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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 106 (9 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.
Hierarchical and Variational Geometric Modeling with Wavelets
 IN PROCEEDINGS SYMPOSIUM ON INTERACTIVE 3D GRAPHICS
, 1995
"... This paper discusses how wavelet techniques may be applied to a variety of geometric modeling tools. In particular, wavelet decompositions are shown to be useful for hierarchical control point or least squares editing. In addition, direct curve and surface manipulation methods using an underlying ge ..."
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Cited by 64 (1 self)
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This paper discusses how wavelet techniques may be applied to a variety of geometric modeling tools. In particular, wavelet decompositions are shown to be useful for hierarchical control point or least squares editing. In addition, direct curve and surface manipulation methods using an underlying geometric variational principle can be solved more efficiently by using a wavelet basis. Because the wavelet basis is hierarchical, iterative solution methods converge rapidly. Also, since the wavelet coefficients indicate the degree of detail in the solution, the number of basis functions needed to express the variational minimum can be reduced, avoiding unnecessary computation. An implementation of a curve and surface modeler based on these ideas is discussed and experimental results are reported.
Free Form Surface Analysis Using a Hybrid of Symbolic and Numeric Computation
, 1992
"... Detailed analysis of many mathematical properties of sculptured models has been hindered by the fact that the properties do not have the same representation as the surface. For example, unit tangents, surface normals, and principal curvatures are typically computed at predefined discrete sets of poi ..."
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Cited by 40 (19 self)
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Detailed analysis of many mathematical properties of sculptured models has been hindered by the fact that the properties do not have the same representation as the surface. For example, unit tangents, surface normals, and principal curvatures are typically computed at predefined discrete sets of points on the surface. As such, aliasing can occur and features between samples can be missed. Synthesizing information about the shape of an object and operating on the model, whether by physical machining tools, graphics display programs, or mathematical analysis, has been treated as either a discrete or local problem in general. The researchbeing reported on here has focused on another approach, that of creating algorithms that construct the mathematical properties in closed form, or construct approximations to those mathematical properties through symbolic computation. Global analysis can then be applied while an accurate error bound is obtained.
Error Bounded Variable Distance Offset Operator for Free Form Curves and Surfaces
 International Journal of Computational Geometry and Applications
, 1991
"... Most offset approximation algorithms for freeform curves and surfaces may be classified into two main groups. The first approximates the curve using simple primitives such as piecewise arcs and lines and then calculates the (exact) offset operator to this approximation. The second offsets the contro ..."
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Cited by 34 (17 self)
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Most offset approximation algorithms for freeform curves and surfaces may be classified into two main groups. The first approximates the curve using simple primitives such as piecewise arcs and lines and then calculates the (exact) offset operator to this approximation. The second offsets the control polygon/mesh and then attempts to estimate the error of the approximated offset over a region. Most of the current offset algorithms estimate the error using a finite set of samples taken from the region and therefore can not guarantee the offset approximation is within a given tolerance over the whole curve or surface. This paper presents new methods to globally bound the error of the approximated offset of freeform curves and surfaces and then automatically derive new approximations with improved accuracy. These tools can also be used to develop a global error bound for a variable distance offset operation and to detect and trim out loops in the offset. 1 Introduction Offset surfaces ar...
A Unified Framework for Primal/Dual Quadrilateral Subdivision Schemes
 CAGD
, 2001
"... Quadrilateral subdivision schemes come in primal and dual varieties, splitting faces or respectively vertices. The scheme of CatmullClark is an example of the former, while the DooSabin scheme exemplifies the latter. In this paper we consider the construction of an increasing sequence of alternati ..."
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Cited by 33 (4 self)
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Quadrilateral subdivision schemes come in primal and dual varieties, splitting faces or respectively vertices. The scheme of CatmullClark is an example of the former, while the DooSabin scheme exemplifies the latter. In this paper we consider the construction of an increasing sequence of alternating primal/dual quadrilateral subdivision schemes based on a simple averaging approach. Beginning with a vertex split step we successively construct variants of DooSabin and CatmullClark schemes followed by novel schemes generalizing Bsplines of bidegree up to nine. We prove the schemes to be C¹ at irregular surface points, and analyze the behavior of the schemes as the number of averaging steps increases. We discuss a number of implementation issues common to all quadrilateral schemes. In particular we show how both primal and dual quadrilateral schemes can be implemented in the same code, opening up new possibilities for more flexible geometric modeling applications and pversions of the Subdivision Element Method. Additionally we describe a simple algorithm for adaptive subdivision of dual schemes.
Second Order Surface Analysis Using Hybrid Symbolic and Numeric Operators
, 1992
"... Results from analyzing the curvature of a surface can be used to improve the implementation, efficiency, and effectiveness of manufacturing and visualization of sculptured surfaces. In this paper ..."
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Cited by 24 (13 self)
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Results from analyzing the curvature of a surface can be used to improve the implementation, efficiency, and effectiveness of manufacturing and visualization of sculptured surfaces. In this paper
Model Fabrication using Surface Layout Projection
 ComputerAided Design
, 1995
"... This paper presents a model fabrication scheme that automatically approximates a model whose boundary consists of several freeform surfaces by developable surfaces and then unroll these developable surfaces onto a plane. The model can then be fabricated by assembling the sets of developable surfaces ..."
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Cited by 17 (4 self)
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This paper presents a model fabrication scheme that automatically approximates a model whose boundary consists of several freeform surfaces by developable surfaces and then unroll these developable surfaces onto a plane. The model can then be fabricated by assembling the sets of developable surfaces which have been cut from planar sheets and rolled back to their proper Euclidean locations. Both the approximation and the rolling methods can be made arbitrarily precise. 1 Introduction It is common to find freeform surfaces manually approximated and assembled as sets of piecewise developable surfaces [9]. In general, freeform surfaces are not developable and cannot be exactly represented as piecewise developable surfaces. Yet, "developable surfaces are of considerable importance to sheetmetal or platemetalbased industries This work was supported in part by DARPA (N0001491J4123). All opinions, findings, conclusions or recommendations expressed in this document are those of the au...
Curvature Integrability of Subdivision Surfaces
 ADVANCES IN COMPUTATIONAL MATHEMATICS
, 2000
"... We examine the properties of the principal curvatures of subdivision surfaces near irregular points. In particular we give an estimate of their... ..."
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Cited by 13 (0 self)
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We examine the properties of the principal curvatures of subdivision surfaces near irregular points. In particular we give an estimate of their...
Multiresolution Surface Approximation for Animation
 In Proceedings of Graphics Interface
, 1993
"... This paper considers the problem of approximating a digitized surface in R 3 with a hierarchical bicubic Bspline to produce a manipulatable surface for further modelling or animation. The 3D data's original mapping from R 2 (multiple rows of cylindrical scans) is mapped into the parametric doma ..."
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Cited by 9 (0 self)
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This paper considers the problem of approximating a digitized surface in R 3 with a hierarchical bicubic Bspline to produce a manipulatable surface for further modelling or animation. The 3D data's original mapping from R 2 (multiple rows of cylindrical scans) is mapped into the parametric domain of the Bspline (also in R 2 ) using a modified chordlength parameterization. This mapping is used to produce a gridded sampling of the surface, and a modified full multigrid (FMG) technique is employed to obtain a highresolution Bspline approximation. The intermediate results of the FMG calculations generate the component overlays of a hierarchical spline surface representation. Storage requirements of the hierarchical representation are reduced by eliminating offsets whereever their removal will not increase the error in the approximation by more than a given amount. The resulting hierarchical spline surface is interactively modifiable (modulo the size of the data set and computing...
Langwidere: A Hierarchical Spline Based Facial Animation System with Simulated Muscles
, 1993
"... This thesis presents Langwidere, a facial animation system. Langwidere is intended to serve as the basis for a flexible system capable of imitating realistic characters and actions, such as speech or expressing emotion, as well as creating the exaggerated and fantastic characters found in traditiona ..."
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Cited by 9 (0 self)
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This thesis presents Langwidere, a facial animation system. Langwidere is intended to serve as the basis for a flexible system capable of imitating realistic characters and actions, such as speech or expressing emotion, as well as creating the exaggerated and fantastic characters found in traditional animation. In the field of computer generated characters, facial animation is difficult because of the complexity of the surface and the fact that humans have entire sections of their brains devoted to facial processing, finding and magnifying any imperfection. Typically, the construction of new characters involves tedious and repetitive labor. An unwieldy level of detail is often needed for a realistic model. Apart from any geometric consideration, animating a new character often requires reprogramming or changing and retuning a large number of arcane parameters. Numerous small, time consuming details go into creating a complete and visually interesting animated sequence. Much of this rep...