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27
Fitting BSpline Curves to Point Clouds by Squared Distance Minimization
 ACM TRANSACTIONS ON GRAPHICS
, 2004
"... 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, c ..."
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Cited by 24 (3 self)
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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 Bspline curve, closed or open, to approximate a target shape defined by a point cloud, i.e., a set of unorganized, possibly noisy data points. We show that SDM outperforms significantly 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 quadratic approximant of squared distances from data points to a fitting curve. The SD
Fitting Subdivision Surfaces to Unorganized Point Data Using SDM
, 2004
"... We study the reconstruction of smooth surfaces from point clouds. We use a new squared distance error term in optimization to fit a subdivision surface to a set of unorganized points, which defines a closed target surface of arbitrary topology. The resulting method is based on the framework of squar ..."
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Cited by 20 (5 self)
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We study the reconstruction of smooth surfaces from point clouds. We use a new squared distance error term in optimization to fit a subdivision surface to a set of unorganized points, which defines a closed target surface of arbitrary topology. The resulting method is based on the framework of squared distance minimization (SDM) proposed by Pottmann et al. Specifically, with an initial subdivision surface having a coarse control mesh as input, we adjust the control points by optimizing an objective function through iterative minimization of a quadratic approximant of the squared distance function of the target shape. Our experiments show that the new method (SDM) converges much faster than the commonly used optimization method using the point distance error function, which is known to have only linear convergence. This observation is further supported by our recent result that SDM can be derived from the Newton method with necessary modifications to make the Hessian positive definite and the fact that the Newton method has quadratic convergence.
Robust realtime segmentation of images and videos uisng a smoothingspline snakebased algorithm
 IEEE Transactions on Image Processing
, 2005
"... Abstract—This paper deals with fast image and video segmentation using active contours. Regionbased active contours using level sets are powerful techniques for video segmentation, but they suffer from large computational cost. A parametric active contour method based on BSpline interpolation has ..."
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Cited by 18 (5 self)
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Abstract—This paper deals with fast image and video segmentation using active contours. Regionbased active contours using level sets are powerful techniques for video segmentation, but they suffer from large computational cost. A parametric active contour method based on BSpline interpolation has been proposed in [26] to highly reduce the computational cost, but this method is sensitive to noise. Here, we choose to relax the rigid interpolation constraint in order to robustify our method in the presence of noise: by using smoothing splines, we trade a tunable amount of interpolation error for a smoother spline curve. We show by experiments on natural sequences that this new flexibility yields segmentation results of higher quality at no additional computational cost. Hence, realtime processing for moving objects segmentation is preserved. I.
Fitting BSpline Curves to Point Clouds by CurvatureBased Squared Distance Minimization
 ACM TRANSACTIONS ON GRAPHICS
, 2006
"... 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, clos ..."
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Cited by 14 (2 self)
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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 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.
A new subdivision based approach for piecewise smooth approximation of 3D polygonal curves”, Pattern Recognition
 In press
, 2005
"... www.elsevier.com/locate/patcog ..."
Approximation Flows in Shape Manifolds
, 2005
"... We consider manifolds of curves and surfaces which are controlled by certain systems of shape parameters. These systems may be given by the control points of a spline curve, the coefficients of an implicit equation, or other parameters controlling the shape. Each system of shape parameters correspon ..."
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Cited by 6 (5 self)
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We consider manifolds of curves and surfaces which are controlled by certain systems of shape parameters. These systems may be given by the control points of a spline curve, the coefficients of an implicit equation, or other parameters controlling the shape. Each system of shape parameters corresponds to a chart of the manifold. In order to fit a curve or surface from such a manifold to given unorganized point data, we define an evolution process which takes an initial solution and modifies it in order to adapt it to the data. We show that this evolution defines a flow on the shape manifold. Consequently, the result of the evolution is independent of the particular choice of the shape parameters / of the chart.
Dual evolution of planar parametric spline curves and T–spline level sets. ComputerAided Design
, 2008
"... Abstract. By simultaneously considering evolution processes for parametric spline curves and implicitly defined curves, we formulate the framework of dual evolution. This allows us to combine the advantages of both representations. On the one hand, the implicit representation is used to guide the to ..."
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Cited by 5 (5 self)
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Abstract. By simultaneously considering evolution processes for parametric spline curves and implicitly defined curves, we formulate the framework of dual evolution. This allows us to combine the advantages of both representations. On the one hand, the implicit representation is used to guide the topology of the parametric curve and to formulate additional constraints, such as range constraints. On the other hand, the parametric representation helps to detect and to eliminate unwanted branches of the implicitly defined curves. Moreover, it is required for many applications, e.g., in Computer Aided Design. 1
Hybrid curve fitting
, 2007
"... We consider a parameterized family of closed planar curves and introduce an evolution process for identifying a member of the family that approximates a given unorganized point cloud {pi} i=1,...,N. The evolution is driven by the normal velocities at the closest (or foot) points (fi) to the data poi ..."
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Cited by 5 (5 self)
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We consider a parameterized family of closed planar curves and introduce an evolution process for identifying a member of the family that approximates a given unorganized point cloud {pi} i=1,...,N. The evolution is driven by the normal velocities at the closest (or foot) points (fi) to the data points, which are found by approximating the corresponding difference vectors pi − fi in the leastsquares sense. In the particular case of parametrically defined curves, this process is shown to be equivalent to normal (or tangent) distance minimization, see [3], [19]. Moreover, it can be generalized to very general representations of curves. These include hybrid curves, which are a collection of parametrically and implicitly defined curve segments, pieced together with certain degrees of geometric continuity.
Rational quadratic approximation to real algebraic curves
 Comput. Aided Geom. Des
"... www.elsevier.com/locate/cagd An algorithm is proposed to give a global approximation of an implicit real plane algebraic curve with rational quadratic Bspline curves. The algorithm consists of four steps: topology determination, curve segmentation, segment approximation and curve tracing. Due to th ..."
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Cited by 4 (0 self)
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www.elsevier.com/locate/cagd An algorithm is proposed to give a global approximation of an implicit real plane algebraic curve with rational quadratic Bspline curves. The algorithm consists of four steps: topology determination, curve segmentation, segment approximation and curve tracing. Due to the detailed geometric analysis, high accuracy of approximation may be achieved with a small number of quadratic segments. The final approximation keeps many important geometric features of the original curve such as the topology, convexity and sharp points. Our method is implemented and experiments show that it may achieve better approximation bound with less segments than previously known methods. We also extend the method to approximate spatial algebraic curves.