Biharmonic Volumetric Mapping Using Fundamental Solutions

We propose a biharmonic model for cross-object volumetric mapping. This new computational model aims to facilitate the mapping of solid models with complicated geometry or heterogeneous inner structures. In order to solve cross-shape mapping between such models through divide and conquer, solid models can be decomposed into subparts upon which mappings is computed individually. The biharmonic volumetric mapping can be performed in each subregion separately. Unlike the widely used harmonic mapping which only allows C0 continuity along the segmentation boundary interfaces, this biharmonic model can provide C1 smoothness. We demonstrate the efficacy of our mapping framework on various geometric models with complex geometry (which are decomposed into subparts with simpler and solvable geometry) or heterogeneous interior structures (whose different material layers can be segmented and processed separately).

[1]  Hong Qin,et al.  Harmonic volumetric mapping for solid modeling applications , 2007, Symposium on Solid and Physical Modeling.

[2]  B. Jin A meshless method for the Laplace and biharmonic equations subjected to noisy boundary data , 2004 .

[3]  Brian T. Helenbrook,et al.  Mesh deformation using the biharmonic operator , 2003 .

[4]  Frank Sottile,et al.  Linear precision for parametric patches , 2007, Adv. Comput. Math..

[5]  Ying He,et al.  Direct-Product Volumetric Parameterization of Handlebodies via Harmonic Fields , 2010, 2010 Shape Modeling International Conference.

[6]  Andreas Karageorghis,et al.  Three-dimensional image reconstruction using the PF/MFS technique , 2009 .

[7]  Elaine Cohen,et al.  Volumetric parameterization and trivariate B-spline fitting using harmonic functions , 2009, Comput. Aided Geom. Des..

[8]  Helen C. Shen,et al.  Linear Neighborhood Propagation and Its Applications , 2009, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  S. Sitharama Iyengar,et al.  Toward More Precise Radiotherapy Treatment of Lung Tumors , 2012, Computer.

[10]  G. Geymonat Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions , 2007 .

[11]  Hong Qin,et al.  Surface Mapping Using Consistent Pants Decomposition , 2009, IEEE Transactions on Visualization and Computer Graphics.

[12]  Elaine Cohen,et al.  Volumetric parameterization of complex objects by respecting multiple materials , 2010, Comput. Graph..

[13]  Graeme Fairweather,et al.  The method of fundamental solutions for the numerical solution of the biharmonic equation , 1987 .

[14]  Charlie C. L. Wang,et al.  Volume Parameterization for Design Automation of Customized Free-Form Products , 2007, IEEE Transactions on Automation Science and Engineering.

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

[16]  Patrick M. Knupp,et al.  Tetrahedral Element Shape Optimization via the Jacobian Determinant and Condition Number , 1999, IMR.

[17]  M. Kasper graphics , 1991, Illustrating Mathematics.

[18]  Michael S. Floater,et al.  Mean value coordinates , 2003, Comput. Aided Geom. Des..

[19]  David S. Watkins,et al.  Fundamentals of matrix computations , 1991 .

[20]  Charlie C. L. Wang,et al.  Efficient Optimization of Common Base Domains for Cross Parameterization , 2012, IEEE Transactions on Visualization and Computer Graphics.

[21]  Daniel Cohen-Or,et al.  Green Coordinates , 2008, ACM Trans. Graph..

[22]  Hong Qin,et al.  Meshless Harmonic Volumetric Mapping Using Fundamental Solution Methods , 2009, IEEE Transactions on Automation Science and Engineering.

[23]  Zhao Yin,et al.  Feature-aligned harmonic volumetric mapping using MFS , 2010, Comput. Graph..

[24]  Paul M. Thompson,et al.  Volumetric harmonic brain mapping , 2004, 2004 2nd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE Cat No. 04EX821).

[25]  Ioannis Pratikakis,et al.  3D Mesh Segmentation Methodologies for CAD applications , 2007 .

[26]  Hong Qin,et al.  Polycube splines , 2007, Comput. Aided Des..

[27]  Hong Qin,et al.  Globally Optimal Surface Mapping for Surfaces with Arbitrary Topology , 2008, IEEE Transactions on Visualization and Computer Graphics.

[28]  Elaine Cohen,et al.  Volumetric parameterization and trivariate b-spline fitting using harmonic functions , 2008, SPM '08.

[29]  W. Yu,et al.  Computing 3D Shape Guarding and Star Decomposition , 2011, Comput. Graph. Forum.

[30]  Alla Sheffer,et al.  Cross-parameterization and compatible remeshing of 3D models , 2004, ACM Trans. Graph..

[31]  Eugene Zhang,et al.  All‐Hex Mesh Generation via Volumetric PolyCube Deformation , 2011, Comput. Graph. Forum.

[32]  Thomas A. Funkhouser,et al.  Biharmonic distance , 2010, TOGS.

[33]  Mark Meyer,et al.  Harmonic coordinates for character articulation , 2007, ACM Trans. Graph..

[34]  Jason F. Shepherd,et al.  New Applications of the Verdict Library for Standardized Mesh Verification Pre, Post, and End-to-End Processing , 2007, IMR.

[35]  Hong Qin,et al.  Generalized PolyCube Trivariate Splines , 2010, 2010 Shape Modeling International Conference.

[36]  P. Grisvard Elliptic Problems in Nonsmooth Domains , 1985 .

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

[38]  Olga Sorkine-Hornung,et al.  Mixed Finite Elements for Variational Surface Modeling , 2010, Comput. Graph. Forum.

[39]  Ariel Shamir,et al.  A survey on Mesh Segmentation Techniques , 2008, Comput. Graph. Forum.

[40]  Paul M. Thompson,et al.  Surface-Constrained Volumetric Brain Registration Using Harmonic Mappings , 2007, IEEE Transactions on Medical Imaging.

[41]  Olga Sorkine-Hornung,et al.  Bounded biharmonic weights for real-time deformation , 2011, Commun. ACM.

[42]  Ying He,et al.  Hexahedral shell mesh construction via volumetric polycube map , 2010, SPM '10.

[43]  Paolo Cignoni,et al.  PolyCube-Maps , 2004, SIGGRAPH 2004.

[44]  Tao Ju,et al.  Mean value coordinates for closed triangular meshes , 2005, ACM Trans. Graph..

[45]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[46]  Giuseppe Patanè,et al.  Topology- and error-driven extension of scalar functions from surfaces to volumes , 2009, TOGS.

[47]  Hassan Ugail,et al.  A general 4th-order PDE method to generate Bézier surfaces from the boundary , 2006, Comput. Aided Geom. Des..