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31
Surface reconstruction from unorganized points
 COMPUTER GRAPHICS (SIGGRAPH ’92 PROCEEDINGS)
, 1992
"... We describe and demonstrate an algorithm that takes as input an unorganized set of points fx1�:::�xng IR 3 on or near an unknown manifold M, and produces as output a simplicial surface that approximates M. Neither the topology, the presence of boundaries, nor the geometry of M are assumed to be know ..."
Abstract

Cited by 645 (8 self)
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We describe and demonstrate an algorithm that takes as input an unorganized set of points fx1�:::�xng IR 3 on or near an unknown manifold M, and produces as output a simplicial surface that approximates M. Neither the topology, the presence of boundaries, nor the geometry of M are assumed to be known in advance — all are inferred automatically from the data. This problem naturally arises in a variety of practical situations such as range scanning an object from multiple view points, recovery of biological shapes from twodimensional slices, and interactive surface sketching.
Piecewise smooth surface reconstruction
, 1994
"... We present a general method for automatic reconstruction of accurate, concise, piecewise smooth surface models from scattered range data. The method can be used in a variety of applications such as reverse engineering — the automatic generation of CAD models from physical objects. Novel aspects of t ..."
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Cited by 269 (13 self)
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We present a general method for automatic reconstruction of accurate, concise, piecewise smooth surface models from scattered range data. The method can be used in a variety of applications such as reverse engineering — the automatic generation of CAD models from physical objects. Novel aspects of the method are its ability to model surfaces of arbitrary topological type and to recover sharp features such as creases and corners. The method has proven to be effective, as demonstrated by a number of examples using both simulated and real data. A key ingredient in the method, and a principal contribution of this paper, is the introduction of a new class of piecewise smooth surface representations based on subdivision. These surfaces have a number of properties that make them ideal for use in surface reconstruction: they are simple to implement, they can model sharp features concisely, and they can be fit to scattered range data using an unconstrained optimization procedure.
Multiresolution Curves
, 1994
"... We describe a multiresolution curve representation, based on wavelets, that conveniently supports a variety of operations: smoothing a curve; editing the overall form of a curve while preserving its details; and approximating a curve within any given error tolerance for scan conversion. We present m ..."
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Cited by 148 (5 self)
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We describe a multiresolution curve representation, based on wavelets, that conveniently supports a variety of operations: smoothing a curve; editing the overall form of a curve while preserving its details; and approximating a curve within any given error tolerance for scan conversion. We present methods to support continuous levels of smoothing as well as direct manipulation of an arbitrary portion of the curve; the control points, as well as the discrete nature of the underlying hierarchical representation, can be hidden from the user. The multiresolution representation requires no extra storage beyond that of the original control points, and the algorithms using the representation are both simple and fast.
Direct LeastSquares Fitting of Algebraic Surfaces
, 1987
"... In the course of developing a system for fitting smooth curves to camera input we have developed several direct (i.e. noniterative) methods for fitting a shape (line, circle, conic, cubic, plane, sphere, quadric, etc.) to a set of points, namely exact fit, simple fit, spherical fit, and blend fit. T ..."
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Cited by 111 (1 self)
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In the course of developing a system for fitting smooth curves to camera input we have developed several direct (i.e. noniterative) methods for fitting a shape (line, circle, conic, cubic, plane, sphere, quadric, etc.) to a set of points, namely exact fit, simple fit, spherical fit, and blend fit. These methods are all dimensionindependent, being just as suitable for 3D surfaces as for the 2D curves they were originally developed for. Exact fit...
Direct Haptic Rendering of Sculptured Models
 IN PROC. 1997 SYMPOSIUM ON INTERACTIVE 3D GRAPHICS
, 1997
"... A new tracing algorithm is described that supports haptic rendering of NURBS surfaces without the use of any intermediate representation. By using this tracing algorithm in conjunction with algorithms for surface proximity testing and surface transitions, a complete haptic rendering system for sculp ..."
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Cited by 50 (9 self)
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A new tracing algorithm is described that supports haptic rendering of NURBS surfaces without the use of any intermediate representation. By using this tracing algorithm in conjunction with algorithms for surface proximity testing and surface transitions, a complete haptic rendering system for sculptured models has been developed. The system links an advanced CAD modeling system with a Sarcos forcereflecting exoskeleton arm. A method for measuring the quality of the tracking component of the haptic rendering separately from the haptic device and force computation is also described.
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
Creating Generative Models from Range Images
 In Proceedings of SIGGRAPH 99
, 1999
"... We describe a new approach for creating concise highlevel generative models from range images or other approximate representations of real objects. Using data from a variety of acquisition techniques and a userdefined class of models, our method produces a compact object representation that is int ..."
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Cited by 21 (0 self)
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We describe a new approach for creating concise highlevel generative models from range images or other approximate representations of real objects. Using data from a variety of acquisition techniques and a userdefined class of models, our method produces a compact object representation that is intuitive and easy to edit. The algorithm has two interrelated phases: recognition, which chooses an appropriate model within a userspecified hierarchy, and parameter estimation, which adjusts the model to best fit the data. Since the approach is modelbased, it is relatively insensitive to noise and missing data. We describe practical heuristics for automatically making tradeoffs between simplicity and accuracy to select the best model in a given hierarchy. We also describe a general and efficient technique for optimizing a model by refining its constituent curves. We demonstrate our approach for model recovery using both real and synthetic data and several generative model hierarchies.
Approximate General Sweep Boundary of a 2D Curved Object
 CVGIP: Graphical Models and Image Processing
, 1993
"... This paper presents an algorithm to compute an approximation to the general sweep boundary of a 2D curved moving object which changes its shape dynamically while traversing a trajectory. In effect, we make polygonal approximations to the trajectory and to the object shape at every appropriate instan ..."
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Cited by 18 (7 self)
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This paper presents an algorithm to compute an approximation to the general sweep boundary of a 2D curved moving object which changes its shape dynamically while traversing a trajectory. In effect, we make polygonal approximations to the trajectory and to the object shape at every appropriate instance along the trajectory so that the approximated polygonal sweep boundary is within a given error bound ffl ?0 from the exact sweep boundary. The algorithm interpolates intermediate polygonal shapes between any two consecutive instances, and constructs polygons which approximate the sweep boundary of the object. Previous algorithms on sweep boundary computation have been mainly concerned about moving objects with fixed shapes; nevertheless, they have involved a fair amount of symbolic and/or numerical computations that have limited their practical uses in graphics modeling systems as well as in many other applications which require fast sweep boundary computation. Although the algorithm pres...
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.