# Lund University Publications

## LUND UNIVERSITY LIBRARIES

### Restricted mesh simplification using edge contractions

(2009) In International Journal of Computational Geometry and Applications 19(3). p.247-265
Abstract
We consider the problem of simplifying a planar triangle mesh using edge contractions, under the restriction that the resulting vertices must be a subset of the input set. That is, contraction of an edge must be made on to one of its adjacent vertices, which results in removing the other adjacent vertex. We show that if the perimeter of the mesh consists of at most five vertices, then we can always find a vertex not on the perimeter which can be removed in this way. If the perimeter consists of more than five vertices such a vertex may not exist. In order to maintain a higher number of removable vertices under the above restriction, we study edge flips which can be performed in a visually smooth way. A removal of a vertex which is preceded... (More)
We consider the problem of simplifying a planar triangle mesh using edge contractions, under the restriction that the resulting vertices must be a subset of the input set. That is, contraction of an edge must be made on to one of its adjacent vertices, which results in removing the other adjacent vertex. We show that if the perimeter of the mesh consists of at most five vertices, then we can always find a vertex not on the perimeter which can be removed in this way. If the perimeter consists of more than five vertices such a vertex may not exist. In order to maintain a higher number of removable vertices under the above restriction, we study edge flips which can be performed in a visually smooth way. A removal of a vertex which is preceded by one such smooth operation is called a 2-step removal. Moreover, we introduce the possibility that the user defines "important" vertices (or edges) which have to remain intact. Given m such important vertices, or edges, we show that a simplification hierarchy of size O(n) and depth O(log(n/m))can be constructed by 2-step removals in O(n) time, such that the simplified graph contains the m important vertices and edges, and at most O(m) other vertices from the input graph. In some triangulations, many vertices may not even be 2-step removable. In order to provide the option to remove such vertices, we also define and examine k-step removals. This increases the lower bound on the number of removable vertices. (Less)
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
edge contractions, Computational geometry, computer graphics
in
International Journal of Computational Geometry and Applications
volume
19
issue
3
pages
247 - 265
publisher
World Scientific
external identifiers
• wos:000267510600004
• scopus:68049144811
ISSN
0218-1959
DOI
10.1142/S0218195909002940
project
VR 2008-4649
language
English
LU publication?
yes
id
f6883800-e775-4f11-991b-15d18d04aea7 (old id 1463174)
2009-08-18 14:24:34
date last changed
2019-02-20 04:55:47
```@article{f6883800-e775-4f11-991b-15d18d04aea7,
abstract     = {We consider the problem of simplifying a planar triangle mesh using edge contractions, under the restriction that the resulting vertices must be a subset of the input set. That is, contraction of an edge must be made on to one of its adjacent vertices, which results in removing the other adjacent vertex. We show that if the perimeter of the mesh consists of at most five vertices, then we can always find a vertex not on the perimeter which can be removed in this way. If the perimeter consists of more than five vertices such a vertex may not exist. In order to maintain a higher number of removable vertices under the above restriction, we study edge flips which can be performed in a visually smooth way. A removal of a vertex which is preceded by one such smooth operation is called a 2-step removal. Moreover, we introduce the possibility that the user defines "important" vertices (or edges) which have to remain intact. Given m such important vertices, or edges, we show that a simplification hierarchy of size O(n) and depth O(log(n/m))can be constructed by 2-step removals in O(n) time, such that the simplified graph contains the m important vertices and edges, and at most O(m) other vertices from the input graph. In some triangulations, many vertices may not even be 2-step removable. In order to provide the option to remove such vertices, we also define and examine k-step removals. This increases the lower bound on the number of removable vertices.},
author       = {Andersson, Mattias and Gudmundsson, Joachim and Levcopoulos, Christos},
issn         = {0218-1959},
keyword      = {edge contractions,Computational geometry,computer graphics},
language     = {eng},
number       = {3},
pages        = {247--265},
publisher    = {World Scientific},
series       = {International Journal of Computational Geometry and Applications},
title        = {Restricted mesh simplification using edge contractions},
url          = {http://dx.doi.org/10.1142/S0218195909002940},
volume       = {19},
year         = {2009},
}

```