Minimization, constraints and composite Bézier curves

Bercovier, Michel; Jacobi, A.

doi:10.1016/0167-8396(94)90303-4

Cited by 10 publications

(5 citation statements)

References 22 publications

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“…Hoschek [1988] proposes an iterative scheme, called intrinsic parameterization, which also uses the PD error term but performs parameter correction using a formula that is a first-order approximation to exact foot point computation. Bercovier and Jacob [1994] prove that the intrinsic parameterization method is equivalent to Uzawa's method for solving a constrained minimization problem, but they do not establish the convergence rate of the intrinsic parameterization method or that of PDM. Higher order approximation or accurate computation of foot points for data parameterization are discussed in Hoschek and Lasser [1993]; Saux and Daniel [2003]; Hu and Wallner [2005].…”

Section: Spline Curve Fitting Techniquesmentioning

confidence: 99%

Fitting B-spline curves to point clouds by curvature-based squared distance minimization

2006

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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.

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Section: Spline Curve Fitting Techniquesmentioning

confidence: 99%

Fitting B-spline curves to point clouds by curvature-based squared distance minimization

2006

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“…The constrained approximation of point sets with Bezier or B-spline curves (cf. (Rogers, 1989;Bercovier and Jacobi, 1994)) may have various aims in mind, such as increasing the quality of the final fitting, simplifying it or ensuring certain geometric properties of the solution. Especially the latter task is addressed in detail in (Hoschek and Kaklis, 1996) where properties such as convexity and monotony of a final approximation are discussed.…”

Section: Related Workmentioning

confidence: 99%

Fitting curves and surfaces to point clouds in the presence of obstacles

Flöry

2009

Computer Aided Geometric Design

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“…Hoschek and Wissel (1988) introduced a method for merging and splitting Bézier curve segments of different degrees that is based on the square error that sum between the original and approximation is minimized. Bercovier and Jacobi (1994) used a method based on a min-max problem which describes approximation and differential geometric characteristics under some constraints to render offset to the Bézier curve. Li (1998) employed Legendre series as approximation for the offset to the original polynomial curve.…”

Section: Introductionmentioning

confidence: 99%

PH-spline approximation for Bézier curve and rendering offset

Zheng¹,

Wang²

2004

J. Zhejiang Univ. Sci.

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In this paper, a G 1 , C 1 , C 2 PH-spline is employed as an approximation for a given Bézier curve within error bound and further renders offset which can be regarded as an approximate offset to the Bézier curve. The errors between PH-spline and the Bézier curve, the offset to PH-spline and the offset to the given Bézier curve are also estimated. A new algorithm for constructing offset to the Bézier curve is proposed.

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Minimization, constraints and composite Bézier curves

Cited by 10 publications

References 22 publications

Fitting B-spline curves to point clouds by curvature-based squared distance minimization

Fitting B-spline curves to point clouds by curvature-based squared distance minimization

Fitting curves and surfaces to point clouds in the presence of obstacles

PH-spline approximation for Bézier curve and rendering offset

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