The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.05 - May (2012 vol.18)
pp: 753-766
T. Martin , Sch. of Comput., Univ. of Utah, Salt Lake City, UT, USA
E. Cohen , Sch. of Comput., Univ. of Utah, Salt Lake City, UT, USA
M. M. Kirby , Sch. of Comput., Univ. of Utah, Salt Lake City, UT, USA
ABSTRACT
In this paper, we present a novel isosurface visualization technique that guarantees the accurate visualization of isosurfaces with complex attribute data defined on (un)structured (curvi)linear hexahedral grids. Isosurfaces of high-order hexahedral-based finite element solutions on both uniform grids (including MRI and CT scans) and more complex geometry representing a domain of interest that can be rendered using our algorithm. Additionally, our technique can be used to directly visualize solutions and attributes in isogeometric analysis, an area based on trivariate high-order NURBS (Non-Uniform Rational B-splines) geometry and attribute representations for the analysis. Furthermore, our technique can be used to visualize isosurfaces of algebraic functions. Our approach combines subdivision and numerical root finding to form a robust and efficient isosurface visualization algorithm that does not miss surface features, while finding all intersections between a view frustum and desired isosurfaces. This allows the use of view-independent transparency in the rendering process. We demonstrate our technique through a straightforward CPU implementation on both complex-structured and complex-unstructured geometries with high-order simulation solutions, isosurfaces of medical data sets, and isosurfaces of algebraic functions.
INDEX TERMS
splines (mathematics), computational geometry, data visualisation, finite element analysis, rendering (computer graphics), algebraic function, direct isosurface visualization technique, hex-based high-order geometry, attribute representation, unstructured curvilinear hexahedral grids, high-order hexahedral-based finite element solution, isogeometric analysis, trivariate high-order NURBS geometry, nonuniform rational B-splines geometry, view-independent transparency, rendering process, complex-structured geometry, complex-unstructured geometry, high-order simulation solution, medical data set, Isosurfaces, Splines (mathematics), Surface reconstruction, Surface topography, Pixel, spline and piecewise polynomial interpolation., Isosurface visualization of hex-based high-order geometry and attribute representations, numerical analysis, roots of nonlinear equations
CITATION
T. Martin, E. Cohen, M. M. Kirby, "Direct Isosurface Visualization of Hex-Based High-Order Geometry and Attribute Representations", IEEE Transactions on Visualization & Computer Graphics, vol.18, no. 5, pp. 753-766, May 2012, doi:10.1109/TVCG.2011.103
REFERENCES
[1] O. Abert, M. Geimer, and S. Müller, "Direct and Fast Ray Tracing of NURBS Surfaces," Proc. IEEE Symp. Interactive Ray Tracing, pp. 161-168, 2006.
[2] J. Amanatides, "Ray Tracing with Cones," SIGGRAPH Computer Graphics, vol. 18, no. 3, pp. 129-135, 1984.
[3] C.L. Bajaj, R.L. Holt, and A.N. Netravali, "Rational Parametrizations of Nonsingular Real Cubic Surfaces," ACM Trans. Graphics, vol. 17, no. 1, pp. 1-31, 1998.
[4] J.F. Blinn, "A Generalization of Algebraic Surface Drawing," ACM Trans. Graphics, vol. 1, no. 3, pp. 235-256, 1982.
[5] X. Chen, R.F. Riesenfeld, and E. Cohen, "Sliding Windows Algorithm for B-Spline Multiplication," SPM '07: Proc. ACM Symp. Solid and Physical Modeling, pp. 265-276, 2007.
[6] P. Cignoni, L.D. Floriani, C. Montani, E. Puppo, and R. Scopigno, "Multiresolution Modeling and Visualization of Volume Data Based on Simplicial Complexes," VVS '94: Proc. Symp. Volume Visualization, pp. 19-26, 1994.
[7] E. Cohen, R.F. Riesenfeld, and G. Elber, Geometric Modeling with Splines: An Introduction. A.K. Peters, Ltd., 2001.
[8] J.A. Cottrell, A. Reali, Y. Bazilevs, and T.R. Hughes, "Isogeometric Analysis of Structural Vibrations," Computer Methods in Applied Mechanics and Eng., vol. 195, nos. 41-43, pp. 5257-5296, 2006.
[9] R. de Toledo, B. Levy, and J.-C. Paul, "Iterative Methods for Visualization of Implicit Surfaces on GPU," ISVC '07: Proc. Int'l Symp. Visual Computing, pp. 598-609, Nov. 2007.
[10] J.E.F. Díaz, "Improvements in the Ray Tracing of Implicit Surfaces Based on Interval Arithmetic," PhD thesis, Universitat de Girona, 2008.
[11] A. Efremov, V. Havran, and H.-P. Seidel, "Robust and Numerically Stable Bézier Clipping Method for Ray Tracing NURBS Surfaces," SCCG '05: Proc. 21st Spring Conf. Computer Graphics, pp. 127-135, 2005.
[12] G. Elber, "Free Form Surface Analysis Using a Hybrid of Symbolic and Numeric Computation," PhD thesis, Computer Science Departmente, Univ. of Utah, 1992.
[13] G. Elber and M.-S. Kim, "Geometric Constraint Solver Using Multivariate Rational Spline Functions," SMA '01: Proc. Sixth ACM Symp. Solid Modeling and Applications, pp. 1-10, 2001.
[14] A. Entezari, R. Dyer, and T. Moller, "Linear and Cubic Box Splines for the Body Centered Cubic Lattice," VIS '04: Proc. Conf. Visualization, pp. 11-18, 2004.
[15] J.C. Hart, "Ray Tracing Implicit Surfaces," Proc. SIGGRAPH Course Notes: Design, Visualization and Animation of Implicit Surfaces, pp. 1-16, 1993.
[16] P.S. Heckbert and P. Hanrahan, "Beam Tracing Polygonal Objects," SIGGRAPH '84: Proc. 11th Ann. Conf. Computer Graphics and Interactive Techniques, pp. 119-127, 1984.
[17] J. Hua, Y. He, and H. Qin, "Multiresolution Heterogeneous Solid Modeling and Visualization Using Trivariate Simplex Splines," SM '04: Proc. Ninth ACM Symp. Solid Modeling and Applications, pp. 47-58, 2004.
[18] T.J.R. Hughes, J.A. Cottrell, and Y. Bazilevs, "Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry, and Mesh Refinement," Computer Methods in Applied Mechanics and Eng., vol. 194, pp. 4135-4195, 2005.
[19] J.T. Kajiya, "Ray Tracing Parametric Patches," SIGGRAPH '82: Proc. Ninth Ann. Conf. Computer Graphics and Interactive Techniques, pp. 245-254, 1982.
[20] T. Kalbe and F. Zeilfelder, "Hardware-Accelerated, High-Quality Rendering Based on Trivariate Splines Approximating Volume Data," Computer Graphics Forum, vol. 27, no. 2, pp. 331-340, 2008.
[21] M. Kim, A. Entezari, and J. Peters, "Box Spline Reconstruction on the Face-Centered Cubic Lattice," IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 6, pp. 1523-1530, Nov./Dec. 2008.
[22] J. Kloetzli, M. Olano, and P. Rheingans, "Interactive Volume Isosurface Rendering Using BT Volumes," I3D '08: Proc. Symp. Interactive 3D Graphics and Games, pp. 45-52, 2008.
[23] A. Knoll, Y. Hijazi, C.D. Hansen, I. Wald, and H. Hagen, "Interactive Ray Tracing of Arbitrary Implicit Functions," Proc. Eurographics/IEEE Symp. Interactive Ray Tracing, 2007.
[24] A. Knoll, I. Wald, S. Parker, and C. Hansen, "Interactive Isosurface Ray Tracing of Large Octree Volumes," Proc. IEEE Symp. Interactive Ray Tracing, pp. 115-124, Sept. 2006.
[25] R. Krawczyk, "Newton Algorithmen Zur Bestimmung Von Nullstellen Mit Fehlerschranken," Computing, vol. 4, pp. 187-201, 1969.
[26] M. Levoy, "Efficient Ray Tracing of Volume Data," ACM Trans. Graphics, vol. 9, no. 3, pp. 245-261, 1990.
[27] C. Loop and J. Blinn, "Real-Time GPU Rendering of Piecewise Algebraic Surfaces," ACM Trans. Graphics, vol. 25, no. 3, pp. 664-670, 2006.
[28] W.E. Lorensen and H.E. Cline, "Marching Cubes: A High Resolution 3D Surface Construction Algorithm," SIGGRAPH Computer Graphics, vol. 21, no. 4, pp. 163-169, 1987.
[29] S.R. Marschner and R.J. Lobb, "An Evaluation of Reconstruction Filters for Volume Rendering," VIS '94: Proc. Conf. Visualization, pp. 100-107, 1994.
[30] M.J. Marsden, "An Identity for Spline Functions with Applications to Variation Diminishing Spline Approximation," J. Approximation Theory, vol. 3, pp. 7-49, 1970.
[31] W. Martin and E. Cohen, "Representation and Extraction of Volumetric Attributes Using Trivariate Splines," Proc. Symp. Solid and Physical Modeling, pp. 234-240, 2001.
[32] M. Meyer, B. Nelson, R. Kirby, and R. Whitaker, "Particle Systems for Efficient and Accurate High-Order Finite Element Visualization," IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 5, pp. 1015-1026, Sept./Oct. 2007.
[33] R.E. Moore, Interval Analysis. Prentice-Hall, 1966.
[34] B. Nelson and R.M. Kirby, "Ray-Tracing Polymorphic Multidomain Spectral/hp Elements for Isosurface Rendering," IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 1, pp. 114-125, Jan./Feb. 2006.
[35] T. Nishita, T.W. Sederberg, and M. Kakimoto, "Ray Tracing Trimmed Rational Surface Patches," SIGGRAPH Computer Graphics, vol. 24, no. 4, pp. 337-345, 1990.
[36] A. Paiva, H. Lopes, T. Lewiner, and L.H. de Figueiredo, "Robust Adaptive Meshes for Implicit Surfaces," Proc. Brazilian Symp. Computer Graphics and Image Processing, pp. 205-212, 2006.
[37] S. Parker, P. Shirley, Y. Livnat, C. Hansen, and P.-P. Sloan, "Interactive Ray Tracing for Isosurface Rendering," VIS '98: Proc. Conf. Visualization, pp. 233-238, 1998.
[38] A. Raviv and G. Elber, "Interactive Direct Rendering of Trivariate B-Spline Scalar Functions," IEEE Trans. Visualization and Computer Graphics, vol. 7, no. 2, pp. 109-119, Apr.-June 2001.
[39] M. Reimers and J. Seland, "Ray Casting Algebraic Surfaces Using the Frustum Form," Computer Graphics Forum, vol. 27, no. 2, pp. 361-370, 2008.
[40] J. Schreiner and C. Scheidegger, "High-Quality Extraction of Isosurfaces from Regular and Irregular Grids," IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 5, pp. 1205-1212, Sept./Oct. 2006.
[41] T. Sederberg and A. Zundel, "Pyramids that Bound Surface Patches," Graphical Models and Image Processing, vol. 58, no. 1, pp. 75-81, Jan. 1996.
[42] P. Shirley, Fundamentals of Computer Graphics. A.K. Peters, Ltd., 2002.
[43] K. Sung and P. Shirley, "Ray Tracing with the BSP Tree," Graphics Gems III, pp. 271-274, Academic Press Professional, Inc., 1992.
[44] D.L. Toth, "On Ray Tracing Parametric Surfaces," SIGGRAPH '85: Proc. 12th Ann. Conf. Computer Graphics and Interactive Techniques, pp. 171-179, 1985.
[45] O. Wilson, A. VanGelder, and J. Wilhelms, "Direct Volume Rendering via 3D Textures," technical report, Univ. of California at Santa Cruz, 1994.
[46] Y. Zhang, Y. Bazilevs, S. Goswami, C.L. Bajaj, and T.J.R. Hughes, "Patient-Specific Vascular NURBS Modeling for Isogeometric Analysis of Blood Flow," Computer Methods in Applied Mechanics and Eng., vol. 196, nos. 29/30, pp. 2943-2959, 2007.
21 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool