An improved Hoschek intrinsic parametrization

Saux, Éric; Daniel, Marc

doi:10.1016/j.cagd.2003.06.004

Cited by 30 publications

(16 citation statements)

References 10 publications

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“…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%

“…PDM, or its simple variants, are the most commonly applied methods for curve fitting in computer graphics and CAD [Plass and Stone 1983;Hoschek 1988;Goshtasby 2000;Saux and Daniel 2003]. The same idea of PDM is also widely used for surface fitting [Hoppe et al 1994;Forsey and Bartels 1995;Hoppe 1996;Ma and Kruth 1995;Haber et al 2001;Wang and Phillips 2002;Djebali et al 2002;Greiner et al 2002;Weiss et al 2002;Maekawa and Ko 2002;Taubin 2002] with B-spline surfaces as well as other types of surfaces.…”

Section: Spline Curve Fitting Techniquesmentioning

confidence: 99%

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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%

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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“…(Hoschek, 1988) tackles the problem of unordered target data with an iterative method of intrinsic parametrization and approximates the foot point computation by a first order term. Approximations of higher order (Saux and Daniel, 2003;Hu and Wallner, 2005) or an accurate foot point computation (Hoschek and Lasser, 1993) in each step are subject of discussion as well. Once these foot points have been obtained, the distance function to the curve is approximated.…”

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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“…The problem assigning t ij to each data point e ij is called the data parameterization problem in the literature [36], [37], [38]. For the time being, we assume that t ij is known.…”

Section: Contour Evolution With Multiple Reference Shapesmentioning

confidence: 99%

Segmentation, Inference and Classification of Partially Overlapping Nanoparticles

Park

Huang

et al. 2013

IEEE Trans. Pattern Anal. Mach. Intell.

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This paper presents a method that enables automated morphology analysis of partially overlapping nanoparticles in electron micrographs. In the undertaking of morphology analysis, three tasks appear necessary: separate individual particles from an agglomerate of overlapping nano-objects; infer the particle's missing contours; and ultimately, classify the particles by shape based on their complete contours. Our specific method adopts a two-stage approach: the first stage executes the task of particle separation, and the second stage conducts simultaneously the tasks of contour inference and shape classification. For the first stage, a modified ultimate erosion process is developed for decomposing a mixture of particles into markers, and an edge-to-marker association method is proposed to identify the set of evidences for the particle contours. We also provided theoretical justification regarding the separation capability of the first stage. In the second stage, the set of evidences become inputs to a Gaussian mixture model on B-splines, the solution of which leads to the joint learning of the missing contour and the particle shape. Using twelve real electron micrographs of overlapping nanoparticles, we compare the proposed method with four state-of-the-art methods. The results show the superiority of the proposed method in terms of particle recognition rate.

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An improved Hoschek intrinsic parametrization

Cited by 30 publications

References 10 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

Segmentation, Inference and Classification of Partially Overlapping Nanoparticles

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