Non-uniform interpolatory subdivision surface

This paper presents an interpolatory subdivision scheme with non-uniform parametrization for arbitrary polygon meshes with arbitrary manifold topology. This is the first attempt to generalize the non-uniform four point interpolatory curve subdivision to surface with extraordinary points. The scheme is constructed from the inspiration of the relation between the non-uniform four-point interpolatory subdivision scheme and the non-uniform B-spline refinement rule. Numerical examples and comparisons with the uniform interpolatory subdivision schemes indicate that the quality of the limit surface can be improved by using non-uniform parameter values for non-uniform initial data.

[1]  Hujun Bao,et al.  Interpolatory v2-Subdivision Surfaces , 2004, GMP.

[2]  Carolina Vittoria Beccari,et al.  Non-uniform non-tensor product local interpolatory subdivision surfaces , 2013, Comput. Aided Geom. Des..

[3]  Zhongxuan Luo,et al.  On interpolatory subdivision from approximating subdivision scheme , 2013, Appl. Math. Comput..

[4]  Weiyin Ma,et al.  Constructing an Interpolatory Subdivision Scheme from Doo-Sabin Subdivision , 2011, 2011 12th International Conference on Computer-Aided Design and Computer Graphics.

[5]  Carolina Vittoria Beccari,et al.  A unified framework for interpolating and approximating univariate subdivision , 2010, Appl. Math. Comput..

[6]  Luiz Velho,et al.  Quasi 4-8 subdivision , 2001, Comput. Aided Geom. Des..

[7]  Ulf Labsik,et al.  Interpolatory √3‐Subdivision , 2000 .

[8]  E. Catmull,et al.  Recursively generated B-spline surfaces on arbitrary topological meshes , 1978 .

[9]  Nira Dyn,et al.  Four-point curve subdivision based on iterated chordal and centripetal parameterizations , 2009, Comput. Aided Geom. Des..

[10]  E. T. Y. Lee,et al.  Choosing nodes in parametric curve interpolation , 1989 .

[11]  Thomas J. R. Hughes,et al.  On linear independence of T-spline blending functions , 2012, Comput. Aided Geom. Des..

[12]  Malcolm A. Sabin,et al.  Behaviour of recursive division surfaces near extraordinary points , 1998 .

[13]  Jörg Peters,et al.  Combining 4- and 3-direction subdivision , 2004, ACM Trans. Graph..

[14]  M. Sabin,et al.  NURBS with extraordinary points: high-degree, non-uniform, rational subdivision schemes , 2009, SIGGRAPH 2009.

[15]  Malcolm A. Sabin,et al.  Non-uniform recursive subdivision surfaces , 1998, SIGGRAPH.

[16]  G. Casciola,et al.  Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines , 2011 .

[17]  Jianmin Zheng,et al.  Making Doo-Sabin surface interpolation always work over irregular meshes , 2005, The Visual Computer.

[18]  Rong-Qing Jia,et al.  Interpolatory Subdivision Schemes Induced by Box Splines , 2000 .

[19]  Jianmin Zheng,et al.  An alternative method for constructing interpolatory subdivision from approximating subdivision , 2012, Comput. Aided Geom. Des..

[20]  Michael S. Floater.  On the deviation of a parametric cubic spline interpolant from its data polygon , 2008, Comput. Aided Geom. Des..

[21]  Hans Hagen,et al.  Dinus: Double insertion, nonuniform, stationary subdivision surfaces , 2010, TOGS.

[22]  I. Daubechies,et al.  Regularity of Irregular Subdivision , 1999 .

[23]  Xunnian Yang,et al.  A simple method for interpolating meshes of arbitrary topology by Catmull–Clark surfaces , 2010, The Visual Computer.

[24]  Jianmin Zheng,et al.  Interpolation over arbitrary topology meshes using a two-phase subdivision scheme , 2006, IEEE Transactions on Visualization and Computer Graphics.

[25]  D. Levin,et al.  Analysis of asymptotically equivalent binary subdivision schemes , 1995 .

[26]  Lucia Romani,et al.  A constructive algebraic strategy for interpolatory subdivision schemes induced by bivariate box splines , 2013, Adv. Comput. Math..

[27]  Leif Kobbelt,et al.  Interpolatory Subdivision on Open Quadrilateral Nets with Arbitrary Topology , 1996, Comput. Graph. Forum.

[28]  Lucia Romani,et al.  From approximating to interpolatory non-stationary subdivision schemes with the same generation properties , 2010, Adv. Comput. Math..

[29]  Weiyin Ma,et al.  A Method for Constructing Interpolatory Subdivision Schemes and Blending Subdivisions , 2007, Comput. Graph. Forum.

[30]  D.W. Fellner,et al.  Isogeometric shell analysis with NURBS compatible subdivision surfaces , 2016, Appl. Math. Comput..

[31]  Zhangjin Huang,et al.  Non-uniform recursive Doo-Sabin surfaces , 2011, Comput. Aided Des..

[32]  D. Zorin,et al.  4-8 Subdivision , 2001 .

[33]  Xiaonan Luo,et al.  Deducing interpolating subdivision schemes from approximating subdivision schemes , 2008, SIGGRAPH 2008.

[34]  Ahmad H. Nasri,et al.  Polyhedral subdivision methods for free-form surfaces , 1987, TOGS.

[35]  Lucia Romani.  From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms , 2009 .

[36]  N. Dyn,et al.  A butterfly subdivision scheme for surface interpolation with tension control , 1990, TOGS.

[37]  Weiyin Ma,et al.  A unified interpolatory subdivision scheme for quadrilateral meshes , 2013, TOGS.

[38]  Thomas W. Sederberg,et al.  G1 non-uniform Catmull-Clark surfaces , 2016, ACM Trans. Graph..

[39]  Tony DeRose,et al.  Subdivision surfaces in character animation , 1998, SIGGRAPH.

[40]  Neil A. Dodgson,et al.  Semi‐sharp Creases on Subdivision Curves and Surfaces , 2014, Comput. Graph. Forum.

[41]  E. Kuznetsov,et al.  The best parameterization for parametric interpolation , 2006 .

[42]  Charles T. Loop,et al.  Quad/Triangle Subdivision , 2003, Comput. Graph. Forum.

[43]  Lucia Romani,et al.  From symmetric subdivision masks of Hurwitz type to interpolatory subdivision masks , 2009 .

[44]  Jos Stam,et al.  A Unified Subdivision Scheme for Polygonal Modeling , 2001, Comput. Graph. Forum.

[45]  Gudrun Albrecht,et al.  Convexity preserving interpolatory subdivision with conic precision , 2012, Appl. Math. Comput..

[46]  Cem Yuksel,et al.  On the parameterization of Catmull-Rom curves , 2009, Symposium on Solid and Physical Modeling.

[47]  Peter Schröder,et al.  Interpolating Subdivision for meshes with arbitrary topology , 1996, SIGGRAPH.

[48]  Carolina Vittoria Beccari,et al.  Polynomial-based non-uniform interpolatory subdivision with features control , 2011, J. Comput. Appl. Math..

[49]  Dieter W. Fellner,et al.  Extended subdivision surfaces: Building a bridge between NURBS and Catmull-Clark surfaces , 2006, TOGS.

[50]  Nira Dyn,et al.  A 4-point interpolatory subdivision scheme for curve design , 1987, Comput. Aided Geom. Des..