Topological quadrangulations of closed triangulated surfaces using the Reeb graph

Abstract

Although surfaces are more and more often represented by dense triangulations, it can be useful to convert them to B-spline surface patches, lying on quadrangles. This paper presents a method for the construction of coarse topological quadrangulations of closed triangulated surfaces, based on Morse theory. In order to construct on the surface a quadrangulation of its canonical polygonal schema, we compute first a Reeb graph then a canonical set of generators embedded on the surface. Some results are shown on different surfaces.

A hybrid parallel algorithm for computing and tracking level set topology

“…Contour trees have also been used in various other applications including topography and GIS [31], [34], for surface segmentation and parameterization in computer graphics [23], [27], [39], image processing and analysis of volume data sets [13], [33], designing transfer functions for volume rendering in scientific visualization [19], [32], [38], [40], and exploring high dimensional data in information visualization [21], [28].…”

A randomized O ( m log m ) time algorithm for computing Reeb graphs of arbitrary simplicial complexes

“…If (r == q) // then t and σ have the same vertices (9) If (e 1 = e 4 ) Then identifyEdges(e 1 , e 4 ); (10) If (e 2 = e 5 ) Then identifyEdges(e 2 , e 5 ); (11) Else (12) create new edge e 6 with endpoints q, r (13) create triangle (e 1 , e 4 , e 6 ) and triangle (e 2 , e 5 , e 6 ) (14)…”

“…It was introduced in graphics applications by Shinagawa et al [22]. Since then, it has been used in a range of shape analysis applications, including shape understanding [2,3,22], segmentation and matching [14,20,25,27], shape repairing and animation [15,29], surface quadrangulations and parametrization [13,18,30]. See [4] for a very nice survey paper on the Reeb graph and its applications.…”

3D Shape Matching through Topological Structures

“…However, the application domain restricts the choice of f ; for instance, a suitable mapping function f has to be independent of rotation, translation, uniform scaling of the object and user's choices. These requirements prevent the use for matching purposes of the height function [1], and the centerline representation [11,5] which respectively depend on the orientation and on the selection of a seed point. The family of continuous or Morse functions is a natural set for identifying f , and in the following we present an overview of possible choices of f for coding triangular meshes without boundary.…”