
This Article  
 
Share  
Bibliographic References  
Add to:  
Digg Furl Spurl Blink Simpy Del.icio.us Y!MyWeb  
Search  
 
ASCII Text  x  
R. Grossmann, N. Kiryati, R. Kimmel, "Computational Surface Flattening: A VoxelBased Approach," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 24, no. 4, pp. 433441, April, 2002.  
BibTex  x  
@article{ 10.1109/34.993552, author = {R. Grossmann and N. Kiryati and R. Kimmel}, title = {Computational Surface Flattening: A VoxelBased Approach}, journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence}, volume = {24}, number = {4}, issn = {01628828}, year = {2002}, pages = {433441}, doi = {http://doi.ieeecomputersociety.org/10.1109/34.993552}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Pattern Analysis and Machine Intelligence TI  Computational Surface Flattening: A VoxelBased Approach IS  4 SN  01628828 SP433 EP441 EPD  433441 A1  R. Grossmann, A1  N. Kiryati, A1  R. Kimmel, PY  2002 KW  surface flattening KW  geodesic distance estimation KW  multidimensional scaling KW  voxel representation KW  texture mapping VL  24 JA  IEEE Transactions on Pattern Analysis and Machine Intelligence ER   
A voxelbased method for flattening a surface in 3D space into 2D while best preserving distances is presented. Triangulation or polyhedral approximation of the voxel data are not required. The problem is divided into two main parts: Voxelbased calculation of the minimal geodesic distances between points on the surface and finding a configuration of points in 2D that has Euclidean distances as close as possible to these distances. The method suggested combines an efficient voxelbased hybrid distance estimation method, that takes the continuity of the underlying surface into account, with classical multidimensional scaling (MDS) for finding the 2D point configuration. The proposed algorithm is efficient, simple, and can be applied to surfaces that are not functions. Experimental results are shown.
[1] S. Angenent, S. Haker, A. Tanenbuam, and R. Kikinis, “Conformal Geometry and Brain Flattening,” Proc. Second Int'l Conf. Medical Image Computing and ComputerAssisted Intervention (MICCAI '99), pp. 269278, 1999.
[2] I. Borg and P. Groenen, Modern Multidimensional Scaling: Theory and Applications. Springer, 1997.
[3] C. Bennis, J.M. Vezien, and G. Iglesias, “Piecewise Surface Flattening for NonDistorted Texture Mapping,” Computer Graphics, vol. 25, pp. 237247, 1991.
[4] F. Critchley, “On Certain Linear Mappings Between InnerProduct and SquaredDistance Matrices,” Linear Algebra and Its Applications, vol. 105, pp. 91107, 1988.
[5] A.M. Dale, B. Fischl, and M.I. Sereno, “Cortical SurfaceBased Analysis: 1. Segmentation and Surface Reconstruction,” NeuroImage, vol. 9, pp. 179194, 1999.
[6] A.M. Dale, B. Fischl, and M.I. Sereno, “Cortical SurfaceBased Analysis: 2. Cortical Surface Based Analysis,” NeuroImage, vol. 9, pp. 195207, 1999.
[7] M.P Do Carmo, Differential Geometry of Curves and Surfaces. PrenticeHall, 1976.
[8] H.A. Drury, D.C. Van Essen, C.A. Anderson, C.W. Lee, T.A. Coogan, and J.W. Lewis, “Computerized Mappings of the Cerebral Cortex: A Multiresolution Flattening Method and a SurfaceBased Coordinate System,” J. Cognitive Neuroscience, vol. 8, pp. 128, 1996.
[9] N. Dyn, “Interpolation of Scattered Data by Radial Functions,” Topics in Multivariate Approximations, C.K. Chui, L.L. Schumaker, and F.I. Utreras, eds., pp. 4762, Academic Press, 1987.
[10] N. Dyn, “Interpolation and Approximation by Radial and Related Functions,” Approximation Theory VI, vol. 1, pp. 211234, 1989.
[11] J. Gomes and O. Faugeras, “Segmentation of the Inner and Outer Surfaces of the Cortex in Man and Monkey: an Approach Based on Partial Differential Equations,” Proc. Human Brain Mapping Conf., June 1999.
[12] G.H Golub and C.F. Van Loan, Matrix Computations. Baltimore: The Johns Hopkins Univ. Press, 1989.
[13] M. Gondran and M. Minoux, Graphs and Algorithms, chapter 2. Chichester: Wiley, 1984.
[14] R. Grossmann, N. Kiryati, and R. Kimmel, “Computational Surface Flattening: A VoxelBased Approach,” Proc. Fourth Int'l Workshop Visual Form, pp. 196204, May 2001.
[15] S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro, and M. Halle, “Conformal Surface Parameterization for Texture Mapping,” IEEE Trans. Visualization and Computer Graphics, vol. 6, no. 2, 2000.
[16] G. Hermosillo, O. Faugeras, and J. Gomes, “Unfolding the Cerebral Cortex Using Level Set Methods,” Proc. Conf. ScaleSpace '99, pp. 5869, 1999.
[17] A. Jonas and N. Kiryati, “Digital Representation Schemes for 3D Curves,” Pattern Recognition, vol. 30, pp. 18031816, 1997.
[18] A. Jonas and N. Kiryati, “Length Estimation in 3D Using Cube Quantization,” J. Mathematical Imaging and Vision, vol. 8, pp. 215238, 1998.
[19] N. Kiryati and O. Kübler, “Chain Code Probabilities and Optimal Length Estimators for Digitized ThreeDimensional Curves,” Pattern Recognition, vol. 28, pp. 361372, 1995.
[20] R. Kimmel and J.A. Sethian, “Computing Geodesic Paths on Manifolds,” Proc. Nat'l Academy of Sciences, vol. 95, pp. 84318435, 1998.
[21] R. Kimmel and J.A. Sethian, “Fast Voronoi Diagrams and Offsets on Triangulated Surfaces,” Curve and Surface Design: SaintMalo 1999, Vanderbilt Univ. Press, 1999.
[22] N. Kiryati and G. Székely, “Estimating Shortest Paths and Minimal Distances on Digitized ThreeDimensional Surfaces,” Pattern Recognition, vol. 26, pp. 16231637, 1993.
[23] W.E. Lorensen and H.E. Cline, “Marching Cubes: A High Resolution 3D Surface Construction Algorithm,” Computer Graphics (SIGGRAPH '87 Proc.), vol. 21, pp. 163169, 1987.
[24] S.D. Ma and H. Lin, “Optimal Texture Mapping,” Eurographics, pp. 421428, 1988.
[25] J.S.B. Mitchell, D.M. Mount, and C.H. Papadimitriou, “The Discrete Geodesic Problem,” SIAM J. Computing, vol. 16, pp. 647668, 1987.
[26] B. O'Neill, Elementary Differential Geometry. Academic Press, 1966.
[27] S.T. Roweis and L.K. Saul, “Nonlinear Dimensionality Reduction by Locally Linear Embedding,” Science, vol. 290, pp. 23232326, 2000.
[28] E.L. Schwartz, A. Shaw, and E. Wolfson, “A Numerical Solution to the Generalized Mapmaker's Problem: Flattening Nonconvex Polyhedral Surfaces,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, pp. 10051008, 1989.
[29] R. Sedgewick, Algorithms, chapters 11 and 31. Reading, Mass.: AddisonWesley, 1988.
[30] J. P. Snyder, Flattening the Earth: Two Thousand Years of Map Projections. The Univ. of Chicago Press, 1993.
[31] J.B. Tenenbaum, V. de Silva, and J.C. Langford, “A Global Geometric Framework for Nonlinear Dimensionality Reduction,” Science, vol. 290, pp. 23192323, 2000.
[32] E. Wolfson and E.L. Schwartz, “Computing Minimal Distances on Polyhedral Surfaces,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, pp. 10011005, 1989.
[33] G. Zigelman, R. Kimmel, and N. Kiryati, “Texture Mapping Using Surface Flattening via MultiDimensional Scaling,” IEEE Trans. Visualization and Computer Graphics, to appear.