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Scattered Data Interpolation with Multilevel Splines
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 1997
"... This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel Bsplines are introduced to compute a C²continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarsetofine hierarchy of control lattices to generate a sequen ..."
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Cited by 158 (10 self)
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This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel Bsplines are introduced to compute a C²continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarsetofine hierarchy of control lattices to generate a sequence of bicubic Bspline functions whose sum approaches the desired interpolation function. Large performance gains are realized by using Bspline refinement to reduce the sum of these functions into one equivalent Bspline function. Experimental results demonstrate that highfidelity reconstruction is possible from a selected set of sparse and irregular samples.
Matrixvalued subdivision schemes for generating surfaces with extraordinary vertices
 Comput. Aided Geom. Design
"... Subdivision templates of numerical values are replaced by templates of matrices in this paper to allow the introduction of shape control parameters for the feasibility of achieving desirable geometric shapes at those points on the subdivision surfaces that correspond to extraordinary control vertice ..."
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Cited by 9 (6 self)
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Subdivision templates of numerical values are replaced by templates of matrices in this paper to allow the introduction of shape control parameters for the feasibility of achieving desirable geometric shapes at those points on the subdivision surfaces that correspond to extraordinary control vertices. Formulation of the matrixvalued subdivision surface is derived. Based on refinable bivariate spline function vectors for matrixvalued subdivisions, the notion of characteristic map introduced by Reif is extended from (scalar) surface subdivisions to matrixvalued subdivisions. The C 1 and C kcontinuity of Reif and Prautzsch for matrixvalued subdivisions are discussed. To illustrate the general theory, the smoothness of matrixvalued triangular subdivision schemes for extraordinary vertices with valences 3 and 4 is analyzed. The issue of effective choices of the shape control parameters will also be discussed in this paper. Keywords: Matrixvalued surface subdivision, matrixvalued templates, surface shape control, extraordinary vertices, characteristic map, C kcontinuity 1
Constructing Curvaturecontinuous Surfaces by Blending
, 2006
"... In this paper we describe an approach to the construction of curvaturecontinuous surfaces with arbitrary control meshes using subdivision. Using a simple modification of the widely used Loop subdivision algorithm we obtain perturbed surfaces which retain the overall shape and appearance of Loop sub ..."
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Cited by 8 (2 self)
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In this paper we describe an approach to the construction of curvaturecontinuous surfaces with arbitrary control meshes using subdivision. Using a simple modification of the widely used Loop subdivision algorithm we obtain perturbed surfaces which retain the overall shape and appearance of Loop subdivision surfaces but no longer have flat spots or curvature singularities at extraordinary vertices. Our method is computationally efficient and can be easily added to any existing subdivision code.
Triangle mesh subdivision with bounded curvature and the convex hull property
"... The masks for Loop’s triangle subdivision surface algorithm are modified resulting in surfaces with bounded curvature and the convex hull property. New edge masks are generated by a cubic polynomial mask equation whose Chebyshev coefficients are closely related to the eigenvalues of the correspond ..."
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Cited by 7 (0 self)
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The masks for Loop’s triangle subdivision surface algorithm are modified resulting in surfaces with bounded curvature and the convex hull property. New edge masks are generated by a cubic polynomial mask equation whose Chebyshev coefficients are closely related to the eigenvalues of the corresponding subdivision matrix. The mask equation is found to satisfy a set of smoothness constraints on these eigenvalues. We observe that controlling the root structure of the mask equation is important for deriving subdivision masks with nonnegative weights.
Analysis and Tuning of Subdivision Algorithms
 Proceedings of the 21st spring conference on Computer Graphics
, 2005
"... Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of ..."
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Cited by 2 (0 self)
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Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee.
Convex triangular subdivision surfaces with bounded curvature
, 2000
"... The edge masks for Loop’s triangular subdivision surface algorithm are modified resulting in surfaces with bounded curvature and the convex hull property. The new edge masks are derived from a polynomial mask equation whose Chebyshev expansion coefficients are closely related to the eigenvalues of t ..."
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The edge masks for Loop’s triangular subdivision surface algorithm are modified resulting in surfaces with bounded curvature and the convex hull property. The new edge masks are derived from a polynomial mask equation whose Chebyshev expansion coefficients are closely related to the eigenvalues of the corresponding subdivision matrix. The mask equation is found to satisfy a set of smoothness constraints on these eigenvalues. 1
Multiple Knot BSpline Representation of Incompressible Flow
, 1996
"... this paper that we are given a set of normal fluid fluxes defined across the face centers of some tensor product cartesian mesh. Other velocity data may require preprocessing to obtain the desired format. x i\Gamma1 x i ..."
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Cited by 1 (1 self)
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this paper that we are given a set of normal fluid fluxes defined across the face centers of some tensor product cartesian mesh. Other velocity data may require preprocessing to obtain the desired format. x i\Gamma1 x i
xcmodel: an aCADemic system
, 2000
"... xcmodel is a CAD system realized and usable in an academic environment. It integrates four packages: a 2D and a 3D modeller, an object composer and a realistic scene renderer; these subsystems can be regarded as being in constant evolution i.e. a continuous work in progress. The system summarises ou ..."
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Cited by 1 (1 self)
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xcmodel is a CAD system realized and usable in an academic environment. It integrates four packages: a 2D and a 3D modeller, an object composer and a realistic scene renderer; these subsystems can be regarded as being in constant evolution i.e. a continuous work in progress. The system summarises our knowledge and experience in geometric modelling and NURBS curves and surfaces acquired over ten years of research. xcmodel and its subsystems were designed to represent a research and teaching laboratory to experiment and learn; it is an ideal environment to develop, perfect and compare methods and algorithms in geometric modelling and graphic visualization.
ManifoldBased Surfaces with Boundaries
"... We present a manifoldbased surface construction extending the C ∞ construction of Ying and Zorin (2004a). Our surfaces allow for pircewisesmooth boundaries, have usercontrolled arbitrary degree of smoothness and improved derivative and visual behavior. 2flexibility of our surface construction is ..."
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We present a manifoldbased surface construction extending the C ∞ construction of Ying and Zorin (2004a). Our surfaces allow for pircewisesmooth boundaries, have usercontrolled arbitrary degree of smoothness and improved derivative and visual behavior. 2flexibility of our surface construction is confirmed numerically for a range of local mesh configurations. Key words: Geometric modeling, manifolds, highorder surfaces
Bspline Curve Generation and Modification based on Specified Radius of Curvature
"... Abstract: A method to generate a quintic Bspline curve which passes through given points is described. In this case, there are four more equations than there are control point positions. Two methods have been developed to compensate for the difference between the number of unknowns and that of the ..."
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Abstract: A method to generate a quintic Bspline curve which passes through given points is described. In this case, there are four more equations than there are control point positions. Two methods have been developed to compensate for the difference between the number of unknowns and that of the equations. These are assuming that the curvatures at both ends of the curve are zero, and assigning four gradients to the given points. In addition to this method, another method to generate a quintic Bspline curve which passes close to given points, and which has the first derivative at these given points is described. In this case, a linear system will be underdetermined, determined or overdetermined depending on the number of given points with gradients. A method to modify a quintic Bspline curve shape according to the specified radius of curvature distribution to realize an aesthetically pleasing freeform curve is described. The difference between the Bspline curve radius of curvature and the specified radius of curvature is minimized by introducing the leastsquares method. Examples of curve generation are given. KeyWords: Bspline curve generation, curvature vector, curve shape modification, given points, given points