Implicit surface reconstruction with total variation regularization

Implicit representations have been widely used for surface reconstruction on account of their capability to describe shapes that exhibit complicated geometry and topology. However, extra zero-level sets or spurious sheets usually emerge in implicit algorithms and damage the reconstruction results. In this paper, we propose a reconstruction approach that involves the total variation (TV) of the implicit representation to minimize the occurrence of spurious sheets. Proof is given to show that the recovered shape has the simplest topology with respect to the input data. By using algebraic spline functions as the implicit representation, an efficient discretization is presented together with effective algorithms to solve it. Hierarchical structures with uniform subdivisions can be applied in the framework for fitting fine details. Numerical experiments demonstrate that our algorithm achieves high quality reconstruction results while reducing the existence of extra sheets.

[1]  E. Stein,et al.  Real Analysis: Measure Theory, Integration, and Hilbert Spaces , 2005 .

[2]  John G. Michopoulos,et al.  Implicit slicing for functionally tailored additive manufacturing , 2016, Comput. Aided Des..

[3]  Michael M. Kazhdan,et al.  Screened poisson surface reconstruction , 2013, TOGS.

[4]  A. Chambolle.  Practical, Unified, Motion and Missing Data Treatment in Degraded Video , 2004, Journal of Mathematical Imaging and Vision.

[5]  Hans-Peter Seidel,et al.  A multi-scale approach to 3D scattered data interpolation with compactly supported basis functions , 2003, 2003 Shape Modeling International..

[6]  M. Nikolova.  An Algorithm for Total Variation Minimization and Applications , 2004 .

[7]  Joachim Giesen,et al.  Delaunay Triangulation Based Surface Reconstruction , 2006 .

[8]  Josiah Manson,et al.  Streaming Surface Reconstruction Using Wavelets , 2008, Comput. Graph. Forum.

[9]  David R. Forsey,et al.  Hierarchical B-spline refinement , 1988, SIGGRAPH.

[10]  Charlie C. L. Wang,et al.  Intersection-Free and Topologically Faithful Slicing of Implicit Solid , 2013, J. Comput. Inf. Sci. Eng..

[11]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[12]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

[13]  Leif Kobbelt,et al.  Robust reconstruction of watertight 3D models from non-uniformly sampled point clouds without normal information , 2006, SGP '06.

[14]  Frédéric Chazal,et al.  Topology guaranteeing manifold reconstruction using distance function to noisy data , 2006, SCG '06.

[15]  Selim Esedoglu,et al.  Analogue of the Total Variation Denoising Model in the Context of Geometry Processing , 2009, Multiscale Model. Simul..

[16]  Jiansong Deng,et al.  Variational Mesh Denoising Using Total Variation and Piecewise Constant Function Space , 2015, IEEE Transactions on Visualization and Computer Graphics.

[17]  Richard K. Beatson,et al.  Reconstruction and representation of 3D objects with radial basis functions , 2001, SIGGRAPH.

[18]  D. Levin,et al.  Mesh-Independent Surface Interpolation , 2004 .

[19]  Tom Goldstein,et al.  The Split Bregman Method for L1-Regularized Problems , 2009, SIAM J. Imaging Sci..

[20]  Brian Wyvill,et al.  Introduction to Implicit Surfaces , 1997 .

[21]  Karl Kunisch,et al.  Total Generalized Variation , 2010, SIAM J. Imaging Sci..

[22]  W. Rudin.  Real and complex analysis , 1968 .

[23]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[24]  Jean-Daniel Boissonnat,et al.  Geometric structures for three-dimensional shape representation , 1984, TOGS.

[25]  Bert Jüttler,et al.  THB-splines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis , 2016 .

[26]  Bert Jüttler,et al.  Least-Squares Fitting of Algebraic Spline Surfaces , 2002, Adv. Comput. Math..

[27]  Joe Warren,et al.  Approximation of dense scattered data using algebraic surfaces , 1991, Proceedings of the Twenty-Fourth Annual Hawaii International Conference on System Sciences.

[28]  Gerald B. Folland,et al.  Real Analysis: Modern Techniques and Their Applications , 1984 .

[29]  Gene H. Golub,et al.  A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration , 1999, SIAM J. Sci. Comput..

[30]  Jiansong Deng,et al.  Fitting unorganized point clouds with active implicit B-spline curves , 2005, The Visual Computer.

[31]  James F. Blinn,et al.  A Generalization of Algebraic Surface Drawing , 1982, TOGS.

[32]  Hans-Peter Seidel,et al.  Multi-level partition of unity implicits , 2005, SIGGRAPH Courses.

[33]  Josef Hoschek,et al.  Fundamentals of computer aided geometric design , 1996 .

[34]  James F. O'Brien,et al.  Interpolating and approximating implicit surfaces from polygon soup , 2005, SIGGRAPH 2005.

[35]  Jun Wang,et al.  Parallel and adaptive surface reconstruction based on implicit PHT-splines , 2011, Comput. Aided Geom. Des..

[36]  Daniel Cohen-Or,et al.  ℓ1-Sparse reconstruction of sharp point set surfaces , 2010, TOGS.

[37]  Tony F. Chan,et al.  Color TV: total variation methods for restoration of vector-valued images , 1998, IEEE Trans. Image Process..

[38]  Falai Chen,et al.  Compact implicit surface reconstruction via low-rank tensor approximation , 2016, Comput. Aided Des..

[39]  Otmar Scherzer,et al.  Denoising with higher order derivatives of bounded variation and an application to parameter estimation , 2007, Computing.

[40]  T. Dokken,et al.  Computational Methods for Algebraic Spline Surfaces , 2008 .

[41]  Yutaka Ohtake,et al.  Multi-scale and Adaptive CS-RBFs for Shape Reconstruction from Clouds of Points , 2005, Advances in Multiresolution for Geometric Modelling.

[42]  Tamal K. Dey,et al.  Provable surface reconstruction from noisy samples , 2004, SCG '04.

[43]  Charlie C. L. Wang,et al.  A closed-form formulation of HRBF-based surface reconstruction by approximate solution , 2015, Comput. Aided Des..

[44]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[45]  Junfeng Yang,et al.  A New Alternating Minimization Algorithm for Total Variation Image Reconstruction , 2008, SIAM J. Imaging Sci..

[46]  Michael M. Kazhdan,et al.  Poisson surface reconstruction , 2006, SGP '06.

[47]  Jiansong Deng,et al.  Polynomial splines over hierarchical T-meshes , 2008, Graph. Model..

[48]  Tamal K. Dey,et al.  Curve and Surface Reconstruction , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[49]  Xue-Cheng Tai,et al.  Augmented Lagrangian Method for Total Variation Based Image Restoration and Segmentation Over Triangulated Surfaces , 2012, J. Sci. Comput..

[50]  Vaughan R. Pratt,et al.  Direct least-squares fitting of algebraic surfaces , 1987, SIGGRAPH.

[51]  Xiaoming Yuan,et al.  Alternating algorithms for total variation image reconstruction from random projections , 2012 .

[52]  Chandrajit L. Bajaj,et al.  Automatic reconstruction of surfaces and scalar fields from 3D scans , 1995, SIGGRAPH.

[53]  Shigeru Muraki,et al.  Volumetric shape description of range data using “Blobby Model” , 1991, SIGGRAPH.

[54]  Tony DeRose,et al.  Surface reconstruction from unorganized points , 1992, SIGGRAPH.

[55]  Irina Voiculescu,et al.  Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms , 2009 .

[56]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[57]  Kalpathi R. Subramanian,et al.  Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions , 2001, Proceedings International Conference on Shape Modeling and Applications.

[58]  Marshall W. Bern,et al.  A new Voronoi-based surface reconstruction algorithm , 1998, SIGGRAPH.

[59]  B. Mercier,et al.  A dual algorithm for the solution of nonlinear variational problems via finite element approximation , 1976 .

[60]  G. Farin.  Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[61]  Hendrik Speleers,et al.  THB-splines: The truncated basis for hierarchical splines , 2012, Comput. Aided Geom. Des..

[62]  A. Chambolle,et al.  An introduction to Total Variation for Image Analysis , 2009 .

[63]  Xue-Cheng Tai,et al.  Augmented Lagrangian Method, Dual Methods, and Split Bregman Iteration for ROF, Vectorial TV, and High Order Models , 2010, SIAM J. Imaging Sci..

[64]  Pierre Alliez,et al.  State of the Art in Surface Reconstruction from Point Clouds , 2014, Eurographics.

[65]  Tony F. Chan,et al.  High-Order Total Variation-Based Image Restoration , 2000, SIAM J. Sci. Comput..

[66]  Thomas W. Sederberg.  Piecewise algebraic surface patches , 1985, Comput. Aided Geom. Des..

[67]  Marc Levoy,et al.  A volumetric method for building complex models from range images , 1996, SIGGRAPH.

[68]  Jonathan Eckstein.  Splitting methods for monotone operators with applications to parallel optimization , 1989 .

[69]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .