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Conformal Surface Parameterization for Texture Mapping
April-June 2000 (vol. 6 no. 2)
pp. 181-189

Abstract—In this paper, we give an explicit method for mapping any simply connected surface onto the sphere in a manner which preserves angles. This technique relies on certain conformal mappings from differential geometry. Our method provides a new way to automatically assign texture coordinates to complex undulating surfaces. We demonstrate a finite element method that can be used to apply our mapping technique to a triangulated geometric description of a surface.

[1] S. Angenent, S. Haker, A. Tannenbaum, and R. Kikinis, “Laplace-Beltrami Operator and Brain Flattening,” IEEE Trans. Medical Imaging, vol. 18, pp. 700-711, 1999.
[2] S. Angenent, S. Haker, A. Tannenbaum, and R. Kikinis, “On Area Preserving Maps of Minimal Distortion,” System Theory: Modeling, Analysis, and Control, T. Djaferis and I. Schick, eds., pp. 275-287, Holland: Kluwer, 1999.
[3] C. Bennis, J.M. Vezien, and G. Iglesias, “Piecewise Surface Flattening for Non-Distorted Texture Mapping,” Computer Graphics, vol. 25, pp. 237-247, 1991.
[4] E.A. Bier and K.R. Sloan Jr., “Two Part Texture Mappings,” IEEE Computer Graphics and Applications, vol. 6, no. 5, pp. 40-53, Sept. 1986.
[5] J.F. Blinn and M.E. Newell, "Texture and Reflection in Computer Generated Images," Comm. ACM, vol. 19, no. 10, pp. 542-547, Oct. 1976.
[6] J.F. Blinn, "Simulation of Wrinkled Surfaces," Proc. Siggraph '78, pp. 286-292, 1978.
[7] E. Catmull, “A Subdivision Algorithm for Computer Display of Curved Surfaces,” PhD thesis, Dept. of Computer Science, Univ. of Utah, Dec. 1974.
[8] C. Davatzikos and R.N. Bryan, “Using a Deformable Surface Model to Obtain a Shape Representation of the Cortex,” IEEE Trans. Medical Imaging, vol. 15, pp. 785-795, 1996.
[9] M.P. Do Carmo, Riemannian Geometry. Englewood Cliffs, N.J.: Prentice Hall, 1992.
[10] J. Dorsey, F. Sillion, and D. Greenberg, “Design and Simulation of Opera Lighting and Projection Effects,” Computer Graphics (SIGGRAPH '91 Proc.), T.W. Sederberg, ed., vol. 25, no. 4, pp. 41-50, July 1991.
[11] G. Dzuik, “Finite Elements for the Beltrami Operator on Arbitary Surfaces,” Lecture Notes in Math., vol. 1,357, pp. 142-155, 1988.
[12] M. Ech, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle, "Multiresolution Analysis of Arbitrary Meshes," Computer Graphics Proc. Ann. Conf. Series (Proc. Siggraph '95), pp. 173-182, 1995.
[13] H. Farkas and I. Kra, Riemann Surfaces. New York: Springer-Verlag, 1991.
[14] J. Foley, A. van Dam, S. Feiner, and J. Hughes, Computer Graphics. Reading, Mass.: Addison-Wesley, 1992.
[15] J. Gomes, L. Darsa, B. Costa, and L. Velho, Warping&Morphing of Graphical Objects. Morgan Kaufman, 1998.
[16] S. Haker, S. Angenent, A. Tannenbaum, and R. Kikinis, “On the 3D Visualization of Colon MR Images,” Proc. Int'l Soc. Computer Aided Surgery '99, 1999.
[17] P. Heckbert, "Survey of Texture Mapping," IEEE Computer Graphics and Applications, Vol. 6, No. 11, Nov. 1986, pp. 56-67.
[18] P.S. Heckbert, “Fundamentals of Texture Mapping and Image Warping,” masters thesis, Dept. of Electrical Eng. and Computer Science, Univ. of California, Berkeley, June 1989.
[19] T. Hughes, The Finite Element Method. Englewood Cliffs, N.J.: Prentice Hall, 1987.
[20] T. Kapur, W. Grimson, W. Wells III, and R. Kikinis, “Segmentation of Brain Tissue from Magnetic Resonance Images,” Medical Image Analysis, vol. 1, pp. 109-127, 1996.
[21] S. Kichenasamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi, “Conformal Curvature Flows: From Phase Transitions to Active Contours,” Archive Rational Mechanics and Analysis, vol. 134, pp. 275-301, 1996.
[22] A.W.F. Lee, W. Sweldens, P. Schröder, L. Cowsar, and D. Dobkin, “MAPS: Multiresolution Adaptive Parameterization of Surfaces,” Computer Graphics (SIGGRAPH '98 Proc.), M. Cohen, ed., vol. 32, pp. 95-104, July 1998.
[23] B. Lévy and J.L. Mallet, “Non-Distorted Texture Mapping for Sheared Triangulated Meshes,” Computer Graphics (SIGGRAPH '98 Proc.), M. Cohen, ed., vol. 32, pp. 343-352, July 1998.
[24] T. McReynolds, “Advanced Graphics Programming Techniques Using OpenGL,” SIGGRAPH '98 Course Notes, pp. 90-99, 1998.
[25] J. Rauch, Partial Differential Equations. New York: Springer-Verlag, 1991.
[26] M. Segal, C. Korobkin, R. van Widenfelt, J. Foran, and P.E. Haeberli, “Fast Shadows and Lighting Effects Using Texture Mapping,” Computer Graphics (SIGGRAPH '92 Proc.), E.E. Catmull, ed., vol. 26, pp. 249-252, July 1992.
[27] P. Teo, G. Sapiro, and B.A. Wandell, “Creating Connected Representations of Cortical Gray Matter for Functional MRI Visualization,” IEEE Trans. Medical Imaging, vol. 17, 1998.
[28] B. Wandell, S. Engel, and H. Hel-Or, “Creating Images of the Flattened Cortical Sheet,” Investments in Opthalmology and Visual Science, vol. 36,S612, 1996.
[29] A. Watt and M. Watt, Advanced Animation and Rendering Techniques. Reading, Mass.: ACM Press and Addison-Wesley, pp. 119-124, 1992.

Index Terms:
Texture mapping, surface parametrization, conformal geometry, finite elements, partial differential equations.
Citation:
Steven Haker, Sigurd Angenent, Allen Tannenbaum, Ron Kikinis, Guillermo Sapiro, Michael Halle, "Conformal Surface Parameterization for Texture Mapping," IEEE Transactions on Visualization and Computer Graphics, vol. 6, no. 2, pp. 181-189, April-June 2000, doi:10.1109/2945.856998
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