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Computing Riemannian Normal Coordinates on Triangle Meshes
Centre for Image Analysis, Swedish University of Agricultural Sciences, Sweden.
Linköping University, Department of Science and Technology. Linköping University, The Institute of Technology.
Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Norway.
Linköping University, Department of Science and Technology, Digital Media. Linköping University, The Institute of Technology.
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(English)Manuscript (preprint) (Other academic)
Abstract [en]

Imagine an ant walking around on the curved surface of a plant, a radio amateur planning to broadcast to a distant location across the globe or a pilot taking o from an airport - all of them are helped by egocentric maps of the world around them that shows directions and distances to various remote places. It is not surprising that this idea has already been used in cartography, where it is known as Azimuthal Equidistant Projection (AEP). If Earth is approximated by a sphere, distances and directions between two places are computed from arcs along great circles. In physics and mathematics, the same idea is known as Riemannian Normal Coordinates (RNC). It has been given a precise and general denition for surfaces (2-D), curved spaces (3-D) and generalized to smooth manifolds (N-D). RNC are the Cartesian coordinates of vectors that index points on the surface (or manifold) through the so called exponential map, which is a well known concept in dierential geometry. They are easily computed for a particular point if the inverse of the exponential map, the logarithm map, is known. Recently, RNC and similar coordinate systems have been used in computer graphics, visualization and related areas of research. In Fig. 1 for instance, RNC are used to produce a texture on the Stanford bunny through decal compositing. Given the growing use of RNC, which is further elaborated on in the next section, it is meaningful to develop accurate and reproducible techniques to compute this parameterization. In this paper, we describe a technique to compute RNC for surfaces represented by triangular meshes, which is the predominant representation of surfaces in computer graphics. The method that we propose has similarities to the Logmap framework, which has previously been developed for dimension reduction of unorganized point clouds in high-dimensional spaces, a.k.a. manifold learning. For this reason we sometimes refer to it as "Logmap for triangular meshes" or simply Logmap.

National Category
Engineering and Technology
URN: urn:nbn:se:liu:diva-54826OAI: diva2:310611
Available from: 2010-04-15 Created: 2010-04-15 Last updated: 2013-08-28
In thesis
1. Level-set methods and geodesic distance functions
Open this publication in new window or tab >>Level-set methods and geodesic distance functions
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The work in this thesis focuses on efficient implementations of level-set methods and geodesic distance functions. The level-set method is a grid based design that inherits many favorable traits from implicit geometry. It is connected to distance functions through its special way of representing geometry: in ìo each point in space stores the closest distance to the surface. To differentiate between the inside and outside of a closed object a signed distance is used. In the discrete form the representation keeps a box around the surface that stores regularly positioned samples of the distance function – i.e. a grid. These samples implicitly encode the surface as the zeroth level-set of the signed distance function, hence the name level-set methods. With this representation of geometry follows a toolbox of operations based on partial differential equations (PDE). The solution to these PDES allows for arbitrary motion and deformation of the surface.

This thesis focuses on two topics: 1) grid storage for level-set methods, and 2) geodesic distance functions and parameterization. These topics are covered in a series of in-depth articles.

Today, level-set methods are becoming widespread in both academia and industry. Data structures and highly accurate methods and numerical schemes are available that allow for efficient handling of topological changes of dynamic curves and surfaces. For some applications, such as the capturing of the air/water interface in free surface fluid simulations, it’s is the only realistic choice. In other areas level-set methods are emerging as a competitive candidate to triangle meshes and other explicit representations.

In particular this work introduces efficient level-set data-structures that allow for extremely detailed simulations and representations. It also presents a parameterization method based on geodesic distance that produces a unique coordinate system, the Riemannian normal coordinates (RNC). Amongst other interesting applications this parameterization can be used for decal compositing, and the translation of vector space algorithms to surfaces. The approximation of the RNC involves one or more distance functions. In this thesis, a method originally presented for triangle meshes is adopted. It is then and extended to compute accurate geodesic distance in anisotropic domains in two and three dimensions. The extension to higher dimensions is also outlined.

To motivate this work several applications based on these novel methods and data structures are presented showing rapid ray-tracing, shape morphing, segmentation, geodesic interpolation, texture mapping, and more.

Place, publisher, year, edition, pages
Linköping: Linköping Universisty Electronic Press, 2009. 92 p.
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1275
National Category
Engineering and Technology
urn:nbn:se:liu:diva-54830 (URN)978-91-7393-524-1 (ISBN)
Public defence
2009-11-19, K3. Kåkenhus, Campus Norrköping, Linköpings universitet, Norrköping, 13:00 (English)
Available from: 2010-04-15 Created: 2010-04-15 Last updated: 2010-06-21Bibliographically approved

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