Low Rank Matrix Approximation for Geometry Filtering

We propose a robust, anisotropic normal estimation method for both point clouds and meshes using a low rank matrix approximation algorithm. First, we compute a local feature descriptor for each point and find similar, non-local neighbors that we organize into a matrix. We then show that a low rank matrix approximation algorithm can robustly estimate normals for both point clouds and meshes. Furthermore, we provide a new filtering method for point cloud data to anisotropically smooth the position data to fit the estimated normals. We show applications of our method to point cloud filtering, point set upsampling, surface reconstruction, mesh denoising, and geometric texture removal. Our experiments show that our method outperforms current methods in both visual quality and accuracy.

[1]  Sébastien Valette,et al.  Sparse Geometric Representation Through Local Shape Probing , 2016, IEEE Transactions on Visualization and Computer Graphics.

[2]  Keenan Crane,et al.  A General Framework for Bilateral and Mean Shift Filtering , 2014, ArXiv.

[3]  Alexandre Boulch,et al.  Fast and Robust Normal Estimation for Point Clouds with Sharp Features , 2012, Comput. Graph. Forum.

[4]  Konrad Polthier,et al.  Anisotropic smoothing of point sets, , 2005, Comput. Aided Geom. Des..

[5]  Alexandre Boulch,et al.  Deep Learning for Robust Normal Estimation in Unstructured Point Clouds , 2016, Comput. Graph. Forum.

[6]  Daniel Cohen-Or,et al.  Edge-aware point set resampling , 2013, ACM Trans. Graph..

[7]  Lei Zhang,et al.  Weighted Nuclear Norm Minimization and Its Applications to Low Level Vision , 2016, International Journal of Computer Vision.

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

[9]  Lei He,et al.  Mesh denoising via L0 minimization , 2013, ACM Trans. Graph..

[10]  Emmanuel J. Candès,et al.  A Singular Value Thresholding Algorithm for Matrix Completion , 2008, SIAM J. Optim..

[11]  Ralph R. Martin,et al.  Fast and Effective Feature-Preserving Mesh Denoising , 2007, IEEE Transactions on Visualization and Computer Graphics.

[12]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2008, Found. Comput. Math..

[13]  Pierre Alliez,et al.  Eurographics Symposium on Geometry Processing (2007) Voronoi-based Variational Reconstruction of Unoriented Point Sets , 2022 .

[14]  Bao Li,et al.  Robust normal estimation for point clouds with sharp features , 2010, Comput. Graph..

[15]  Daniel Cohen-Or,et al.  Consolidation of unorganized point clouds for surface reconstruction , 2009, ACM Trans. Graph..

[16]  Julie Digne,et al.  Similarity based filtering of point clouds , 2012, 2012 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops.

[17]  Marc Alexa,et al.  Point set surfaces , 2001, Proceedings Visualization, 2001. VIS '01..

[18]  Nathan Halko,et al.  Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions , 2009, SIAM Rev..

[19]  Jie Zhang,et al.  Quality point cloud normal estimation by guided least squares representation , 2015, Comput. Graph..

[20]  N. Mitra,et al.  Non-local scan consolidation for 3D urban scenes , 2010, ACM Trans. Graph..

[21]  Ligang Liu,et al.  Multi-Normal Estimation via Pair Consistency Voting , 2019, IEEE Transactions on Visualization and Computer Graphics.

[22]  Charlie C. L. Wang,et al.  Bilateral recovering of sharp edges on feature-insensitive sampled meshes , 2006, IEEE Transactions on Visualization and Computer Graphics.

[23]  Konrad Polthier,et al.  Mesh Denoising Based on Normal Voting Tensor and Binary Optimization , 2016, IEEE Transactions on Visualization and Computer Graphics.

[24]  Yuzhong Shen,et al.  Fuzzy vector median-based surface smoothing , 2004, IEEE Transactions on Visualization and Computer Graphics.

[25]  Lei Zhang,et al.  Weighted Nuclear Norm Minimization with Application to Image Denoising , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[26]  Niloy J. Mitra,et al.  Estimating surface normals in noisy point cloud data , 2003, SCG '03.

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

[28]  Youyi Zheng,et al.  IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 1 Bilateral Normal Filtering for Mesh Denoising , 2022 .

[29]  Wenping Wang,et al.  Denoising point sets via L0 minimization , 2015, Comput. Aided Geom. Des..

[30]  Michael Wimmer,et al.  Continuous projection for fast L1 reconstruction , 2014, ACM Trans. Graph..

[31]  Yong Yu,et al.  Robust Subspace Segmentation by Low-Rank Representation , 2010, ICML.

[32]  Yi Ma,et al.  TILT: Transform Invariant Low-Rank Textures , 2010, ACCV 2010.

[33]  Wenzhi Chen,et al.  Robust mesh denoising via vertex pre-filtering and L1-median normal filtering , 2017, Comput. Aided Geom. Des..

[34]  Matthias Zwicker,et al.  GPF: GMM-Inspired Feature-Preserving Point Set Filtering , 2018, IEEE Transactions on Visualization and Computer Graphics.

[35]  Xin Tong,et al.  Mesh denoising via cascaded normal regression , 2016, ACM Trans. Graph..

[36]  Gérard G. Medioni,et al.  A Closed-Form Solution to Tensor Voting: Theory and Applications , 2012, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[37]  Frédo Durand,et al.  Non-iterative, feature-preserving mesh smoothing , 2003, ACM Trans. Graph..

[38]  John Wright,et al.  Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization , 2009, NIPS.

[39]  Bailin Deng,et al.  Guided Mesh Normal Filtering , 2015, Comput. Graph. Forum.

[40]  Jian Liu,et al.  Point cloud normal estimation via low-rank subspace clustering , 2013, Comput. Graph..

[41]  Daniel Cohen-Or,et al.  Bilateral mesh denoising , 2003 .

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

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

[44]  Yutaka Ohtake,et al.  Mesh smoothing via mean and median filtering applied to face normals , 2002, Geometric Modeling and Processing. Theory and Applications. GMP 2002. Proceedings.

[45]  Markus H. Gross,et al.  Feature Preserving Point Set Surfaces based on Non‐Linear Kernel Regression , 2009, Comput. Graph. Forum.

[46]  Wenping Wang,et al.  Feature-preserving mesh denoising via bilateral normal filtering , 2005, Ninth International Conference on Computer Aided Design and Computer Graphics (CAD-CG'05).

[47]  Yutaka Ohtake,et al.  Mesh denoising via iterative alpha-trimming and nonlinear diffusion of normals with automatic thresholding , 2003, Proceedings Computer Graphics International 2003.

[48]  Ralph R. Martin,et al.  Random walks for feature-preserving mesh denoising , 2008, Computer Aided Geometric Design.

[49]  Baining Guo,et al.  Rolling guidance normal filter for geometric processing , 2015, ACM Trans. Graph..