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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.
Nonuniform BSpline subdivision using refine and smooth
 In IMA Conference on the Mathematics of Surfaces
, 2007
"... Abstract. Subdivision surfaces would be useful in a greater number of applications if an arbitrarydegree, nonuniform scheme existed that was a generalisation of NURBS. As a step towards building such a scheme, we investigate nonuniform analogues of the LaneRiesenfeld ‘refine and smooth ’ subdivi ..."
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Cited by 4 (0 self)
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Abstract. Subdivision surfaces would be useful in a greater number of applications if an arbitrarydegree, nonuniform scheme existed that was a generalisation of NURBS. As a step towards building such a scheme, we investigate nonuniform analogues of the LaneRiesenfeld ‘refine and smooth ’ subdivision paradigm. We show that the assumptions made in constructing such an analogue are critical, and conclude that Schaefer’s global knot insertion algorithm is the most promising route for further investigation in this area. 1
A unified approach to evaluation algorithms for multivariate polynomials
 Math. Comp
, 1997
"... Abstract. We present a unified framework for most of the known and a few new evaluation algorithms for multivariate polynomials expressed in a wide variety of bases including the BernsteinBézier, multinomial (or Taylor), Lagrange and Newton bases. This unification is achieved by considering evaluat ..."
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Cited by 4 (1 self)
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Abstract. We present a unified framework for most of the known and a few new evaluation algorithms for multivariate polynomials expressed in a wide variety of bases including the BernsteinBézier, multinomial (or Taylor), Lagrange and Newton bases. This unification is achieved by considering evaluation algorithms for multivariate polynomials expressed in terms of Lbases, a class of bases that include the BernsteinBézier, multinomial, and a rich subclass of Lagrange and Newton bases. All of the known evaluation algorithms can be generated either by considering up recursive evaluation algorithms for Lbases or by examining change of basis algorithms for Lbases. For polynomials of degree n in s variables, the class of up recursive evaluation algorithms includes a parallel up recurrence algorithm with computational complexity O(n s+1), a nested multiplication algorithm with computational complexity O(n s log n) and a ladder recurrence algorithm with computational complexity O(n s). These algorithms also generate a new generalization of the AitkenNeville algorithm for evaluation of multivariate polynomials expressed in terms of Lagrange Lbases. The second class of algorithms, based on certain change of basis algorithms between Lbases, include a nested multiplication algorithm with computational complexity O(n s), a divided difference algorithm, a forward difference algorithm, and a Lagrange evaluation algorithm with computational complexities O(n s), O(n s)andO(n) per point respectively for the evaluation of multivariate polynomials at several points. 1.
Biorthogonal Spline Wavelets on the Interval
"... Dedicated to Charles Chui on the occasion of his 65th birthday Abstract. We investigate biorthogonal spline wavelets on the interval. We give sufficient and necessary conditions for the reconstruction and decomposition matrices to be sparse. Furthermore, we give numerical estimates for the Riesz sta ..."
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Cited by 2 (0 self)
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Dedicated to Charles Chui on the occasion of his 65th birthday Abstract. We investigate biorthogonal spline wavelets on the interval. We give sufficient and necessary conditions for the reconstruction and decomposition matrices to be sparse. Furthermore, we give numerical estimates for the Riesz stability of such bases. In [7], Dahmen, Kunoth, and Urban introduced locally supported spline wavelets for the interval with locally supported dual wavelets. These wavelet bases consist of the biorthogonal spline wavelets of Cohen, Daubechies, and Feauveau [5] as ‘inner wavelets’, supplemented by boundary
Shape Recovery Of Volume Data With Deformable BSpline Models
, 1996
"... In many fields today such as radiology, images of interesting structures are obtained. From these images the physician or scientist attempts to make decisions using a variety of techniques. The existing techniques for representing and analyzing particular structures include volume rendering, and sur ..."
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In many fields today such as radiology, images of interesting structures are obtained. From these images the physician or scientist attempts to make decisions using a variety of techniques. The existing techniques for representing and analyzing particular structures include volume rendering, and surface renderings from contours, freeform surfaces and geometric primitives. Several of these techniques are inadequate for accurate representations and studying changes in the structure over time. Further, some of these techniques have large data requirements that prevent interactive viewing. It is believed that if the structures of interest can be extracted from the image background, viewed and analyzed in an interactive setting, more accurate decisions can be made. The research described in this dissertation explores a new technique for shape recovery with deformable models using Bspline surfaces. The current literature shows that there have been many successful attempts to create deform...
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
c c Submitted to a special issue of Algorithmica on Shape Algorithmics. Discrete Curve Evolution with Hausdorff Distance
"... S &,9 ( J4'7B _TY! " & ( * S ")1]!" ' # 4 4 3 (5 1 General strategy for knot removal ..."
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S &,9 ( J4'7B _TY! " & ( * S ")1]!" ' # 4 4 3 (5 1 General strategy for knot removal