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Article: Fitting Bspline curves to point clouds by curvaturebased squared distance minimization
Title  Fitting Bspline curves to point clouds by curvaturebased squared distance minimization 

Authors  
Keywords  BSpline Curve Curve Fitting GaussNewton Method Least Squares Problem Optimization Point Cloud QuasiNewton Method Scatter Data Approximation Shape Reconstruction Squared Distance 
Issue Date  2006 
Citation  Acm Transactions On Graphics, 2006, v. 25 n. 2, p. 214238 How to Cite? 
Abstract  Computing a curve to approximate data points is a problem encountered frequently in many applications in computer graphics, computer vision, CAD/CAM, and image processing. We present a novel and efficient method, called squared distance minimization (SDM), for computing a planar Bspline curve, closed or open, to approximate a target shape defined by a point cloud, that is, a set of unorganized, possibly noisy data points. We show that SDM significantly outperforms other optimization methods used currently in common practice of curve fitting. In SDM, a Bspline curve starts from some properly specified initial shape and converges towards the target shape through iterative quadratic minimization of the fitting error. Our contribution is the introduction of a new fitting error term, called the squared distance (SD) error term, defined by a curvaturebased quadratic approximant of squared distances from data points to a fitting curve. The SD error term faithfully measures the geometric distance between a fitting curve and a target shape, thus leading to faster and more stable convergence than the point distance (PD) error term, which is commonly used in computer graphics and CAGD, and the tangent distance (TD) error term, which is often adopted in the computer vision community. To provide a theoretical explanation of the superior performance of SDM, we formulate the Bspline curve fitting problem as a nonlinear least squares problem and conclude that SDM is a quasiNewton method which employs a curvaturebased positive definite approximant to the true Hessian of the objective function. Furthermore, we show that the method based on the TD error term is a GaussNewton iteration, which is unstable for target shapes with high curvature variations, whereas optimization based on the PD error term is the alternating method that is known to have linear convergence. © 2006 ACM. 
Persistent Identifier  http://hdl.handle.net/10722/152338 
ISSN  2015 Impact Factor: 4.218 2015 SCImago Journal Rankings: 2.552 
ISI Accession Number ID  
References 
DC Field  Value  Language 

dc.contributor.author  Wang, W  en_US 
dc.contributor.author  Pottmann, H  en_US 
dc.contributor.author  Liu, Y  en_US 
dc.date.accessioned  20120626T06:37:18Z   
dc.date.available  20120626T06:37:18Z   
dc.date.issued  2006  en_US 
dc.identifier.citation  Acm Transactions On Graphics, 2006, v. 25 n. 2, p. 214238  en_US 
dc.identifier.issn  07300301  en_US 
dc.identifier.uri  http://hdl.handle.net/10722/152338   
dc.description.abstract  Computing a curve to approximate data points is a problem encountered frequently in many applications in computer graphics, computer vision, CAD/CAM, and image processing. We present a novel and efficient method, called squared distance minimization (SDM), for computing a planar Bspline curve, closed or open, to approximate a target shape defined by a point cloud, that is, a set of unorganized, possibly noisy data points. We show that SDM significantly outperforms other optimization methods used currently in common practice of curve fitting. In SDM, a Bspline curve starts from some properly specified initial shape and converges towards the target shape through iterative quadratic minimization of the fitting error. Our contribution is the introduction of a new fitting error term, called the squared distance (SD) error term, defined by a curvaturebased quadratic approximant of squared distances from data points to a fitting curve. The SD error term faithfully measures the geometric distance between a fitting curve and a target shape, thus leading to faster and more stable convergence than the point distance (PD) error term, which is commonly used in computer graphics and CAGD, and the tangent distance (TD) error term, which is often adopted in the computer vision community. To provide a theoretical explanation of the superior performance of SDM, we formulate the Bspline curve fitting problem as a nonlinear least squares problem and conclude that SDM is a quasiNewton method which employs a curvaturebased positive definite approximant to the true Hessian of the objective function. Furthermore, we show that the method based on the TD error term is a GaussNewton iteration, which is unstable for target shapes with high curvature variations, whereas optimization based on the PD error term is the alternating method that is known to have linear convergence. © 2006 ACM.  en_US 
dc.language  eng  en_US 
dc.relation.ispartof  ACM Transactions on Graphics  en_US 
dc.rights  ACM Transactions on Graphics. Copyright © Association for Computing Machinery, Inc.   
dc.subject  BSpline Curve  en_US 
dc.subject  Curve Fitting  en_US 
dc.subject  GaussNewton Method  en_US 
dc.subject  Least Squares Problem  en_US 
dc.subject  Optimization  en_US 
dc.subject  Point Cloud  en_US 
dc.subject  QuasiNewton Method  en_US 
dc.subject  Scatter Data Approximation  en_US 
dc.subject  Shape Reconstruction  en_US 
dc.subject  Squared Distance  en_US 
dc.title  Fitting Bspline curves to point clouds by curvaturebased squared distance minimization  en_US 
dc.type  Article  en_US 
dc.identifier.email  Wang, W:wenping@cs.hku.hk  en_US 
dc.identifier.authority  Wang, W=rp00186  en_US 
dc.description.nature  link_to_subscribed_fulltext  en_US 
dc.identifier.doi  10.1145/1138450.1138453  en_US 
dc.identifier.scopus  eid_2s2.033745771700  en_US 
dc.identifier.hkuros  141115   
dc.relation.references  http://www.scopus.com/mlt/select.url?eid=2s2.033745771700&selection=ref&src=s&origin=recordpage  en_US 
dc.identifier.volume  25  en_US 
dc.identifier.issue  2  en_US 
dc.identifier.spage  214  en_US 
dc.identifier.epage  238  en_US 
dc.identifier.eissn  15577368   
dc.identifier.isi  WOS:000238447200003   
dc.publisher.place  United States  en_US 
dc.identifier.scopusauthorid  Wang, W=35147101600  en_US 
dc.identifier.scopusauthorid  Pottmann, H=7003351050  en_US 
dc.identifier.scopusauthorid  Liu, Y=27172089200  en_US 
dc.identifier.citeulike  2461073   