Direct Fitting of Gaussian Mixture Models

When fitting Gaussian Mixture Models to 3D geometry, the model is typically fit to point clouds, even when the shapes were obtained as 3D meshes. Here we present a formulation for fitting Gaussian Mixture Models (GMMs) directly to geometric objects, using the triangles of triangular mesh instead of using points sampled from its surface. We demonstrate that this modification enables fitting higher-quality GMMs under a wider range of initialization conditions. Additionally, models obtained from this fitting method are shown to produce an improvement in 3D registration for both meshes and RGB-D frames.

[1]  Nathan Michael,et al.  Fast Monte-Carlo Localization on Aerial Vehicles Using Approximate Continuous Belief Representations , 2017, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition.

[2]  Jarek Rossignac,et al.  Multi-resolution 3D approximations for rendering complex scenes , 1993, Modeling in Computer Graphics.

[3]  Nathan Michael,et al.  On-Manifold GMM Registration , 2018, IEEE Robotics and Automation Letters.

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

[5]  Andrew W. Moore,et al.  X-means: Extending K-means with Efficient Estimation of the Number of Clusters , 2000, ICML.

[6]  Paolo Cignoni,et al.  Ieee Transactions on Visualization and Computer Graphics 1 Efficient and Flexible Sampling with Blue Noise Properties of Triangular Meshes , 2022 .

[7]  Heinrich Niemann,et al.  Using Quaternions for Parametrizing 3-D Rotations in Unconstrained Nonlinear Optimization , 2001, VMV.

[8]  Paul J. Besl,et al.  Method for registration of 3-D shapes , 1992, Other Conferences.

[9]  V. Hasselblad.  Estimation of parameters for a mixture of normal distributions , 1966 .

[10]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[11]  Jan Kautz,et al.  MLMD: Maximum Likelihood Mixture Decoupling for Fast and Accurate Point Cloud Registration , 2015, 2015 International Conference on 3D Vision.

[12]  Joachim Hertzberg,et al.  Evaluation of 3D registration reliability and speed - A comparison of ICP and NDT , 2009, 2009 IEEE International Conference on Robotics and Automation.

[13]  R. Feynman.  An Operator calculus having applications in quantum electrodynamics , 1951 .

[14]  Andriy Myronenko,et al.  Point Set Registration: Coherent Point Drift , 2009, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[15]  Anders Grunnet-Jepsen,et al.  Intel RealSense Stereoscopic Depth Cameras , 2017, CVPR 2017.

[16]  Danilo P. Mandic,et al.  Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms , 2016, IEEE Transactions on Neural Networks and Learning Systems.

[17]  Nathan Michael,et al.  Approximate continuous belief distributions for precise autonomous inspection , 2016, 2016 IEEE International Symposium on Safety, Security, and Rescue Robotics (SSRR).

[18]  Sergei Vassilvitskii,et al.  k-means++: the advantages of careful seeding , 2007, SODA '07.

[19]  Achim J. Lilienthal,et al.  Point set registration through minimization of the L2 distance between 3D-NDT models , 2012, 2012 IEEE International Conference on Robotics and Automation.

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

[21]  Andrew W. Fitzgibbon,et al.  KinectFusion: Real-time dense surface mapping and tracking , 2011, 2011 10th IEEE International Symposium on Mixed and Augmented Reality.

[22]  P. Alliez,et al.  Principal Component Analysis in CGAL , 2008 .

[23]  Agamirza E. Bashirov,et al.  Multiplicative calculus and its applications , 2008 .

[24]  Nuno Vasconcelos,et al.  Learning Mixture Hierarchies , 1998, NIPS.

[25]  E. Polak,et al.  Note sur la convergence de méthodes de directions conjuguées , 1969 .

[26]  Raymond A. Guenther.  Product integrals and sum integrals , 1983 .

[27]  Baba C. Vemuri,et al.  Robust Point Set Registration Using Gaussian Mixture Models , 2011, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[28]  Martin Magnusson,et al.  The three-dimensional normal-distributions transform : an efficient representation for registration, surface analysis, and loop detection , 2009 .

[29]  Richard I. Hartley,et al.  Global Optimization through Rotation Space Search , 2009, International Journal of Computer Vision.

[30]  R. Sridharan,et al.  Gaussian mixture models and the EM algorithm , 2014 .

[31]  Brian Mirtich,et al.  Fast and Accurate Computation of Polyhedral Mass Properties , 1996, J. Graphics, GPU, & Game Tools.

[32]  Achim J. Lilienthal,et al.  Fast and accurate scan registration through minimization of the distance between compact 3D NDT representations , 2012, Int. J. Robotics Res..

[33]  Jan Kautz,et al.  Accelerated Generative Models for 3D Point Cloud Data , 2016, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[34]  J. Shewchuk.  An Introduction to the Conjugate Gradient Method Without the Agonizing Pain , 1994 .

[35]  Wolfram Burgard,et al.  A benchmark for the evaluation of RGB-D SLAM systems , 2012, 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[36]  Michael Garland,et al.  Surface simplification using quadric error metrics , 1997, SIGGRAPH.

[37]  Gaël Varoquaux,et al.  Scikit-learn: Machine Learning in Python , 2011, J. Mach. Learn. Res..

[38]  Jan Kautz,et al.  HGMR: Hierarchical Gaussian Mixtures for Adaptive 3D Registration , 2018, ECCV.

[39]  Jonathan A. Tawn,et al.  The multivariate Gaussian tail model: an application to oceanographic data , 2000 .