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Efficiently Computing Exact Geodesic Loops within Finite Steps
June 2012 (vol. 18 no. 6)
pp. 879-889
ASCII Text
x 
 
Shi-Qing Xin, Ying He, Chi-Wing Fu, "Efficiently Computing Exact Geodesic Loops within Finite Steps," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 6, pp. 879-889, June, 2012.
 
 
BibTex
x
 
@article{ 10.1109/TVCG.2011.119,
author = {Shi-Qing Xin and Ying He and Chi-Wing Fu},
title = {Efficiently Computing Exact Geodesic Loops within Finite Steps},
journal ={IEEE Transactions on Visualization and Computer Graphics},
volume = {18},
number = {6},
issn = {1077-2626},
year = {2012},
pages = {879-889},
doi = {http://doi.ieeecomputersociety.org/10.1109/TVCG.2011.119},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Visualization and Computer Graphics
TI - Efficiently Computing Exact Geodesic Loops within Finite Steps
IS - 6
SN - 1077-2626
SP879
EP889
EPD - 879-889
A1 - Shi-Qing Xin,
A1 - Ying He,
A1 - Chi-Wing Fu,
PY - 2012
KW - Discrete geodesic
KW - geodesic loop
KW - triangular mesh.
VL - 18
JA - IEEE Transactions on Visualization and Computer Graphics
ER -
 
Shi-Qing Xin, Nanyang Technological University, Singapore
Ying He, Nanyang Technological University, Singapore
Chi-Wing Fu, Nanyang Technological University, Singapore
Closed geodesics, or geodesic loops, are crucial to the study of differential topology and differential geometry. Although the existence and properties of closed geodesics on smooth surfaces have been widely studied in mathematics community, relatively little progress has been made on how to compute them on polygonal surfaces. Most existing algorithms simply consider the mesh as a graph and so the resultant loops are restricted only on mesh edges, which are far from the actual geodesics. This paper is the first to prove the existence and uniqueness of geodesic loop restricted on a closed face sequence; it contributes also with an efficient algorithm to iteratively evolve an initial closed path on a given mesh into an exact geodesic loop within finite steps. Our proposed algorithm takes only an O(k) space complexity and an O(mk) time complexity (experimentally), where m is the number of vertices in the region bounded by the initial loop and the resultant geodesic loop, and k is the average number of edges in the edge sequences that the evolving loop passes through. In contrast to the existing geodesic curvature flow methods which compute an approximate geodesic loop within a predefined threshold, our method is exact and can apply directly to triangular meshes without needing to solve any differential equation with a numerical solver; it can run at interactive speed, e.g., in the order of milliseconds, for a mesh with around 50K vertices, and hence, significantly outperforms existing algorithms. Actually, our algorithm could run at interactive speed even for larger meshes. Besides the complexity of the input mesh, the geometric shape could also affect the number of evolving steps, i.e., the performance. We motivate our algorithm with an interactive shape segmentation example shown later in the paper.

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Index Terms:
Discrete geodesic, geodesic loop, triangular mesh.
Citation:
Shi-Qing Xin, Ying He, Chi-Wing Fu, "Efficiently Computing Exact Geodesic Loops within Finite Steps," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 6, pp. 879-889, June 2012, doi:10.1109/TVCG.2011.119
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