Abstract
We present a technique for texture mapping arbitrary sphere-like surfaces with minimal distortions by spherical embedding. The embedding is computed using spherical multi-dimensional scaling (MDS). MDS is a family of methods that map a set of points into a finite dimensional domain by minimizing the difference in distances between every pair of points in the original and the new embedding domains. In this paper spherical embedding is derived using geodesic distances on triangulated domains, computed by the fast marching method. The MDS is formulated as a non-linear optimization problem and a fast multi-resolution solution is derived. Finally, we show that the embedding of complex objects which are not sphere-like, can be improved by defining a texture dependent scale factor. This scale is the maximal distance to be preserved by the embedding and can be estimated using spherical harmonics. Experimental results show the benefits of the proposed approach.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Arad, M.: Isometric texture mapping for free-form surfaces. Computer Graphics Forum 16, 247–256 (1997), ISSN 1067-7055
Azariadis, P.N., Aspragathos, N.A.: On using planar developments to perform texture mapping on arbitrarily curved surfaces. Computers and Graphics 24, 539–554 (2000)
Neyret, F., Cani, M.P.: Pattern-based texturing revisited. In: SIGGRAPH, pp. 235–242 (1999)
Praun, E., Finkelstein, A., Hoppe, H.: Lapped textures. In: Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pp. 465–470. ACM Press/Addison-Wesley Publishing Co. (2000)
Gu, X., Wang, Y., Chan, T.F., Thompson, P.M., Yau, S.T.: Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Transaction on Medical Imaging 23 (2004)
Haker, S., Angenent, S., Tannenbaum, A., Kikinis, R., Sapiro, G., Halle, M.: Conformal surface parameterization for texture mapping. IEEE Trans. on Visualization and Computer Graphics 6, 181–189 (2000)
Sheffer, A., de Sturler, E.: Parameterization of faceted surfaces for meshing using angle based flattening. Engineering with Computers 17, 326–337 (2001)
Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., Stuetzle, W.: Multiresolution analysis of arbitrary meshes. Computer Graphics 29, 173–182 (1995)
Maillot, J., Yahia, H., Verroust, A.: Interactive texture mapping. In: Proceedings of the 20th annual conference on Computer graphics and interactive techniques, pp. 27–34. ACM Press, New York (1993)
Sheffer, A.: Spanning tree seams for reducing parameterization distortion of triangulated surfaces. In: Proceedings of the Shape Modeling International 2002 (SMI 2002), p. 61. IEEE Computer Society, Los Alamitos (2002)
Wolfson, E., Schwartz, E.L.: Computing minimal distances on polyhedral surfaces 11, 1001–1005 (1989)
Schwartz, E.L., Shaw, A., Wolfson, E.: A numerical solution to the generalized mapmaker’s problem: Flattening nonconvex polyhedral surfaces 11, 1005–1008 (1989)
Grossman, R., Kiryati, N., Kimmel, R.: Computational surface flattening: A voxel-based approach 24, 433–441 (2002)
Zigelman, G., Kimmel, R., Kiryati, N.: Texture mapping using surface flattening via MDS. IEEE Trans. on Visualization and Computer Graphics 8, 198–207 (2002)
Wandell, B.A., Chial, S., Backus, B.: Visualization and measurements of the cortical surface. Journal of Cognitive Neuroscience (2000)
Cox, M., Cox, T.: Multidimensional Scaling. Chapman and Hall, Boca Raton (1994)
Borg, I., Groenen, P.: Modern Multidimensional Scaling - Theory and Applications. Springer, Heidelberg (1997)
Kruskal, J.B., Wish, M.: Multidimensional Scaling. Sage, Thousand Oaks (1978)
Kimmel, R., Sethian, J.: Computing geodesic paths on manifolds. Proc. of National Academy of Science 95, 8431–8435 (1998)
Elad, A., Kimmel, R.: Spherical flattening of the cortex surface. In: Malladi, R. (ed.) Geometric Methods in Biomedical Image Processing, pp. 77–90. Springer, Heidelberg (2002)
Hastie, T., Ribshirani, R., Friedman, J.H.: The elements of statistical learning: data mining, inference and prediction. Springer, Heidelberg (2002)
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000)
Sethian, J.: A review of the theory, algorithms, and applications of level set method for propagating surfaces. In: Acta Numerica. Cambridge University Press, Cambridge (1996)
Gill, P.: Practical Optimization. Academic Press, London (1982)
Mann, S., Picard, R.: Virtual bellows: constructing high quality stills from video. In: IEEE International Conference Image Processing, Austin, TX, pp. 363–367 (1994)
Melax, S.: A simple, fast and effective polygon reduction algorithm. Game Developer Journal (1998)
Schroeder, W., Martin, K., Lorensen, B.: The Visualization Toolkit: An Object-Oriented Approach to 3D Graphics. Prentice-Hall, Englewood Cliffs (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Elad, A., Keller, Y., Kimmel, R. (2005). Texture Mapping via Spherical Multi-dimensional Scaling. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds) Scale Space and PDE Methods in Computer Vision. Scale-Space 2005. Lecture Notes in Computer Science, vol 3459. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11408031_38
Download citation
DOI: https://doi.org/10.1007/11408031_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25547-5
Online ISBN: 978-3-540-32012-8
eBook Packages: Computer ScienceComputer Science (R0)