Volume and shape preservation via moving frame manipulation

This article introduces a method for mesh editing that is aimed at preserving shape and volume. We present two new developments: The first is a minimization of a functional expressing a geometric distance measure between two isometric surfaces. The second is a local volume analysis linking the volume of an object to its surface curvature. Our method is based upon the moving frames representation of meshes. Applying a rotation field to the moving frames defines an isometry. Given rotational constraints, the mesh is deformed by an optimal isometry defined by minimizing the distance measure between original and deformed meshes. The resulting isometry nicely preserves the surface details, but when large rotations are applied, the volumetric behavior of the model may be unsatisfactory. Using the local volume analysis, we define a scalar field by which we scale the moving frames. Scaled and rotated moving frames restore volumetric properties of the original mesh, while properly maintaining the surface details. Our results show that even extreme deformations can be applied to meshes, with only minimal distortion of surface details and object volume.

[1]  H. Saunders,et al.  Finite element procedures in engineering analysis , 1982 .

[2]  J. Stoker.  Differential Geometry: Stoker/Differential , 1988 .

[3]  Ulrich Pinkall,et al.  Computing Discrete Minimal Surfaces and Their Conjugates , 1993, Exp. Math..

[4]  Peisheng Gao,et al.  2-D shape blending: an intrinsic solution to the vertex path problem , 1993, SIGGRAPH.

[5]  M. D. Carmo.  Differential forms and applications , 1994 .

[6]  Alla Sheffer,et al.  Volume-preserving free-form solid , 1995, SMA '95.

[7]  Gabriel Taubin,et al.  A signal processing approach to fair surface design , 1995, SIGGRAPH.

[8]  Dominique Bechmann,et al.  Volume-preserving space deformation , 1997, Comput. Graph..

[9]  Daniel Cohen-Or,et al.  Three-dimensional distance field metamorphosis , 1998, TOGS.

[10]  Hans-Peter Seidel,et al.  Interactive multi-resolution modeling on arbitrary meshes , 1998, SIGGRAPH.

[11]  Ming C. Lin,et al.  Fast volume-preserving free form deformation using multi-level optimization , 1999, SMA '99.

[12]  Marc Alexa,et al.  As-rigid-as-possible shape interpolation , 2000, SIGGRAPH.

[13]  Leonard McMillan,et al.  Stable real-time deformations , 2002, SCA '02.

[14]  K. Polthier.  Computational Aspects of Discrete Minimal Surfaces , 2002 .

[15]  J. Landsberg,et al.  Moving frames and exterior differential systems , 2003 .

[16]  J. Landsberg,et al.  Cartan for beginners , 2003 .

[17]  M. Floater.  Mean value coordinates , 2003, Computer Aided Geometric Design.

[18]  Leif Kobbelt,et al.  Multiresolution Surface Representation Based on Displacement Volumes , 2003, Comput. Graph. Forum.

[19]  Christian Rössl,et al.  Differential coordinates for interactive mesh editing , 2004, Proceedings Shape Modeling Applications, 2004..

[20]  Leif Kobbelt,et al.  An intuitive framework for real-time freeform modeling , 2004, SIGGRAPH 2004.

[21]  Christian Rössl,et al.  Laplacian surface editing , 2004, SGP '04.

[22]  John Hart,et al.  ACM Transactions on Graphics , 2004, SIGGRAPH 2004.

[23]  Geoff Wyvill,et al.  Swirling-sweepers: constant-volume modeling , 2004, 12th Pacific Conference on Computer Graphics and Applications, 2004. PG 2004. Proceedings..

[24]  Kun Zhou,et al.  Mesh editing with poisson-based gradient field manipulation , 2004, SIGGRAPH 2004.

[25]  D. Levin,et al.  Linear rotation-invariant coordinates for meshes , 2005, SIGGRAPH 2005.

[26]  Large mesh deformation using the volumetric graph Laplacian , 2005, SIGGRAPH 2005.

[27]  Christian Rössl,et al.  Harmonic Guidance for Surface Deformation , 2005, Comput. Graph. Forum.

[28]  J. Warren,et al.  Mean value coordinates for closed triangular meshes , 2005, SIGGRAPH 2005.

[29]  Takeo Igarashi,et al.  As-rigid-as-possible shape manipulation , 2005, ACM Trans. Graph..

[30]  H. Shum,et al.  Subspace gradient domain mesh deformation , 2006, SIGGRAPH 2006.

[31]  Markus H. Gross,et al.  PriMo: coupled prisms for intuitive surface modeling , 2006, SGP '06.

[32]  Hans-Peter Seidel,et al.  Vector field based shape deformations , 2006, ACM Trans. Graph..

[33]  Hujun Bao,et al.  Poisson shape interpolation , 2006, Graph. Model..