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Abstract

Point clouds are a common representation of three-dimensional shapes in computer graphics and 3D-data processing. However, in some applications, other representations are more useful. Gaussian Mixture Models (GMMs) can be used as such an alternative representation. A GMM is a convex sum of normal distributions, which aims to describe a point cloud’s density. In this thesis, we investigate both visualization and construction of GMMs. For visualization, we have implemented a tool that enables both isoellipsoid and density visualization of GMMs. We describe the mathematical backgrounds, the algorithms, and our implementation of this tool. Regarding GMM construction, we investigate several algorithms used in previous papers for constructing GMMs for 3D-data processing tasks. We present our implementations of the expectation-maximization (EM) algorithm and top-down HEM. Additionally, we have adapted the implementation of geometrically regularized bottom-up HEM to produce a fixed number of Gaussians. We evaluate these three algorithms in terms of the quality of their generated GMMs. In many cases, the statistical likelihood, which is maximized by the EM algorithm, is not a reliable indicator for a GMM’s quality. Therefore, we instead rely on the reconstruction error of a reconstructed point cloud based on the Chamfer distance. Additionally, we provide metrics for measuring the reconstructed point cloud’s uniformity and the GMM’s variation of Gaussians. We demonstrate that EM provides the best results in terms of these metrics. Top-down HEM is a fast alternative, and can produce even better results when using fewer input points. The results of geometrically regularized bottom-up HEM are inferior for lower numbers of Gaussians but it can create good GMMs consisting of high numbers of Gaussians very eciently.

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BibTeX

@mastersthesis{FRAISS-2022-CGMM,
  title =      "Construction and Visualization of Gaussian Mixture Models
               from Point Clouds for 3D Object Representation",
  author =     "Simon Maximilian  Fraiss",
  year =       "2022",
  abstract =   "Point clouds are a common representation of
               three-dimensional shapes in computer graphics and 3D-data
               processing. However, in some applications, other
               representations are more useful. Gaussian Mixture Models
               (GMMs) can be used as such an alternative representation. A
               GMM is a convex sum of normal distributions, which aims to
               describe a point cloud’s density. In this thesis, we
               investigate both visualization and construction of GMMs. For
               visualization, we have implemented a tool that enables both
               isoellipsoid and density visualization of GMMs. We describe
               the mathematical backgrounds, the algorithms, and our
               implementation of this tool. Regarding GMM construction, we
               investigate several algorithms used in previous papers for
               constructing GMMs for 3D-data processing tasks. We present
               our implementations of the expectation-maximization (EM)
               algorithm and top-down HEM. Additionally, we have adapted
               the implementation of geometrically regularized bottom-up
               HEM to produce a fixed number of Gaussians. We evaluate
               these three algorithms in terms of the quality of their
               generated GMMs. In many cases, the statistical likelihood,
               which is maximized by the EM algorithm, is not a reliable
               indicator for a GMM’s quality. Therefore, we instead rely
               on the reconstruction error of a reconstructed point cloud
               based on the Chamfer distance. Additionally, we provide
               metrics for measuring the reconstructed point cloud’s
               uniformity and the GMM’s variation of Gaussians. We
               demonstrate that EM provides the best results in terms of
               these metrics. Top-down HEM is a fast alternative, and can
               produce even better results when using fewer input points.
               The results of geometrically regularized bottom-up HEM are
               inferior for lower numbers of Gaussians but it can create
               good GMMs consisting of high numbers of Gaussians very
               eciently.",
  month =      mar,
  address =    "Favoritenstrasse 9-11/E193-02, A-1040 Vienna, Austria",
  school =     "Research Unit of Computer Graphics, Institute of Visual
               Computing and Human-Centered Technology, Faculty of
               Informatics, TU Wien ",
  URL =        "https://www.cg.tuwien.ac.at/research/publications/2022/FRAISS-2022-CGMM/",
}