research-article

Mesh parameterization: theory and practice

Authors:

Konrad Polthier,

Alia ShefferAuthors Info & Claims

SIGGRAPH Asia '08: ACM SIGGRAPH ASIA 2008 courses

Article No.: 12, Pages 1 - 87

https://doi.org/10.1145/1508044.1508091

Published: 10 December 2008 Publication History

Abstract

Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and inter-surface mapping, and demonstrates a variety of parameterization applications.

References

[1]

J. H. Ahlberg, E. N. Nilson, and J. L. Walsh. The theory of splines and their applications. Academic Press, New York, 1967.

[2]

B. Aksoylu, A. Khodakovsky, and P. Schröder. Multilevel solvers for unstructured surface meshes. SIAM Journal on Scientific Computing, 26(4):1146--1165, 2005.

Digital Library

[3]

M. Alexa. Merging polyhedral shapes with scattered features. The Visual Computer, 16 (1):26--37, 2000.

Digital Library

[4]

M. Alexa. Recent advances in mesh morphing. Computer Graphics Forum, 21(2):173--196, 2002.

[5]

B. Allen, B. Curless, and Z. Popović. The space of human body shapes: reconstruction and parameterization from range scans. ACM Transactions on Graphics, 22(3):587--594, 2003. Proceedings of SIGGRAPH 2003.

Digital Library

[6]

P. Alliez and C. Gotsman. Recent advances in compression of 3D meshes. In N. A. Dodgson, M. S. Floater, and M. A. Sabin, editors, Advances in Multiresolution for Geometric Modelling, Mathematics and Visualization, pages 3--26. Springer, Berlin, Heidelberg, 2005.

[7]

P. Alliez, D. Cohen-Steiner, O. Devillers, B. Lévy, and M. Desbrun. Anisotropic polygonal remeshing. ACM Transactions on Graphics, 22(3):485--493, 2003. Proceedings of SIGGRAPH 2003.

Digital Library

[8]

P. Alliez, M. Attene, C. Gotsman, and G. Ucelli. Recent advances in remeshing of surfaces. In L. De Floriani and M. Spagnuolo, editors, Shape Analysis and Structuring, Mathematics and Visualization, pages 53--82. Springer, Berlin, Heidelberg, 2008.

[9]

D. Anguelov, P. Srinivasan, D. Koller, S. Thrun, J. Rodgers, and J. Davis. SCAPE: Shape completion and animation of people. ACM Transactions on Graphics, 24(3): 408--416, 2005. Proceedings of SIGGRAPH 2005.

Digital Library

[10]

N. Arad and G. Elber. Isometric texture mapping for free-form surfaces. Computer Graphics Forum, 16(5):247--256, 1997.

[11]

P. N. Azariadis and N. A. Aspragathos. On using planar developments to perform texture mapping on arbitrarily curved surfaces. Computers & Graphics, 24(4):539--554, 2000.

[12]

L. Balmelli, G. Taubin, and F. Bernardini. Space-optimized texture maps. Computer Graphics Forum, 21(3):411--420, 2002. Proceedings of Eurographics 2002.

[13]

A. Belyaev. On transfinite barycentric coordinates. In Proceedings of the 4th Eurographics Symposium on Geometry Processing (SGP 2006), pages 89--99. Eurographics Association, 2006.

Digital Library

[14]

M. Ben-Chen, C. Gotsman, and G. Bunin. Conformal flattening by curvature prescription and metric scaling. Computer Graphics Forum, 27(2):449--458, 2008. Proceedings of Eurographics 2008.

[15]

C. Bennis, J.-M. Vezien, and G. Iglésias. Piecewise surface flattening for non-distorted texture mapping. ACM SIGGRAPH Computer Graphics, 25(4):237--246, 1991. Proceedings of SIGGRAPH '91.

Digital Library

[16]

E. Bier and K. Sloan. Two-part texture mappings. IEEE Computer Graphics and Applications, 6(9):40--53, 1986.

Digital Library

[17]

H. Biermann, I. Martin, F. Bernardini, and D. Zorin. Cut-and-paste editing of multiresolution surfaces. ACM Transactions on Graphics, 21(3):312--321, 2002. Proceedings of SIGGRAPH 2002.

Digital Library

[18]

H. Birkholz. Shape-preserving parametrization of genus 0 surfaces. In Proceedings of the 12th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision (WSCG 2004), pages 57--64, 2004.

[19]

V. Blanz and T. Vetter. A morphable model for the synthesis of 3D faces. In Proceedings of SIGGRAPH '99, pages 187--194. ACM Press, 1999.

Digital Library

[20]

V. Blanz, C. Basso, T. Poggio, and T. Vetter. Reanimating faces in images and video. Computer Graphics Forum, 22(3):641--650, 2003. Proceedings of Eurographics 2003.

[21]

V. Blanz, K. Scherbaum, T. Vetter, and H.-P. Seidel. Exchanging faces in images. Computer Graphics Forum, 23(3):669--676, 2004. Proceedings of Eurographics 2004.

[22]

J. F. Blinn. Simulation of wrinkled surfaces. ACM SIGGRAPH Computer Graphics, 12 (3):286--292, 1978. Proceedings of SIGGRAPH '78.

Digital Library

[23]

T. Bobach, G. Farin, D. Hansford, and G. Umlauf. Discrete harmonic functions from local coordinates. In R. Martin, M. Sabin, and J. Winkler, editors, Mathematics of Surfaces XII, volume 4647 of Lecture Notes in Computer Science, pages 93--103. Springer, 2007. Proceedings of the 12th IMA International Conference.

Digital Library

[24]

A. I. Bobenko and B. A. Springborn. Variational principles for circle patterns and koebe's theorem. Transactions of the American Mathematical Society, 356(2):659--689, 2004.

[25]

I. Boier-Martin, H. Rushmeier, and J. Jin. Parameterization of triangle meshes over quadrilateral domains. In Proceedings of the Second Eurographics Symposium on Geometry Processing (SGP 2004), pages 193--203. Eurographics Association, 2004.

Digital Library

[26]

M. Botsch, D. Bommes, and L. Kobbelt. Efficient linear system solvers for mesh processing. In Mathematics of Surfaces XI, volume 3604 of Lecture Notes in Computer Science, pages 62--83. Springer, Berlin, Heidelberg, 2005. Proceedings of the 11th IMA International Conference.

Digital Library

[27]

P. L. Bowers and M. K. Hurdal. Planar conformal mappings of piecewise flat surfaces. In H.-C. Hege and K. Polthier, editors, Visualization and Mathematics III, pages 3--34. Springer, Heidelberg, 2003.

[28]

S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer, second edition, 2002.

[29]

S. Campagna and H.-P. Seidel. Parameterizing meshes with arbitrary topology. In Proceedings of Image and Multidimensional Digital Signal Processing '98, pages 287--290, 1998.

[30]

N. A. Carr and J. C. Hart. Meshed atlases for real-time procedural solid texturing. ACM Transactions on Graphics, 21(2):106--131, 2002.

Digital Library

[31]

N. A. Carr and J. C. Hart. Painting detail. ACM Transactions on Graphics, 23(3): 587--594, 2004. Proceedings of SIGGRAPH 2004.

Digital Library

[32]

N. A. Carr, J. Hoberock, K. Crane, and J. C. Hart. Rectangular multi-chart geometry images. In Proceedings of the 4th Eurographics Symposium on Geometry Processing (SGP 2006), pages 181--190. Eurographics Association, 2006.

Digital Library

[33]

Z. Chen, L. Liu, Z. Zhang, and G. Wang. Surface parameterization via aligning optimal local flattening. In Proceedings of the 2007 Symposium on Solid and Physical Modeling (SPM '07), pages 291--296. ACM Press, 2007.

Digital Library

[34]

G. Choquet. Sur un type de transformation analytique généralisant la représentation conforme et définé au moyen de fonctions harmoniques. Bulletin des Sciences Mathématiques, 69:156--165, 1945.

[35]

B. Chow and F. Luo. Combinatorial ricci flows on surfaces. Journal of Differential Geometry, 63(1):97--129, 2003.

[36]

F. R. K. Chung. Spectral Graph Theory. Number 92 in CBMS Regional Conference Series in Mathematics. American Mathematical Society, 1997.

[37]

B. Cipra. You can't always hear the shape of a drum. In What's Happening in the Mathematical Sciences, volume 1. American Mathematical Society, Providence, RI, 1993.

[38]

U. Clarenz, N. Litke, and M. Rumpf. Axioms and variational problems in surface parameterization. Computer Aided Geometric Design, 21(8):727--749, 2004.

Digital Library

[39]

D. Cohen-Steiner, P. Alliez, and M. Desbrun. Variational shape approximation. ACM Transactions on Graphics, 23(3):905--914, 2004. Proceedings of SIGGRAPH 2004.

Digital Library

[40]

C. R. Collins and K. Stephenson. A circle packing algorithm. Computational Geometry: Theory and Applications, 25(3):233--256, 2003.

Digital Library

[41]

R. Courant. Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces. Inter-science, New York, 1950.

[42]

Y. C. de Verdière. On a new graph invariant and a criterion for planarity. In Graph Structure Theory, volume 147 of Contemporary Mathematics, pages 137--148. American Mathematical Society, 1993.

[43]

P. Degener, J. Meseth, and R. Klein. An adaptable surface parameterization method. In Proceedings of the 12th International Meshing Roundtable (IMR 2003), pages 201--213, 2003.

[44]

M. Desbrun, M. Meyer, and P. Alliez. Intrinsic parameterizations of surface meshes. Computer Graphics Forum, 21(3):209--218, 2002. Proceedings of Eurographics 2002.

[45]

M. do Carmo. Differential geometry of curves and surfaces. Prentice Hall, 1976.

[46]

S. Dong and M. Garland. Iterative methods for improving mesh parameterizations. In Proceedings of Shape Modeling International (SMI '07), pages 199--210. IEEE Computer Society, 2007.

Digital Library

[47]

S. Dong, P.-T. Bremer, M. Garland, V. Pascucci, and J. C. Hart. Quadrangulating a mesh using Laplacian eigenvectors. Technical Report UIUCDCS-R-2005-2583, University of Illinois, June 2005a.

[48]

S. Dong, S. Kircher, and M. Garland. Harmonic functions for quadrilateral remeshing of arbitrary manifolds. Comp. Aided Design, 22(4):392--423, 2005b.

Digital Library

[49]

S. Dong, P.-T. Bremer, M. Garland, V. Pascucci, and J. C. Hart. Spectral surface quadrangulation. ACM Transactions on Graphics, 25(3): 1057--1066, 2006. Proceedings of SIGGRAPH 2006.

Digital Library

[50]

J. Douglas. Solution of the problem of Plateau. Transactions of the American Mathematical Society, 33(1):263--321, 1931.

[51]

T. Duchamp, A. Certain, T. DeRose, and W. Stuetzle. Hierarchical computation of PL harmonic embeddings. Technical report, University of Washington, July 1997.

[52]

M. Eck, T. D. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle. Multiresolution analysis of arbitrary meshes. In Proceedings of SIGGRAPH '95, pages 173--182. ACM Press, 1995.

Digital Library

[53]

I. Eckstein, V. Surazhsky, and C. Gotsman. Texture mapping with hard constraints. Computer Graphics Forum, 20(3):95--104, 2001. Proceedings of Eurographics 2001.

[54]

J. Erickson and S. Har-Peled. Optimally cutting a surface into a disk. Discrete & Computational Geometry, 31(1):37--59, 2004.

Digital Library

[55]

J. Erickson and K. Whittlesey. Greedy optimal homotopy and homology generators. In Proceedings of the 16th Symposium on Discrete Algorithms (SODA '05), pages 1038--1046. SIAM, 2005.

Digital Library

[56]

G. Farin. Surfaces over Dirichlet tessellations. Computer Aided Geometric Design, 7 (1--4):281--292, 1990.

Digital Library

[57]

H. M. Farkas and I. Kra. Riemann Surfaces. Springer Verlag, 1980.

[58]

H. Ferguson and A. Rockwood. Multiperiodic functions for surface design. Computer Aided Geometric Design, 10(3--4):315--328, 1993.

Digital Library

[59]

M. Fiedler. Algebraic connectivity of graphs. Czechoslovak Mathematical Journal, 23: 298--305, 1973.

[60]

M. Fiedler. A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czechoslovak Mathematical Journal, 25:619--633, 1975.

[61]

E. Fiume, A. Fournier, and V. Canale. Conformal texture mapping. In Proceedings of Eurographics '87, pages 53--64, 1987.

[62]

M. S. Floater. Parameterization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14(3):231--250, 1997.

Digital Library

[63]

M. S. Floater. Parametric tilings and scattered data approximation. International Journal of Shape Modeling, 4(3--4):165--182, 1998.

[64]

M. S. Floater. Meshless parameterization and B-spline surface approximation. In R. Cipolla and R. Martin, editors, The Mathematics of Surfaces IX, pages 1--18, London, 2000. Springer.

Digital Library

[65]

M. S. Floater. Convex combination maps. In J. Levesley, I. J. Anderson, and J. C. Mason, editors, Algorithms for Approximation IV, pages 18--23, 2002.

[66]

M. S. Floater. Mean value coordinates. Computer Aided Geometric Design, 20(1):19--27, 2003a.

Digital Library

[67]

M. S. Floater. One-to-one piecewise linear mappings over triangulations. Mathematics of Computation, 72(242):685--696, 2003b.

Digital Library

[68]

M. S. Floater and K. Hormann. Parameterization of triangulations and unorganized points. In A. Iske, E. Quak, and M. S. Floater, editors, Tutorials on Multiresolution in Geometric Modelling, Mathematics and Visualization, pages 287--316. Springer, Berlin, Heidelberg, 2002.

[69]

M. S. Floater and K. Hormann. Surface parameterization: a tutorial and survey. In N. A. Dodgson, M. S. Floater, and M. A. Sabin, editors, Advances in Multiresolution for Geometric Modelling, Mathematics and Visualization, pages 157--186. Springer, Berlin, Heidelberg, 2005.

[70]

M. S. Floater and M. Reimers. Meshless parameterization and surface reconstruction. Computer Aided Geometric Design, 18(2):77--92, 2001.

Digital Library

[71]

M. S. Floater, K. Hormann, and M. Reimers. Parameterization of manifold triangulations. In C. K. Chui, L. L. Schumaker, and J. Stöckler, editors, Approximation Theory X: Abstract and Classical Analysis, Innovations in Applied Mathematics, pages 197--209. Vanderbilt University Press, Nashville, TN, 2002.

[72]

M. S. Floater, G. Kós, and M. Reimers. Mean value coordinates in 3D. Computer Aided Geometric Design, 22(7):623--631, 2005.

Digital Library

[73]

M. S. Floater, K. Hormann, and G. Kós. A general construction of barycentric coordinates over convex polygons. Advances in Computational Mathematics, 24(1--4): 311--331, 2006.

[74]

J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes. Computer Graphics: Principles and Practice. Addison-Wesley, Boston, second edition, 1995.

Digital Library

[75]

I. Friedel, P. Schröder, and M. Desbrun. Unconstrained spherical parameterization. Journal of Graphics Tools, 12(1):17--26, 2007.

[76]

W. Fulton. Algebraic Topology, A first course. Springer Verlag, 1995.

[77]

V. A. Garanzha. Maximum norm optimization of quasi-isometric mappings. Numerical Linear Algebra with Applications, 9(6, 7):493--510, 2002.

[78]

M. Garland, A. Willmott, and P. S. Heckbert. Hierarchical face clustering on polygonal surfaces. In Proceedings of the 2001 Symposium on Interactive 3D Graphics (SI3D '01), pages 49--58. ACM Press, 2001.

Digital Library

[79]

C. F. Gauß. Disquisitiones generales circa superficies curvas. Commentationes Societatis Regi&oelig; Scientiarum Gottingensis Recentiores, 6:99--146, 1827.

[80]

W. J. Gordon and J. A. Wixom. Pseudo-harmonic interpolation on convex domains. SIAM Journal on Numerical Analysis, 11(5):909--933, 1974.

Digital Library

[81]

S. J. Gortler, C. Gotsman, and D. Thurston. Discrete one-forms on meshes and applications to 3D mesh parameterization. Computer Aided Geometric Design, 23(2):83--112, 2006.

Digital Library

[82]

C. Gotsman. On graph partitioning, spectral analysis, and digital mesh processing. In Proceedings of Shape Modeling International (SMI '03), pages 165--171. IEEE Computer Society, 2003.

Digital Library

[83]

C. Gotsman, X. Gu, and A. Sheffer. Fundamentals of spherical parameterization for 3D meshes. ACM Transactions on Graphics, 22(3):358--363, 2003. Proceedings of SIGGRAPH 2003.

Digital Library

[84]

G. Greiner. Variational design and fairing of spline surfaces. Computer Graphics Forum, 13(3):143--154, 1994. Proceedings of Eurographics '94.

[85]

G. Greiner and K. Hormann. Interpolating and approximating scattered 3D-data with hierarchical tensor product B-splines. In A. L. Méhauté, C. Rabut, and L. L. Schumaker, editors, Surface Fitting and Multiresolution Methods, Innovations in Applied Mathematics, pages 163--172. Vanderbilt University Press, Nashville, 1997.

[86]

C. M. Grimm and J. F. Hughes. Modeling surfaces of arbitrary topology using manifolds. In Proceedings of SIGGRAPH '95, pages 359--368. ACM Press, 1995.

Digital Library

[87]

C. M. Grimm and J. F. Hughes. Parameterizing N-holed tori. In Mathematics of Surfaces, volume 2768 of Lecture Notes in Computer Science, pages 14--29. Springer, Berlin, Heidelberg, 2003. Proceedings of the 10th IMA International Conference.

[88]

L. Gross and G. Farin. A transfinite form of Sibson's interpolant. Discrete Applied Mathematics, 93(1):33--50, 1999.

Digital Library

[89]

X. Gu and S.-T. Yau. Computing conformal structures of surfaces. Communications in Information and Systems, 2(2):121--146, 2002.

[90]

X. Gu and S.-T. Yau. Global conformal surface parameterization. In Proceedings of the First Eurographics Symposium on Geometry Processing (SGP 2003), pages 127--137. Eurographics Association, 2003.

Digital Library

[91]

X. Gu, S. J. Gortler, and H. Hoppe. Geometry images. ACM Transactions on Graphics, 21(3):355--361, 2002. Proceedings of SIGGRAPH 2002.

Digital Library

[92]

X. Gu, Y. Wang, and S.-T. Yau. Computing conformal invariants: Period matrices. Communications in Information and Systems, 3(3):153--170, 2003a.

[93]

X. Gu, Y. Wang, and S.-T. Yau. Geometric compression using Riemann surface structure. Communications in Information and Systems, 3(3): 171--182, 2003b.

[94]

X. Gu, Y. Wang, T. F. Chan, P. M. Thompson, and S.-T. Yau. Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Transaction on Medical Imaging, 23(8):949--958, 2004a.

[95]

X. Gu, Y. Wang, and S.-T. Yau. Multiresolution computation of conformal structures of surfaces. Journal of Systemics, Cybernetics and Informatics, 1(5):45--50, 2004b.

[96]

I. Guskov. An anisotropic mesh parameterization scheme. Engineering with Computers, 20(2):129--135, 2004.

Digital Library

[97]

I. Guskov, W. Sweldens, and P. Schröder. Multiresolution signal processing for meshes. In Proceedings of SIGGRAPH '99, pages 325--334. ACM Press, 1999.

Digital Library

[98]

I. Guskov, K. Vidimče, W. Sweldens, and P. Schröder. Normal meshes. In Proceedings of SIGGRAPH 2000, pages 95--102. ACM Press, 2000.

Digital Library

[99]

I. Guskov, A. Khodakovsky, P. Schröder, and W. Sweldens. Hybrid meshes: Multiresolution using regular and irregular refinement. In Proceedings of the 18th Annual Symposium on Computational Geometry (SCG '02), pages 264--272. ACM Press, 2002.

Digital Library

[100]

S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro, and M. Halle. Conformal surface parameterization for texture mapping. IEEE Transactions on Visualization and Computer Graphics, 6(2): 181--189, 2000.

Digital Library

[101]

Y. He, X. Gu, and H. Qin. Rational spherical splines for genus zero shape modeling. In Proceedings of the 2005 International Conference on Shape Modeling and Applications (SMI '05), pages 82--91. IEEE Computer Society, 2005.

Digital Library

[102]

A. Hertzmann and D. Zorin. Illustrating smooth surfaces. In ACM SIGGRAPH, pages 517--526, 2000.

Digital Library

[103]

A. Hirani. Discrete Exterior Calculus. PhD thesis, California Institute of Technology, Pasadena, CA, 2003.

Digital Library

[104]

H. Hiyoshi and K. Sugihara. Voronoi-based interpolation with higher continuity. In Proceedings of the 16th Annual Symposium on Computational Geometry (SCG '00), pages 242--250. ACM Press, 2000.

Digital Library

[105]

H. Hoppe. Progressive meshes. In Proceedings of SIGGRAPH '96, pages 99--108. ACM Press, 1996.

Digital Library

[106]

H. Hoppe and E. Praun. Shape compression using spherical geometry images. In N. A. Dodgson, M. S. Floater, and M. A. Sabin, editors, Advances in Multiresolution for Geometric Modelling, Mathematics and Visualization, pages 27--46. Springer, Berlin, Heidelberg, 2005.

[107]

K. Hormann. Fitting free form surfaces. In B. Girod, G. Greiner, and H. Niemann, editors, Principles of 3D Image Analysis and Synthesis, volume 556 of The International Series in Engineering and Computer Science, chapter 4.7, pages 192--202. Kluwer Academic Publishers, Boston, MA, 2000.

[108]

K. Hormann. Theory and Applications of Parameterizing Triangulations. PhD thesis, Department of Computer Science, University of Erlangen, Nov. 2001.

[109]

K. Hormann and M. S. Floater. Mean value coordinates for arbitrary planar polygons. ACM Transactions on Graphics, 25(4):1424--1441, 2006.

Digital Library

[110]

K. Hormann and G. Greiner. MIPS: An efficient global parametrization method. In P.-J. Laurent, P. Sablonnière, and L. L. Schumaker, editors, Curve and Surface Design: Saint-Malo 1999, Innovations in Applied Mathematics, pages 153--162. Vanderbilt University Press, Nashville, TN, 2000a.

[111]

K. Hormann and G. Greiner. Quadrilateral remeshing. In B. Girod, G. Greiner, H. Niemann, and H.-P. Seidel, editors, Proceedings of Vision, Modeling, and Visualization 2000, pages 153--162, Saarbrücken, Germany, 2000b. infix.

[112]

K. Hormann and M. Reimers. Triangulating point clouds with spherical topology. In T. Lyche, M.-L. Mazure, and L. L. Schumaker, editors, Curve and Surface Design: Saint-Malo 2002, Modern Methods in Applied Mathematics, pages 215--224. Nashboro Press, Brentwood, TN, 2003.

[113]

K. Hormann and N. Sukumar. Maximum entropy coordinates for arbitrary polytopes. Computer Graphics Forum, 27(5):1513--1520, 2008. Proceedings of SGP 2008.

Digital Library

[114]

K. Hormann and M. Tarini. A quadrilateral rendering primitive. In Proceedings of the 2004 Workshop on Graphics Hardware (GH 2004), Eurographics Symposium Proceedings, pages 7--14. Eurographics Association, 2004.

Digital Library

[115]

K. Hormann, G. Greiner, and S. Campagna. Hierarchical parametrization of triangulated surfaces. In B. Girod, H. Niemann, and H.-P. Seidel, editors, Proceedings of Vision, Modeling, and Visualization 1999, pages 219--226, Erlangen, Germany, 1999. infix.

[116]

K. Hormann, U. Labsik, and G. Greiner. Remeshing triangulated surfaces with optimal parameterizations. Computer-Aided Design, 33(11):779--788, 2001.

[117]

M. K. Hurdal, P. L. Bowers, K. Stephenson, D. W. L. Sumners, K. Rehm, K. Schaper, and D. A. Rottenberg. Quasi-conformally flat mapping the human cerebellum. In Medical Image Computing and Computer-Assisted Intervention (MICCAI '99), volume 1679 of Lecture Notes in Computer Science, pages 279--286. Springer, Berlin, Heidelberg, 1999.

Digital Library

[118]

T. Igarashi and D. Cosgrove. Adaptive unwrapping for interactive texture painting. In Proceedings of the 2001 Symposium on Interactive 3D Graphics (SI3D '01), pages 209--216. ACM Press, 2001.

Digital Library

[119]

M. Isenburg and P. Lindstrom. Streaming meshes. In Proceedings of IEEE Visualization 2005, pages 231--238. IEEE Computer Society, 2005.

[120]

M. Isenburg, S. Gumhold, and C. Gotsman. Connectivity shapes. In Proceedings of IEEE Visualization 2001, pages 135--142. IEEE Computer Society, 2001.

Digital Library

[121]

M. Jin, Y. Wang, S.-T. Yau, and X. Gu. Optimal global conformal surface parameterization. In Proceedings of IEEE Visualization 2004, pages 267--274. IEEE Computer Society, 2004.

Digital Library

[122]

M. Jin, F. Luo, and X. Gu. Computing surface hyperbolic structure and real projective structure. In Proceedings of the 2006 Symposium on Solid and Physical Modeling (SPM '06), pages 105--116. ACM Press, 2006.

Digital Library

[123]

M. Jin, J. Kim, and X. Gu. Discrete surface ricci flow: Theory and applications. In R. Martin, M. Sabin, and J. Winkler, editors, Mathematics of Surfaces XII, volume 4647 of Lecture Notes in Computer Science, pages 209--232. Springer, 2007. Proceedings of the 12th IMA International Conference.

Digital Library

[124]

P. Joshi, M. Meyer, T. DeRose, B. Green, and T. Sanocki. Harmonic coordinates for character articulation. ACM Trans. Gr., 26(3):Article 71, 2007. Proceedings of ACM SIGGRAPH 2007.

Digital Library

[125]

J. Jost. Compact Riemann Surfaces. Springer, 2002.

[126]

T. Ju, S. Schaefer, and J. Warren. Mean value coordinates for closed triangular meshes. ACM Transactions on Graphics, 24(3):561--566, 2005a. Proceedings of ACM SIGGRAPH 2005.

Digital Library

[127]

T. Ju, S. Schaefer, J. Warren, and M. Desbrun. A geometric construction of coordinates for convex polyhedra using polar duals. In Proceedings of the Third Eurographics Symposium on Geometry Processing (SGP 2005), pages 181--186. Eurographics Association, 2005b.

Digital Library

[128]

T. Ju, P. Liepa, and J. Warren. A general geometric construction of coordinates in a convex simplicial polytope. Computer Aided Geometric Design, 24(3): 161--178, 2007.

Digital Library

[129]

D. Julius, V. Kraevoy, and A. Sheffer. D-charts: Quasi-developable mesh segmentation. Computer Graphics Forum, 24(3):581--590, 2005. Proceedings of Eurographics 2005.

[130]

M. Kac. Can one hear the shape of a drum? American Mathematical Monthly, 73(4), 1966.

[131]

F. Kälberer, M. Nieser, and K. Polthier. Quadcover--surface parameterization using branched coverings. Computer Graphics Forum, 26(3):375--384, 2007. Proceedings of Eurographics 2007.

[132]

Z. Karni and C. Gotsman. Spectral compression of mesh geometry. In Proceedings of SIGGRAPH 2000, pages 279--286. ACM Press, 2000.

Digital Library

[133]

Z. Karni, C. Gotsman, and S. J. Gortler. Free-boundary linear parameterization of 3D meshes in the presence of constraints. In Proceedings of the 2005 International Conference on Shape Modeling and Applications (SMI '05), pages 266--275. IEEE Computer Society, 2005.

Digital Library

[134]

L. Kharevych, B. Springborn, and P. Schröder. Discrete conformal mappings via circle patterns. ACM Transactions on Graphics, 25(2):412--438, 2006.

Digital Library

[135]

A. Khodakovsky, N. Litke, and P. Schröder. Globally smooth parameterizations with low distortion. ACM Transactions on Graphics, 22(3):350--357, 2003. Proceedings of SIGGRAPH 2003.

Digital Library

[136]

W. Klingenberg. A Course in Differential Geometry. Springer, Berlin, Heidelberg, 1978.

[137]

H. Kneser. Lösung der Aufgabe 41. Jahresbericht der Deutschen Mathematiker-Vereinigung, 35:123--124, 1926.

[138]

L. P. Kobbelt, J. Vorsatz, U. Labsik, and H.-P. Seidel. A shrink-wrapping approach to remeshing polygonal surfaces. Computer Graphics Forum, 18(3):119--130, 1999. Proceedings of Eurographics 1999.

[139]

S. Kolmanič and N. Guid. The flattening of arbitrary surfaces by approximation with developable stripes. In U. Cugini and M. J. Wozny, editors, From geometric modeling to shape modeling, volume 80 of International Federation for Information Processing, pages 35--46. Kluwer Academic Publishers, Boston, 2001.

Digital Library

[140]

Y. Koren. On spectral graph drawing. In Computing and Combinatorics, volume 2697 of Lecture Notes in Computer Science, pages 496--508. Springer, Berlin, Heidelberg, 2003. Proceedings of the 9th Annual International Conference (COCOON 2003).

Digital Library

[141]

G. Kós and T. Várady. Parameterizing complex triangular meshes. In T. Lyche, M.-L. Mazure, and L. L. Schumaker, editors, Curve and Surface Design: Saint-Malo 2002, Modern Methods in Mathematics, pages 265--274. Nashboro Press, Brentwood, TN, 2003.

[142]

V. Kraevoy and A. Sheffer. Cross-parameterization and compatible remeshing of 3D models. ACM Transactions on Graphics, 23(3):861--869, 2004. Proceedings of SIGGRAPH 2004.

Digital Library

[143]

V. Kraevoy and A. Sheffer. Template based mesh completion. In Proceedings of the Third Eurographics Symposium on Geometry Processing (SGP 2005), pages 13--22. Eurographics Association, 2005.

Digital Library

[144]

V. Kraevoy, A. Sheffer, and C. Gotsman. Matchmaker: constructing constrained texture maps. ACM Transactions on Graphics, 22(3):326--333, 2003. Proceedings of SIGGRAPH 2003.

Digital Library

[145]

E. Kreyszig. Differential Geometry. Dover, New York, 1991.

[146]

U. Labsik, K. Hormann, and G. Greiner. Using most isometric parametrizations for remeshing polygonal surfaces. In R. Martin and W. Wang, editors, Proceedings of Geometric Modeling and Processing 2000, pages 220--228, Hong Kong, China, 2000. IEEE Computer Society.

Digital Library

[147]

J. H. Lambert. Beyträge zum Gebrauche der Mathematik und deren Anwendung, volume 3. Buchhandlung der Realschule, Berlin, 1772.

[148]

T. Langer and H.-P. Seidel. Higher order barycentric coordinates. Computer Graphics Forum, 27(2):459--466, 2008. Proceedings of Eurographics 2008.

[149]

T. Langer, A. Belyaev, and H.-P. Seidel. Spherical barycentric coordinates. In Proceedings of the 4th Eurographics Symposium on Geometry Processing (SGP 2006), pages 81--88. Eurographics Association, 2006.

Digital Library

[150]

T. Langer, A. Belyaev, and H.-P. Seidel. Mean value coordinates for arbitrary spherical polygons and polyhedra in R³. In P. Chenin, T. Lyche, and L. L. Schumaker, editors, Curve and Surface Design: Avignon 2006, Modern Methods in Applied Mathematics, pages 193--202. Nashboro Press, Brentwood, TN, 2007.

[151]

F. Lazarus, M. Pocchiola, G. Vegter, and A. Verroust. Computing a canonical polygonal schema of an orientable triangulated surface. In Proceedings of the 17th Annual Symposium on Computational Geometry (SCG '01), pages 80--89. ACM Press, 2001.

Digital Library

[152]

A. Lee, H. Moreton, and H. Hoppe. Displaced subdivision surfaces. In Proceedings of SIGGRAPH 2000, pages 85--94. ACM Press, 2000.

Digital Library

[153]

A. W. F. Lee, W. Sweldens, P. Schröder, L. Cowsar, and D. Dobkin. MAPS: Multiresolution adaptive parameterization of surfaces. In Proceedings of SIGGRAPH '98, pages 95--104. ACM Press, 1998.

Digital Library

[154]

A. W. F. Lee, D. Dobkin, W. Sweldens, and P. Schröder. Multiresolution mesh morphing. In Proceedings of SIGGRAPH '99, pages 343--350. ACM Press, 1999.

Digital Library

[155]

E. T. Y. Lee. Choosing nodes in parametric curve interpolation. Computer-Aided Design, 21(6):363--370, 1989.

Digital Library

[156]

Y. Lee, H. S. Kim, and S. Lee. Mesh parameterization with a virtual boundary. Computers & Graphics, 26(5):677--686, 2002.

[157]

J. Lengyel, E. Praun, A. Finkelstein, and H. Hoppe. Real-time fur over arbitrary surfaces. In Proceedings of the 2001 Symposium on Interactive 3D Graphics (SI3D '01), pages 227--232. ACM Press, 2001.

Digital Library

[158]

B. Lévy. Constrained texture mapping for polygonal meshes. In Proceedings of SIGGRAPH 2001, pages 417--424. ACM Press, 2001.

Digital Library

[159]

B. Lévy. Dual domain extrapolation. ACM Transactions on Graphics, 22(3):364--369, 2003. Proceedings of SIGGRAPH 2003.

Digital Library

[160]

B. Lévy. Laplace-Beltrami eigenfunctions: Towards an algorithm that "understands" geometry. In Proceedings of the 2006 International Conference on Shape Modeling and Applications (SMI '06), pages 65--72. IEEE Computer Society, 2006.

Digital Library

[161]

B. Lévy and J.-L. Mallet. Non-distorted texture mapping for sheared triangulated meshes. In Proceedings of SIGGRAPH '98, pages 343--352. ACM Press, 1998.

Digital Library

[162]

B. Lévy, S. Petitjean, N. Ray, and J. Maillot. Least squares conformal maps for automatic texture atlas generation. ACM Transactions on Graphics, 21(3):362--371, 2002. Proceedings of SIGGRAPH 2002.

Digital Library

[163]

W.-C. Li, N. Ray, and B. Lévy. Automatic and interactive mesh to T-spline conversion. In Proceedings of the 4th Eurographics Symposium on Geometry Processing (SGP 2006), pages 191--200. Eurographics Association, 2006.

Digital Library

[164]

J. Liesen, E. de Sturler, A. Sheffer, Y. Aydin, and C. Siefert. Preconditioners for indefinite linear systems arising in surface parameterization. In Proceedings of the 9th International Meshing Roundtable, pages 71--82, 2001.

[165]

Y. Lipman, J. Kopf, D. Cohen-Or, and D. Levin. GPU-assisted positive mean value coordinates for mesh deformations. In A. Belyaev and M. Garland, editors, Geometry Processing, Eurographics Symposium Proceedings, pages 117--123, Barcelona, Spain, 2007. Eurographics Association.

Digital Library

[166]

Y. Lipman, D. Levin, and D. Cohen-Or. Green coordinates. ACM Transactions on Graphics, 27(3):Article 78, 2008. Proceedings of SIGGRAPH 2008.

Digital Library

[167]

N. Litke, M. Droske, M. Rumpf, and P. Schröder. An image processing approach to surface matching. In Proceedings of the Third Eurographics Symposium on Geometry Processing (SGP 2005), pages 207--216. Eurographics Association, 2005.

Digital Library

[168]

L. Liu, L. Zhang, Y. Xu, C. Gotsman, and S. J. Gortler. A local/global approach to mesh parameterization. Computer Graphics Forum, 27(5): 1495--1504, 2008. Proceedings of SGP 2008.

Digital Library

[169]

F. Losasso, H. Hoppe, S. Schaefer, and J. Warren. Smooth geometry images. In Proceedings of the First Eurographics Symposium on Geometry Processing (SGP 2003), pages 138--145. Eurographics Association, 2003.

Digital Library

[170]

L. Lovász and A. Schrijver. On the null space of a Colin de Verdière matrix. Annales de l'institut Fourier, 49(3):1017--1026, 1999.

[171]

S. D. Ma and H. Lin. Optimal texture mapping. In Proceedings of Eurographics '88, pages 421--428, 1988.

[172]

W. Ma and J. P. Kruth. Parameterization of randomly measured points for least squares fitting of B-spline curves and surfaces. Computer-Aided Design, 27(9):663--675, 1995.

[173]

J. Maillot, H. Yahia, and A. Verroust. Interactive texture mapping. In Proceedings of SIGGRAPH '93, pages 27--34. ACM Press, 1993.

Digital Library

[174]

E. A. Malsch and G. Dasgupta. Interpolations for temperature distributions: A method for all non-concave polygons. International Journal of Solids and Structures, 41(8): 2165--2188, 2004.

[175]

E. A. Malsch and G. Dasgupta. Algebraic construction of smooth interpolants on polygonal domains. The Mathematica Journal, 9(3):641--658, 2005.

[176]

E. A. Malsch, J. J. Lin, and G. Dasgupta. Smooth two dimensional interpolants: a recipe for all polygons. Journal of Graphics Tools, 10(2):27--39, 2005.

[177]

M. Marinov and L. Kobbelt. Direct anisotropic quad-dominant remeshing. In Proc. of the 12th Pacific Conf. on Comp. Graph, and Appl., pages 207--216, 2004.

Digital Library

[178]

M. Marinov and L. Kobbelt. A robust two-step procedure for quad-dominant remeshing. Computer Graphics Forum, 25(3):537--546, 2006.

[179]

S. R. Marschner, B. K. Guenter, and S. Raghupathy. Modeling and rendering for realistic facial animation. In B. Péroche and H. Rushmeier, editors, Rendering Techniques 2000, pages 231--242. Springer, Wien, New York, 2000. Proceedings of the 11th Eurographics Workshop on Rendering.

Digital Library

[180]

J. McCartney, B. K. Hinds, and B. L. Seow. The flattening of triangulated surfaces incorporating darts and gussets. Computer-Aided Design, 31(4):249--260, 1999.

[181]

W. H. Meeks. A survey of the geometric results in the classical theory of minimal surfaces. Bulletin of the Brazilian Mathematical Society, 12(1):29--86, 1981.

[182]

G. Mercator. Nova et aucta orbis terrae descriptio ad usum navigantium emendate accommodata. Duisburg, 1569.

[183]

O. Meshar, D. Irony, and S. Toledo. An out-of-core sparse symmetric indefinite factorization method. ACM Transactions on Mathematical Software, 32(3):445--471, 2006.

Digital Library

[184]

M. Meyer, H. Lee, A. H. Barr, and M. Desbrun. Generalized barycentric coordinates on irregular polygons. Journal of Graphics Tools, 7(1):13--22, 2002.

Digital Library

[185]

V. J. Milenkovic. Rotational polygon containment and minimum enclosure. In Proceedings of the 14th Annual Symposium on Computational Geometry (SCG '98), pages 1--8. ACM Press, 1998.

Digital Library

[186]

J. Mitani and H. Suzuki. Making papercraft toys from meshes using strip-based approximate unfolding. ACM Transactions on Graphics, 23(3):259--263, 2004. Proceedings of SIGGRAPH 2004.

Digital Library

[187]

F. Morgan. Riemannian Geometry: A Beginner's Guide. A K Peters, Wellesley, MA, second edition, 1998.

[188]

P. Mullen, Y. Tong, P. Alliez, and M. Desbrun. Spectral conformal parameterization. Computer Graphics Forum, 27(5):1487--1494, 2008. Proceedings of SGP 2008.

Digital Library

[189]

T. Needham. Visual Complex Analysis. Oxford University Press, Oxford, New York, 1997.

[190]

X. Ni, M. Garland, and J. C. Hart. Fair morse functions for extracting the topological structure of a surface mesh. ACM Transactions on Graphics, 23(3):613--622, 2004. Proceedings of SIGGRAPH 2004.

Digital Library

[191]

J. Nocedal and S. J. Wright. Numerical Optimization. Springer Series in Operations Research. Springer, New York, second edition, 2006.

[192]

J. Pach and R. Wenger. Embedding planar graphs at fixed vertex locations. In Graph Drawing, volume 1547 of Lecture Notes in Computer Science, pages 263--274. Springer, Berlin, Heidelberg, 1998. Proceedings of the 6th International Symposium (GD '98).

Digital Library

[193]

L. Parida and S. P. Mudur. Constraint-satisfying planar development of complex surfaces. Computer-Aided Design, 25(4):225--232, 1993.

[194]

H. K. Pedersen. Decorating implicit surfaces. In Proceedings of SIGGRAPH '95, pages 291--300. ACM Press, 1995.

Digital Library

[195]

J. Peng, D. Kristjansson, and D. Zorin. Interactive modeling of topologically complex geometric detail. ACM Transactions on Graphics, 23(3):635--643, 2004. Proceedings of SIGGRAPH 2004.

Digital Library

[196]

U. Pinkall and K. Polthier. Computing discrete minimal surfaces and their conjugates. Experimental Mathematics, 2(1):15--36, 1993.

[197]

D. Piponi and G. Borshukov. Seamless texture mapping of subdivision surfaces by model pelting and texture blending. In Proceedings of SIGGRAPH 2000, pages 471--478. ACM Press, 2000.

Digital Library

[198]

J. A. F. Plateau. Statistique Expérimental et Théorique des Liquides Soumis aux Seules Forces Moléculaires. Gauthier-Villars, Paris, 1873.

[199]

K. Polthier and E. Preuss. Identifying vector field singularities using a discrete Hodge decomposition. In Visualization and Mathematics III, pages 113--134. Springer Verlag, 2003.

[200]

S. D. Porumbescu, B. Budge, L. Feng, and K. I. Joy. Shell maps. ACM Transactions on Graphics, 24(3):626--633, 2005. Proceedings of SIGGRAPH 2005.

Digital Library

[201]

E. Praun and H. Hoppe. Spherical parametrization and remeshing. ACM Transactions on Graphics, 22(3):340--349, 2003. Proceedings of SIGGRAPH 2003.

Digital Library

[202]

E. Praun, A. Finkelstein, and H. Hoppe. Lapped textures. In Proceedings of SIGGRAPH 2000, pages 465--470. ACM Press, 2000.

Digital Library

[203]

E. Praun, W. Sweldens, and P. Schroder. Consistent mesh parameterizations. In Proceedings of SIGGRAPH 2001, pages 179--184. ACM Press, 2001.

Digital Library

[204]

C. Ptolemy. The Geography. Dover, 1991. Translated by E. L. Stevenson.

[205]

B. Purnomo, J. D. Cohen, and S. Kumar. Seamless texture atlases. In Proceedings of the Second Eurographics Symposium on Geometry Processing (SGP 2004), pages 67--76. Eurographics Association, 2004.

Digital Library

[206]

T. Radó. Aufgabe 41. Jahresbericht der Deutschen Mathematiker-Vereinigung, 35:49, 1926.

[207]

T. Radó. The problem of least area and the problem of Plateau. Mathematische Zeitschrift, 32:763--796, 1930.

[208]

N. Ray and B. Lévy. Hierarchical least squares conformal maps. In Proceedings of the 11th Pacific Conference on Computer Graphics and Applications (PG 2003), pages 263--270. IEEE Computer Society, 2003.

Digital Library

[209]

N. Ray, W. C. Li, B. Vallet, F. Boutantin, B. Lévy, and C. Borgese. Graphite, http://alice.loria.fr/software/graphite/, 2003.

[210]

N. Ray, W.-C. Li, B. Lévy, A. Sheffer, and P. Alliez. Periodic global parameterization. ACM Transactions on Graphics, 25(4):1460--1485, 2006.

Digital Library

[211]

N. Ray, B. Vallet, W. C. Li, and B. Lévy. N-symmetry direction field design. ACM Transactions on Graphics, 27(2):Article 10, 2008.

Digital Library

[212]

N. Ray, B. Vallet, W.-C. Li, and B. Lévy. N-symmetry direction fields on surfaces of arbitrary genus. Technical report, January 2006.

[213]

M. Reuter, F.-E. Wolter, and N. Peinecke. Laplace-Beltrami spectra as 'Shape-DNA' of surfaces and solids. Computer-Aided Design, 38(4):342--366, 2006.

Digital Library

[214]

B. Riemann. Grundlagen für eine allgemeine Theorie der Functionen einer veränder-lichen complexen Größe. PhD thesis, Universitat Gättingen, 1851.

[215]

A. Rockwood and H. Park. Interactive design of smooth genus N objects using multi-periodic functions and applications. International Journal of Shape Modeling, 5(2): 135--157, 1999.

[216]

B. Rodin and D. Sullivan. The convergence of circle packings to the Riemann mapping. Journal of Differential Geometry, 26(2):349--360, 1987.

[217]

S. T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323--2326, 2000.

[218]

S. Saba, I. Yavneh, C. Gotsman, and A. Sheffer. Practical spherical embedding of manifold triangle meshes. In Proceedings of the 2005 International Conference on Shape Modeling and Applications (SMI '05), pages 256--265. IEEE Computer Society, 2005.

Digital Library

[219]

P. V. Sander, J. Snyder, S. J. Gortler, and H. Hoppe. Texture mapping progressive meshes. In Proceedings of SIGGRAPH 2001, pages 409--416. ACM Press, 2001.

Digital Library

[220]

P. V. Sander, S. J. Gortler, J. Snyder, and H. Hoppe. Signal-specialized parametrization. In Proceedings of the 13th Eurographics Workshop on Rendering (EGWR '02), pages 87--98. Eurographics Association, 2002.

Digital Library

[221]

P. V. Sander, Z. J. Wood, S. J. Gortler, J. Snyder, and H. Hoppe. Multi-chart geometry images. In Proceedings of the First Eurographics Symposium on Geometry Processing (SGP 2003), pages 138--145. Eurographics Association, 2003.

Digital Library

[222]

S. Schaefer, T. Ju, and J. Warren. A unified, integral construction for coordinates over closed curves. Computer Aided Geometric Design, 24(8--9):481--493, 2007.

Digital Library

[223]

J. Schreiner, A. Asirvatham, E. Praun, and H. Hoppe. Inter-surface mapping. ACM Transactions on Graphics, 23(3):870--877, 2004. Proceedings of SIGGRAPH 2004.

Digital Library

[224]

E. L. Schwartz, B. Merker, E. Wolfson, and A. Shaw. Applications of computer graphics and image processing to 2D and 3D modeling of the functional architecture of visual cortex. IEEE Computer Graphics and Applications, 8(4): 13--23, 1988.

Digital Library

[225]

E. L. Schwartz, A. Shaw, and E. Wolfson. A numerical solution to the generalized mapmaker's problem: flattening nonconvex polyhedral surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(9):1005--1008, 1989.

Digital Library

[226]

A. Shamir. A formulation of boundary mesh segmentation. In Proceedings of the Second International Symposium on 3D Data Processing, Visualization and Transmission (3DPVT 2004), Pages 82--89. IEEE Computer Society, 2004.

Digital Library

[227]

L. Shapira and A. Shamir. Local geodesic parametrization: An ant's perspective. In T. Möller, B. Hamann, and R. Russel, editors, Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration, Mathematics and Visualization. Springer, 2008.

[228]

A. Shapiro and A. Tal. Polyhedron realization for shape transformation. The Visual Computer, 14(8--9):429--444, 1998.

[229]

A. Sheffer. Spanning tree seams for reducing parameterization distortion of triangulated surfaces. In Proceedings of Shape Modeling International (SMI J02), pages 61--66. IEEE Computer Society, 2002.

Digital Library

[230]

A. Sheffer. Non-optimal parameterization and user control. In T. Lyche, M.-L. Mazure, and L. L. Schumaker, editors, Curve and Surface Design: Saint-Malo 2002, Modern Methods in Applied Mathematics, pages 355--364. Nashboro Press, Brentwood, TN, 2003a.

[231]

A. Sheffer. Skinning 3D meshes. Graphical Models, 65(5):274--285, 2003b.

Digital Library

[232]

A. Sheffer and E. de Sturler. Surface parameterization for meshing by triangulation flattening. In Proceedings of the 9th International Meshing Roundtable (IMR 2000), pages 161--172, 2000.

[233]

A. Sheffer and E. de Sturler. Parameterization of faceted surfaces for meshing using angle-based flattening. Engineering with Computers, 17(3):326--337, 2001.

[234]

A. Sheffer and E. de Sturler. Smoothing an overlay grid to minimize linear distortion in texture mapping. ACM Transactions on Graphics, 21(4):874--890, 2002.

Digital Library

[235]

A. Sheffer and J. C. Hart. Seamster: inconspicuous low-distortion texture seam layout. In Proceedings of IEEE Visualization 2002, pages 291--298. IEEE Computer Society, 2002.

Digital Library

[236]

A. Sheffer, C. Gotsman, and N. Dyn. Robust spherical parametrization of triangular meshes. In D. Cohen-Or, N. Dyn, G. Elber, and A. Shamir, editors, Proceedings of the 4th Israel-Korea Bi-National Conference on Geometric Modeling and Computer Graphics, pages 94--99, Tel Aviv, Israel, 2003.

[237]

A. Sheffer, B. Lévy, M. Mogilnitsky, and A. Bogomyakov. ABF++: fast and robust angle based flattening. ACM Transactions on Graphics, 24(2):311--330, 2005.

Digital Library

[238]

A. Sheffer, E. Praun, and K. Rose. Mesh parameterization methods and their applications. Foundations and Trends in Computer Graphics and Vision, 2(2):105--171, 2006.

Digital Library

[239]

J. R. Shewchuk. An introduction to the conjugate method without the agonizing pain. Technical report, School of Computer Science, Carnegie Mellon University, 1994. http://www.es.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf.

Digital Library

[240]

T. Shimada and Y. Tada. Approximate transformation of an arbitrary curved surface into a plane using dynamic programming. Computer-Aided Design, 23(2):153--159, 1991.

Digital Library

[241]

R. Sibson. A vector identity for the Dirichlet tesselation. Mathematical Proceedings of the Cambridge Philosophical Society, 87:151--155, 1980.

[242]

R. Sibson. A brief description of natural neighbour interpolation. In V. Barnett, editor, Interpolating Multivariate Data, pages 21--36. Wiley, New York, 1981.

[243]

C. Soler, M.-P. Cani, and A. Angelidis. Hierarchical pattern mapping. ACM Transactions on Graphics, 21(3):673--680, 2002. Proceedings of SIGGRAPH 2002.

Digital Library

[244]

O. Sorkine, D. Cohen-Or, R. Goldenthal, and D. Lischinski. Bounded-distortion piece-wise mesh parametrization. In Proceedings of IEEE Visualization 2002, pages 355--362. IEEE Computer Society, 2002.

Digital Library

[245]

O. Sorkine, D. Cohen-Or, Y. Lipman, M. Alexa, C. Rössl, and H.-P. Seidel. Laplacian surface editing. In Proceedings of the Second Eurographics Symposium on Geometry Processing (SGP 2004), pages 179--188. Eurographics Association, 2004.

Digital Library

[246]

B. Springborn, P. Schröder, and U. Pinkall. Conformal equivalence of triangle meshes. ACM Transactions on Graphics, 27(3):Article 77, 2008. Proceedings of SIGGRAPH 2008.

Digital Library

[247]

D. Steiner and A. Fischer. Planar parameterization for closed manifold genus-g meshes using any type of positive weights. Journal of Computing and Information Science in Engineering, 5(2):118--125, 2005.

[248]

K. Stephenson. Introduction to Circle Packing: The theory of discrete analytic functions. Cambridge University Press, New York, 2005.

[249]

N. Sukumar. Construction of polygonal interpolants: a maximum entropy approach. International Journal for Numerical Methods in Engineering, 61(12):2159--2181, Nov. 2004.

[250]

N. Sukumar and E. A. Malsch. Recent advances in the construction of polygonal finite element interpolants. Archives of Computational Methods in Engineering, 13(1):129--163, 2006.

[251]

R. W. Sumner and J. Popović. Deformation transfer for triangle meshes. ACM Transactions on Graphics, 23(3):399--405, 2004. Proceedings of ACM SIGGRAPH 2004.

Digital Library

[252]

V. Surazhsky and C. Gotsman. Explicit surface remeshing. In Proceedings of the First Eurographics Symposium on Geometry Processing (SGP 2003), pages 20--30. Eurographics Association, 2003.

Digital Library

[253]

M. Tarini, K. Hormann, P. Cignoni, and C. Montani. PolyCube-Maps. ACM Transactions on Graphics, 23(3):853--860, 2004. Proceedings of ACM SIGGRAPH 2004.

Digital Library

[254]

G. Taubin. A signal processing approach to fair surface design. In Proceedings of SIGGRAPH '95, pages 351--358. ACM Press, 1995.

Digital Library

[255]

J. B. Tenenbaum, V. de Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290(5500):2319--2323, 2000.

[256]

G. Tewari, J. Snyder, P. V. Sander, S. J. Gortler, and H. Hoppe. Signal-specialized parameterization for piecewise linear reconstruction. In Proceedings of the Second Eurographics Symposium on Geometry Processing (SGP 2004), pages 57--66. Eurographics Association, 2004.

Digital Library

[257]

S. Toledo. TAUCS: A library of sparse linear solvers, http://www.tau.ac.il/~stoledo/taucs/, 2003.

[258]

Y. Tong, P. Alliez, D. Cohen-Steiner, and M. Desbrun. Designing quadrangulations with discrete harmonic forms. In Proceedings of the 4th Eurographics Symposium on Geometry Processing (SGP 2006), pages 201--210. Eurographics Association, 2006.

Digital Library

[259]

G. Turk. Texture synthesis on surfaces. In Proceedings of SIGGRAPH 2001, pages 347--354. ACM Press, 2001.

Digital Library

[260]

W. T. Tutte. Convex representations of graphs. Proceedings of the London Mathematical Society, 10:304--320, 1960.

[261]

W. T. Tutte. How to draw a graph. Proceedings of the London Mathematical Society, 13:743--767, 1963.

[262]

B. Vallet and B. Lévy. Manifold harmonics. Computer Graphics Forum, 27(2):251--260, 2008. Proceedings of Eurographics 2008.

[263]

E. L. Wachspress. A Rational Finite Element Basis. Academic Press, New York, 1975.

[264]

C. C. L. Wang, S. S.-F. Smith, and M. M. F. Yuen. Surface flattening based on energy model. Computer-Aided Design, 34(11):823--833, 2002.

[265]

J. Warren. Barycentric coordinates for convex polytopes. Advances in Computational Mathematics, 6(2):97--108, 1996.

[266]

J. Warren. On the uniqueness of barycentric coordinates. In R. Goldman and R. Krasauskas, editors, Topics in Algebraic Geometry and Geometric Modeling, volume 334 of Contemporary Mathematics, pages 93--99. American Mathematical Society, 2003.

[267]

J. Warren, S. Schaefer, A. N. Hirani, and M. Desbrun. Barycentric coordinates for convex sets. Advances in Computational Mathematics, 27(3):319--338, 2007.

[268]

L.-Y. Wei and M. Levoy. Texture synthesis over arbitrary manifold surfaces. In Proceedings of SIGGRAPH 2001, pages 355--360. ACM Press, 2001.

Digital Library

[269]

Y.-L. Yang, J. Kim, F. Luo, S.-M. Hu, and X. Gu. Optimal surface parameterization using inverse curvature map. IEEE Transactions on Visualization and Computer Graphics, 14(5): 1054--1066, 2008.

Digital Library

[270]

L. Ying and D. Zorin. A simple manifold-based construction of surfaces of arbitrary smoothness. ACM Transactions on Graphics, 23(3):271--275, 2004. Proceedings of SIGGRAPH 2004.

Digital Library

[271]

L. Ying, A. Hertzmann, H. Biermann, and D. Zorin. Texture and shape synthesis on surfaces. In S. J. Gortler and K. Myszkowski, editors, Rendering Techniques 2001, pages 301--312. Springer, Wien, New York, 2001. Proceedings of the 12th Eurographics Workshop on Rendering.

Digital Library

[272]

S. Yoshizawa, A. G. Belyaev, and H.-P. Seidel. A fast and simple stretch-minimizing mesh parameterization. In Proceedings of the 2004 International Conference on Shape Modeling and Applications (SMI '04), pages 200--208. IEEE Computer Society, 2004.

Digital Library

[273]

F. W. Young. Multidimensional scaling. In S. Kotz, N. L. Johnson, and C. B. Read, editors, Encyclopedia of Statistical Sciences, volume 5, pages 649--659. Wiley, New York, 1985.

[274]

R. Zayer, C. Rössl, and H.-P. Seidel. Convex boundary angle based flattening. In T. Ertl, B. Girod, G. Greiner, H. Niemann, H.-P. Seidel, E. Steinbach, and R. Westermann, editors, Proceedings of Vision, Modeling, and Visualization 2003, pages 281--288, München, Germany, 2003. infix.

[275]

R. Zayer, C. Rössl, and H.-P. Seidel. Discrete tensorial quasi-harmonic maps. In Proceedings of the 2005 International Conference on Shape Modeling and Applications (SMI '05), pages 276--285. IEEE Computer Society, 2005a.

Digital Library

[276]

R. Zayer, C. Rössl, and H.-P. Seidel. Setting the boundary free: A composite approach to surface parameterization. In Proceedings of the Third Eurographics Symposium on Geometry Processing (SGP 2005), pages 91--100. Eurographics Association, 2005b.

Digital Library

[277]

R. Zayer, C. Rössl, and H.-P. Seidel. Variations on angle based flattening. In N. A. Dodgson, M. S. Floater, and M. A. Sabin, editors, Advances in Multiresolution for Geometric Modelling, Mathematics and Visualization, pages 187--199. Springer, Berlin, Heidelberg, 2005c.

[278]

R. Zayer, C. Rössl, and H.-P. Seidel. Curvilinear spherical parameterization. In Proceedings of the 2006 International Conference on Shape Modeling and Applications (SMI '06), pages 57--64. IEEE Computer Society, 2006.

Digital Library

[279]

R. Zayer, B. Lévy, and H.-P. Seidel. Linear angle based parameterization. In Proceedings of the 5th Eurographics Symposium on Geometry Processing (SGP 2007), pages 135--141. Eurographics Association, 2007.

Digital Library

[280]

E. Zhang, K. Mischaikow, and G. Turk. Feature-based surface parameterization and texture mapping. ACM Transactions on Graphics, 24(1):1--27, 2005.

Digital Library

[281]

L. Zhang, L. Liu, Z. Ji, and G. Wang. Manifold parameterization. In T. Nishita, Q. Peng, and H.-P. Seidel, editors, Advances in Computer Graphics, volume 4035 of Lecture Notes in Computer Science, pages 160--171. Springer, 2006. Proceedings of the 24th Computer Graphics International Conference.

Digital Library

[282]

K. Zhou, J. Synder, B. Guo, and H.-Y. Shum. Iso-charts: Stretch-driven mesh parameterization using spectral analysis. In Proceedings of the Second Eurographics Symposium on Geometry Processing (SGP 2004), pages 47--56. Eurographics Association, 2004.

Digital Library

[283]

K. Zhou, X. Wang, Y. Tong, M. Desbrun, B. Guo, and H.-Y. Shum. TextureMontage: Seamless texturing of arbitrary surfaces from multiple images. ACM Transactions on Graphics, 24(3):1148--1155, 2005. Proceedings of SIGGRAPH 2005.

Digital Library

[284]

G. Zigelman, R. Kimmel, and N. Kiryati. Texture mapping using surface flattening via multidimensional scaling. IEEE Transactions on Visualization and Computer Graphics, 8(2):198--207, 2002.

Digital Library

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Mesh parameterization: theory and practice

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cover image ACM Conferences

SIGGRAPH Asia '08: ACM SIGGRAPH ASIA 2008 courses

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2261 pages

ISBN:9781450379243

DOI:10.1145/1508044

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https://doi.org/10.2514/6.2025-0216
Yan XLi SHuang WWang HZhao THuang MZhou NShen YLi X(2025)MPicker: visualizing and picking membrane proteins for cryo-electron tomographyNature Communications10.1038/s41467-024-55767-w16:1Online publication date: 8-Jan-2025
https://doi.org/10.1038/s41467-024-55767-w
Morales AAlomar APorras ALinguraru MPiella GSukno F(2025)BabyFM: Towards accurate 3D baby facial models using spectral decomposition and asymmetry swappingComputers in Biology and Medicine10.1016/j.compbiomed.2025.109652186(109652)Online publication date: Mar-2025
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