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Projected Tetrahedra Revisited: A Barycentric Formulation Applied to Digital Radiograph Reconstruction Using Higher-Order Attenuation Functions
July/August 2006 (vol. 12 no. 4)
pp. 461-473

Abstract—This paper presents a novel method for volume rendering of unstructured grids. Previously [CHECK END OF SENTENCE], we introduced an algorithm for perspective-correct interpolation of barycentric coordinates and computing polynomial attenuation integrals for a projected tetrahedron using graphics hardware. Here, we enhance the algorithm by providing a simple and efficient method to compute the projected shape (silhouette) and tessellation of a tetrahedron, in perspective and orthographic projection models. Our tessellation algorithm is published here for the first time. Compared with works of other groups on rendering unstructured grids, the main contributions of this work are: 1) A new algorithm for finding the silhouette of a projected tetrahedron. 2) A method for interpolating barycentric coordinates and thickness on the faces of the tetrahedron. 3) Visualizing higher-order attenuation functions using GPU without preintegration. 4) Capability of applying shape deformations to a rendered tetrahedral mesh without significant performance loss. Our visualization model is independent of depth-sorting of the cells. We present imaging and timing results of our implementation, and an application in time-critical "2D-3D” deformable registration of anatomical models. We discuss the impact of using higher-order functions on quality and performance.

[1] O. Sadowsky, J.D. Cohen, and R.H. Taylor, “Rendering Tetrahedral Meshes with Higher Order Attenuation Functions for Digital Radiograph Reconstruction,” Proc. Conf. IEEE Visualization, pp. 303-310, 2005.
[2] D. Laney, S.P. Callahan, N. Max, C.T. Silva, S. Langer, and R. Frank, “Hardware Accelerated Simulated Radiography,” Proc. Conf. IEEE Visualization, pp. 343-350, 2005.
[3] P. Shirley and A. Tuchman, “A Polygonal Approximation to Direct Scalar Volume Rendering,” Proc. Workshop Volume Visualization, 1990.
[4] J. Yao, “A Statistical Bone Density Atlas and Deformable Medical Image Registration,” PhD dissertation, Johns Hopkins Univ., 2002.
[5] M. Kraus, W. Qiao, and D.S. Ebert, “Projecting Tetrahedra without Rendering Artifacts,” Proc. Conf. IEEE Visualization, pp. 27-33, Oct. 2004.
[6] B. Wylie, K. Moreland, L.A. Fisk, and P. Crossno, “Tetrahedral Projection Using Vertex Shaders,” Proc. 2002 IEEE Symp. Volume Visualization and Graphics, pp. 7-12, 2002.
[7] M. Levoy and P. Hanrahan, “Light Field Rendering,” Proc. SIGGRAPH '96, pp. 31-42, 1996.
[8] S.J. Gortler, R. Grzeszczuk, R. Szeliski, and M.F. Cohen, “The Lumigraph,” Proc. SIGGRAPH '96, pp. 43-54, 1996.
[9] D.A. LaRose, “Iterative X-Ray/CT Registration Using Accelerated Volume Rendering,” PhD dissertation, Carnegie Mellon Univ., 2001.
[10] S. Röttger, M. Kraus, and T. Ertl, “Hardware-Accelerated Volume and Isosurface Rendering Based on Cell-Projection,” Proc. Conf. IEEE Visualization, pp. 109-116, 2000.
[11] M. Weiler, M. Kraus, and T. Ertl, “Hardware-Based View-Independent Cell Projection,” Proc. IEEE Volume Visualization and Graphics Symp., pp. 13-22, 2002.
[12] K. Moreland and E. Angel, “A Fast High Accuracy Volume Renderer for Unstructured Data,” Proc. IEEE Symp. Volume Visualization and Graphics, pp. 9-16, Oct. 2004.
[13] P.L. Williams, “Visibility Ordering Meshed Polyhedra,” ACM Trans. Graphics, vol. 11, no. 2, pp. 103-126, Apr. 1992.
[14] C.T. Silva, J.S. Mitchell, and P.L. Williams, “An Exact Interactive Time Visibility Ordering Algorithm for Polyhedral Cell Complexes,” Proc. IEEE Symp. Volume Visualization, pp. 87-94, 1998.
[15] R. Farias, J. Mitchell, and C. Silva, “Zsweep: An Efficient and Exact Projection Algorithm for Unstructured Volume Rendering,” Proc. 2000 ACM/IEEE Volume Visualization and Graphics Symp., pp. 91-99, 2000.
[16] S. Callahan, M. Iktis, J. Comba, and C.T. Silva, “Hardware-Assisted Visibility Sorting for Unstructured Volume Rendering,” IEEE Trans. Visualization and Computer Graphics, vol. 11, no. 3, May/June 2005.
[17] A. Mohamed and C. Davatzikos, “An Approach to 3D Finite Element Mesh Generation from Segmented Medical Images,” Proc. IEEE Int'l Symp. Biomedical Imaging (ISBI), pp. 420-423, 2004.
[18] L. Ellingsen and J. Prince, “Mjolnir: Deformable Image Registration Using Feature Diffusion,” Proc. Conf. SPIE Medical Imaging, 2006.
[19] T.F. Cootes, C.J. Taylor, D.H. Cooper, and J. Graham, “Active Shape Models— Their Training and Application,” Computer Vision and Image Understanding, vol. 6, no. 1, pp. 38-59, 1995.
[20] J. Muller-Merbach, “Simulation of X-Ray Projections for Experimental 3D Tomography,” Technical Report, Image Processing Laboratory, Dept. of Electrical Eng., Linkoping Univ., SE-581 83, Sweden, 1996.
[21] R. Gupta and R. Hartley, “Linear Pushbroom Cameras,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 9, pp. 963-975, Sept. 1997.

Index Terms:
Volume rendering, unstructured grids, projected tetrahedra, DRR, higher-order volumetric functions.
Ofri Sadowsky, Jonathan D. Cohen, Russell H. Taylor, "Projected Tetrahedra Revisited: A Barycentric Formulation Applied to Digital Radiograph Reconstruction Using Higher-Order Attenuation Functions," IEEE Transactions on Visualization and Computer Graphics, vol. 12, no. 4, pp. 461-473, July-Aug. 2006, doi:10.1109/TVCG.2006.77
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