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Fitting Smooth Surfaces to Dense Polygon Meshes
 Proceedings of SIGGRAPH 96
, 1996
"... Recent progress in acquiring shape from range data permits the acquisition of seamless millionpolygon meshes from physical models. In this paper, we present an algorithm and system for converting dense irregular polygon meshes of arbitrary topology into tensor product Bspline surface patches with ..."
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Cited by 208 (5 self)
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Recent progress in acquiring shape from range data permits the acquisition of seamless millionpolygon meshes from physical models. In this paper, we present an algorithm and system for converting dense irregular polygon meshes of arbitrary topology into tensor product Bspline surface patches with accompanying displacement maps. This choice of representation yields a coarse but efficient model suitable for animation and a fine but more expensive model suitable for rendering. The first step in our process consists of interactively painting patch boundaries over a rendering of the mesh. In many applications, interactive placement of patch boundaries is considered part of the creative process and is not amenable to automation. The next step is gridded resampling of eachboundedsection of the mesh. Our resampling algorithm lays a grid of springs acrossthe polygonmesh, then iterates between relaxing this grid and subdividing it. This grid provides a parameterization for the mesh section, w...
A Review of Current Geometric Tolerancing Theories and Inspection Data Analysis Algorithms

, 1991
"... ..."
Hybrid Curve Fitting
, 2006
"... 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 poin ..."
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Cited by 6 (6 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 least–squares 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.
GaussNewtontype Techniques for Robustly Fitting Implicitly Defined Curves and Surfaces to Unorganized Data Points
"... Abstract — We describe GaussNewton type methods for fitting implicitly defined curves and surfaces to given unorganized data points. The methods can deal with general error functions, such as approximations to the ℓ1 or ℓ ∞ norm of the vector of residuals. Depending on the definition of the residua ..."
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Cited by 1 (1 self)
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Abstract — We describe GaussNewton type methods for fitting implicitly defined curves and surfaces to given unorganized data points. The methods can deal with general error functions, such as approximations to the ℓ1 or ℓ ∞ norm of the vector of residuals. Depending on the definition of the residuals, we distinguish between direct and data–based methods. In addition, we show that these methods can either be seen as (discrete) iterative methods, where an update of the unknown shape parameters is computed in each step, or as continuous evolution processes, that generate a time–dependent family of curves or surfaces, which converges towards the final result. It is shown that the data–based methods – which are less costly, as they work without the need of computing the closest points – can efficiently deal with error functions that are adapted to noisy and uncertain data. In addition, we observe that the interpretation as evolution process allows to deal with the issues of regularization and with additional constraints. I.
A BSpline Curve Fitting Approach by Implementing the Parameter Correction Terms
"... Abstract Fitting a set of points with a BSpline curve is a usual CADG application, which remains an open problem due to the choice of parameter values. The crucial point is to find optimal parameter values which lead to an optimal approximation curve. Since these parameter values are only a first ..."
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Cited by 1 (0 self)
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Abstract Fitting a set of points with a BSpline curve is a usual CADG application, which remains an open problem due to the choice of parameter values. The crucial point is to find optimal parameter values which lead to an optimal approximation curve. Since these parameter values are only a first guess, parameter correction can be used to improve parameterization. This paper discusses iterative solutions in leastsquares BSpline curve fitting sense. And the initial results are presented. 1.
unknown title
, 2010
"... should determine the number n of the control points and the corresponding knot vector Un of C(t) as well. The approximation result depends on the selection of n and Un. There are some notable works based on the optimization approach [13,14], but many challenges still remain to be tackled. When n is ..."
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should determine the number n of the control points and the corresponding knot vector Un of C(t) as well. The approximation result depends on the selection of n and Un. There are some notable works based on the optimization approach [13,14], but many challenges still remain to be tackled. When n is equal to m, the curve fitting problem degenerates to a general interpolation problem [15]. A knot removal approach can then be used to reduce the number of control points, which is to progressively remove a selected number of knots that have the least significance to the approximation curve until the error reaches the tolerance level [4,16,17]. A knot increment approach can also be used for the curve fitting problem, which is to use less number of knots at first, and then add more knots to obtain the desired accuracy. Usually, the knot increment approach tends to require less number of control points than the knot removal approach [1]. When the number of the control points
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
, 1995
"... Let a family of curves or surfaces be given in parametric form via the model equation x =J‘(s, /I), where x E R”, p E KY’, and s E S c [wd. d < n. We present an algorithm for solving the problem: Find a shape cec’tor p * such that the manlfold M * = (f(s, /I*): s E S) is a best fir to scattered dat ..."
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Let a family of curves or surfaces be given in parametric form via the model equation x =J‘(s, /I), where x E R”, p E KY’, and s E S c [wd. d < n. We present an algorithm for solving the problem: Find a shape cec’tor p * such that the manlfold M * = (f(s, /I*): s E S) is a best fir to scattered data:z, j,‘ = , c R ” in the Sense that, jbr some is *)y=, , the sum of the squared least distances)y=,;I:,j (ST. fi*) 1):,f rom the datu points to the munifold M * is minimal umong all such mun$olds. For robustness. our algorithm uses a globally convergent trust region approach in which, at each iteration, an approximation to the objective function is minimized in a given neighborhood of the current iterate. The set S may be all of Rd or a closed, convex subset, In particular, it may be chosen so that our theory is applicable to splines.
unknown title
, 2004
"... Optimized stereo reconstruction of freeform space curves based on a nonuniform rational Bspline model ..."
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Optimized stereo reconstruction of freeform space curves based on a nonuniform rational Bspline model
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"... High rate compression of CAD meshes based on subdivision inversion Guillaume LAVOUÉ*, Florent DUPONT*, Atilla BASKURT* In this paper we present a new framework, based on subdivision surface approximation, for efficient compression and coding of 3D models represented by polygonal meshes. Our algorith ..."
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High rate compression of CAD meshes based on subdivision inversion Guillaume LAVOUÉ*, Florent DUPONT*, Atilla BASKURT* In this paper we present a new framework, based on subdivision surface approximation, for efficient compression and coding of 3D models represented by polygonal meshes. Our algorithm fits the input 3D model with a piecewise smooth subdivision surface represented by a coarse control polyhedron, near optimal in terms of control points number and connectivity. Our algorithm, which remains independent of the connectivity of the input mesh, is particularly suited for meshes issued from mechanical or cad parts. The found subdivision control polyhedron is much more compact than the original mesh and visually represents the same shape after several subdivision steps, without artifacts or cracks, like traditional lossy compression schemes. This control polyhedron is then encoded specifically to give the final compressed stream. Experiments conducted on several cad models have proven the coherency and the efficiency of our algorithm, compared with existing methods.