Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 450460, 17 pages
Research Article

Point Ordering with Natural Distance Based on Brownian Motion

1Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea
2Department of Multimedia Engineering, Dongeui University, Busan 614-714, South Korea

Received 7 August 2009; Revised 24 November 2009; Accepted 12 February 2010

Academic Editor: Gradimir V. Milovanović

Copyright © 2010 Philsu Kim and Hyoungseok Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider a curve reconstruction problem from unorganized point clouds with noise. In general, the result of curve reconstruction depends on how to select and order the representative points to resemble the shape of the clouds. We exploit a natural distance based on a property of one-dimensional Brownian motion to order sample points, which simultaneously reflect smoothness and nearness of points, so that our algorithm is able to reconstruct not only simple curves but also nonsimple curves. Numerous examples show that this algorithm is effective. The natural distance proposed in this paper is able to play an important role in a variety of fields of measuring the distance of points with considering direction.