Bounded distortion parametrization in the space of metrics

We present a framework for global parametrization that utilizes the edge lengths (squared) of the mesh as variables. Given a mesh with arbitrary topology and prescribed cone singularities, we flatten the original metric of the surface under strict bounds on the metric distortion (various types of conformal and isometric measures are supported). Our key observation is that the space of bounded distortion metrics (given any particular bounds) is convex, and a broad range of useful and well-known distortion energies are convex as well. With the addition of nonlinear Gaussian curvature constraints, the parametrization problem is formulated as a constrained optimization problem, and a solution gives a locally injective map. Our method is easy to implement. Sequential convex programming (SCP) is utilized to solve this problem effectively. We demonstrate the flexibility of the method and its uncompromised robustness and compare it to state-of-the-art methods.

[1]  Pierre Alliez,et al.  Polygon Mesh Processing , 2010 .

[2]  Denis Zorin,et al.  Locally injective parametrization with arbitrary fixed boundaries , 2014, ACM Trans. Graph..

[3]  Denis Zorin,et al.  Strict minimizers for geometric optimization , 2014, ACM Trans. Graph..

[4]  Yaron Lipman,et al.  Bounded distortion mapping spaces for triangular meshes , 2012, ACM Trans. Graph..

[5]  Neil A. Dodgson,et al.  Advances in Multiresolution for Geometric Modelling , 2005 .

[6]  Konrad Polthier,et al.  QuadCover ‐ Surface Parameterization using Branched Coverings , 2007, Comput. Graph. Forum.

[7]  Ronen Basri,et al.  Controlling singular values with semidefinite programming , 2014, ACM Trans. Graph..

[8]  Peter Schröder,et al.  Discrete conformal mappings via circle patterns , 2005, TOGS.

[9]  Bruno Lévy,et al.  Quad‐Mesh Generation and Processing: A Survey , 2013, Comput. Graph. Forum.

[10]  David Eppstein,et al.  Quasiconvex Programming , 2004, ArXiv.

[11]  Ligang Liu,et al.  A Local/Global Approach to Mesh Parameterization , 2008, Comput. Graph. Forum.

[12]  Roi Poranne,et al.  Provably good planar mappings , 2014, ACM Trans. Graph..

[13]  Roi Poranne,et al.  Lifted bijections for low distortion surface mappings , 2014, ACM Trans. Graph..

[14]  David Bommes,et al.  Quantized global parametrization , 2015, ACM Trans. Graph..

[15]  Denis Zorin,et al.  Controlled-distortion constrained global parametrization , 2013, ACM Trans. Graph..

[16]  Bruno Lévy,et al.  Least squares conformal maps for automatic texture atlas generation , 2002, ACM Trans. Graph..

[17]  Pierre Alliez,et al.  Integer-grid maps for reliable quad meshing , 2013, ACM Trans. Graph..

[18]  Olga Sorkine-Hornung,et al.  Locally Injective Mappings , 2013 .

[19]  Dani Lischinski,et al.  Bounded-distortion piecewise mesh parameterization , 2002, IEEE Visualization, 2002. VIS 2002..

[20]  D. Bommes,et al.  Mixed-integer quadrangulation , 2009, SIGGRAPH 2009.

[21]  Denis Zorin,et al.  Global parametrization by incremental flattening , 2012, ACM Trans. Graph..

[22]  Mirela Ben-Chen,et al.  Planar shape interpolation with bounded distortion , 2013, ACM Trans. Graph..

[23]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[24]  John M. Lee Riemannian Manifolds: An Introduction to Curvature , 1997 .

[25]  Eftychios Sifakis,et al.  Fast and Robust Inversion‐Free Shape Manipulation , 2016, Comput. Graph. Forum.

[26]  Alla Sheffer,et al.  Mesh parameterization: theory and practice Video files associated with this course are available from the citation page , 2007, SIGGRAPH Courses.

[27]  Peter Schröder,et al.  Conformal equivalence of triangle meshes , 2008, ACM Trans. Graph..

[28]  Anne Verroust-Blondet,et al.  Interactive texture mapping , 1993, SIGGRAPH.

[29]  Ofir Weber,et al.  Bounded distortion harmonic shape interpolation , 2016, ACM Trans. Graph..

[30]  Ofir Weber,et al.  Bounded distortion harmonic mappings in the plane , 2015, ACM Trans. Graph..

[31]  Ronen Basri,et al.  Large-scale bounded distortion mappings , 2015, ACM Trans. Graph..

[32]  Yaron Lipman,et al.  Injective and bounded distortion mappings in 3D , 2013, ACM Trans. Graph..

[33]  Bruno Lévy,et al.  Mesh parameterization: theory and practice , 2007, SIGGRAPH Courses.

[34]  Denis Zorin,et al.  Computing Extremal Quasiconformal Maps , 2012, Comput. Graph. Forum.

[35]  Craig Gotsman,et al.  Conformal Flattening by Curvature Prescription and Metric Scaling , 2008, Comput. Graph. Forum.

[36]  Alla Sheffer,et al.  Parameterization of Faceted Surfaces for Meshing using Angle-Based Flattening , 2001, Engineering with Computers.

[37]  Shi-Min Hu,et al.  Metric-Driven RoSy Field Design and Remeshing , 2010, IEEE Transactions on Visualization and Computer Graphics.

[38]  Moritz Diehl,et al.  Local Convergence of Sequential Convex Programming for Nonconvex Optimization , 2010 .

[39]  Denis Zorin,et al.  Robust field-aligned global parametrization , 2014, ACM Trans. Graph..

[40]  Hugues Hoppe,et al.  Inter-surface mapping , 2004, ACM Trans. Graph..

[41]  U. Pinkall,et al.  Discrete conformal maps and ideal hyperbolic polyhedra , 2010, 1005.2698.

[42]  Daniele Panozzo,et al.  Directional Field Synthesis, Design, and Processing , 2016, Comput. Graph. Forum.

[43]  Kai Hormann,et al.  Surface Parameterization: a Tutorial and Survey , 2005, Advances in Multiresolution for Geometric Modelling.