Intrinsic parametrization for approximation
Computer Aided Geometric Design
Choosing nodes in parametric curve interpolation
Computer-Aided Design
Turning point preserving planar interpolation
ACM Transactions on Graphics (TOG)
Parameter optimization in approximating curves and surfaces to measurement data
Computer Aided Geometric Design
Curves and surfaces for computer aided geometric design (3rd ed.): a practical guide
Curves and surfaces for computer aided geometric design (3rd ed.): a practical guide
Fundamentals of computer aided geometric design
Fundamentals of computer aided geometric design
Minimization, constraints and composite Be´zier curves
Computer Aided Geometric Design
Piecewise smooth surface reconstruction
SIGGRAPH '94 Proceedings of the 21st annual conference on Computer graphics and interactive techniques
Shape Modeling with Front Propagation: A Level Set Approach
IEEE Transactions on Pattern Analysis and Machine Intelligence
Surface fitting with hierarchical splines
ACM Transactions on Graphics (TOG)
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
Global reparametrization for curve approximation
Computer Aided Geometric Design
SIGGRAPH '85 Proceedings of the 12th annual conference on Computer graphics and interactive techniques
Grouping and parameterizing irregularly spaced points for curve fitting
ACM Transactions on Graphics (TOG)
Curve Fitting with Conic Splines
ACM Transactions on Graphics (TOG)
Advanced surface fitting techniques
Computer Aided Geometric Design
Multi-weight enveloping: least-squares approximation techniques for skin animation
Proceedings of the 2002 ACM SIGGRAPH/Eurographics symposium on Computer animation
Range-Image segmentation and model reconstruction based on a fit-and-merge strategy
Proceedings of the seventh ACM symposium on Solid modeling and applications
Computational Line Geometry
Smooth approximation and rendering of large scattered data sets
Proceedings of the conference on Visualization '01
Graphical Models - Pacific graphics 2001
Surface construction by fitting unorganized curves
Graphical Models
Curve-fitting with piecewise parametric cubics
SIGGRAPH '83 Proceedings of the 10th annual conference on Computer graphics and interactive techniques
Approximation with Active B-Spline Curves and Surfaces
PG '02 Proceedings of the 10th Pacific Conference on Computer Graphics and Applications
An improved Hoschek intrinsic parametrization
Computer Aided Geometric Design
A second order algorithm for orthogonal projection onto curves and surfaces
Computer Aided Geometric Design
Spline curve approximation and design by optimal control over the knots
Computing - Geometric modelling dagstuhl 2002
B-spline curve fitting based on adaptive curve refinement using dominant points
Computer-Aided Design
Evolution-based least-squares fitting using Pythagorean hodograph spline curves
Computer Aided Geometric Design
Accurate integration of multi-view range images using k-means clustering
Pattern Recognition
Moving parabolic approximation of point clouds
Computer-Aided Design
Dual evolution of planar parametric spline curves and T-spline level sets
Computer-Aided Design
Constrained curve fitting on manifolds
Computer-Aided Design
Approximate computation of curves on B-spline surfaces
Computer-Aided Design
ACM SIGGRAPH 2008 papers
Semantic fitting and reconstruction
Journal on Computing and Cultural Heritage (JOCCH)
Fitting curves and surfaces to point clouds in the presence of obstacles
Computer Aided Geometric Design
A clustering approach to free form surface reconstruction from multi-view range images
Image and Vision Computing
Circular spline fitting using an evolution process
Journal of Computational and Applied Mathematics
Surface fitting with NURBS: a gauss Newton with trust region approach
MATH'08 Proceedings of the 13th WSEAS international conference on Applied mathematics
Finding positively invariant sets of a class of nonlinear loops via curve fitting
Proceedings of the 2009 conference on Symbolic numeric computation
Analytic drawing of 3D scaffolds
ACM SIGGRAPH Asia 2009 papers
Computer Aided Geometric Design
Distance regression by Gauss–Newton-type methods and iteratively re-weighted least-squares
Computing - Geometric Modelling, Dagstuhl 2008
Weighted progressive iteration approximation and convergence analysis
Computer Aided Geometric Design
Fitting sharp features with loop subdivision surfaces
SGP '08 Proceedings of the Symposium on Geometry Processing
Cubic B-spline curve approximation by curve unclamping
Computer-Aided Design
A revisit to least squares orthogonal distance fitting of parametric curves and surfaces
GMP'08 Proceedings of the 5th international conference on Advances in geometric modeling and processing
Algorithm for orthogonal projection of parametric curves onto B-spline surfaces
Computer-Aided Design
Technical Section: An extended iterative format for the progressive-iteration approximation
Computers and Graphics
Two-view curve reconstruction based on the snake model
Journal of Computational and Applied Mathematics
Technical Section: Quadratic curve and surface fitting via squared distance minimization
Computers and Graphics
Efficient particle swarm optimization approach for data fitting with free knot B-splines
Computer-Aided Design
Least–squares approximation by pythagorean hodograph spline curves via an evolution process
GMP'06 Proceedings of the 4th international conference on Geometric Modeling and Processing
Volumetric geometry reconstruction of turbine blades for aircraft engines
Proceedings of the 7th international conference on Curves and Surfaces
SMI 2012: Short A revisit to fitting parametric surfaces to point clouds
Computers and Graphics
Fast B-spline curve fitting by L-BFGS
Computer Aided Geometric Design
G1 continuous approximate curves on NURBS surfaces
Computer-Aided Design
Computer Graphics in China: Convergence analysis for B-spline geometric interpolation
Computers and Graphics
A framework for 3D model reconstruction in reverse engineering
Computers and Industrial Engineering
A new iterative mutually coupled hybrid GA-PSO approach for curve fitting in manufacturing
Applied Soft Computing
Curvature-guided adaptive T-spline surface fitting
Computer-Aided Design
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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 B-spline 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 B-spline 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 curvature-based 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 B-spline curve fitting problem as a nonlinear least squares problem and conclude that SDM is a quasi-Newton method which employs a curvature-based 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 Gauss-Newton 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.