efficient grids. In Proc. of IEEE Symposium on Volume Visualization and Graphics, pages 97–106, 2002.
[32] A. M. Noll, A computer technique for displaying N-dimensional hyper-objects. Communication ACM, 10 (8): 469–473, 1967.
[33] K. Proudfoot, W. R. Mark, S. Tzvetkov, and P. Hanrahan, A real-time procedural shading system for programmable graphics hardware. In SIGGRAPH 2001, pages 159–170, 2001.
[34] P. Shirley and A. Tuchman, A polygonal approximation to direct scalar volume rendering. volume 24, pages 63–70, 1990. SIGGRAPH 1990.
[35] K. V. Steiner and R. P. Burton, Hidden volumes: The 4th dimension. Computer Graphics World, pages 71–74, February 1987.
[36] H. Zhang and A. J. Hanson, Shadow-driven 4D haptic visualization. In Proc. of IEEE Visualization 2007, pages 1688–1695, 2007.
[37] Y. Zhou, W. Chen, and Z. Tang, An elaborate ambiguity detection method for constructing isosurfaces within tetrahedral meshes. Computers & Graphics, 19 (3): 355–364, 1995.
27 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool