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202
Radial Basis Functions
, 2003
"... papproximation orders with scattered centres ..."
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Approximation properties of multivariate wavelets
 Math. Comp
, 1998
"... Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of ..."
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Cited by 75 (10 self)
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Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in W k−1 1 (Rs) provides approximation order k. 1.
4–8 Subdivision
, 2001
"... In this paper we introduce 4–8 subdivision, a new scheme that generalizes the fourdirectional box spline of class C4 to surfaces of arbitrary topological type. The crucial advantage of the proposed scheme is that it uses bisection refinement as an elementary refinement operation, rather than more co ..."
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Cited by 66 (6 self)
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In this paper we introduce 4–8 subdivision, a new scheme that generalizes the fourdirectional box spline of class C4 to surfaces of arbitrary topological type. The crucial advantage of the proposed scheme is that it uses bisection refinement as an elementary refinement operation, rather than more commonly used face or vertex splits. In the uniform case, bisection refinement results in doubling, rather than quadrupling of the number of faces in a mesh. Adaptive bisection refinement automatically generates conforming variableresolution meshes in contrast to face and vertex split methods which require a postprocessing step to make an adaptively refined mesh conforming. The fact that the size of faces decreases more gradually with refinement allows one to have greater control over the resolution of a refined mesh. It also makes it possible to achieve higher smoothness while using small stencils (the size of the stencils used by our scheme is similar to Loop subdivision). We show that the subdivision surfaces produced by the 4–8 scheme are C^4 continuous almost everywhere, except at extraordinary vertices where they are is C¹continuous.
Evaluation of Loop Subdivision Surfaces
 In Proceedings of SIGGRAPH ’98
, 1998
"... This paper describes a technique to evaluate Loop subdivision surfaces at arbitrary parameter values. The method is a straightforward extension of our evaluation work for CatmullClark surfaces. The same ideas are applied here, with the differences being in the details only. 1 ..."
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Cited by 51 (1 self)
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This paper describes a technique to evaluate Loop subdivision surfaces at arbitrary parameter values. The method is a straightforward extension of our evaluation work for CatmullClark surfaces. The same ideas are applied here, with the differences being in the details only. 1
Multidimensional Interpolatory Subdivision Schemes
"... : This paper presents a general construction of multidimensional interpolatory subdivision schemes. In particular, we provide a concrete method for the construction of bivariate interpolatory subdivision schemes of increasing smoothness by finding an appropriate mask to convolve with the mask of a t ..."
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Cited by 47 (10 self)
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: This paper presents a general construction of multidimensional interpolatory subdivision schemes. In particular, we provide a concrete method for the construction of bivariate interpolatory subdivision schemes of increasing smoothness by finding an appropriate mask to convolve with the mask of a threedirection box spline B r;r;r of equal multiplicities. The resulting mask for the interpolatory subdivision exhibits all the symmetries of the threedirection box spline and with this increased symmetry comes increased smoothness. Several examples are computed (for r = 2; : : : ; 8). Regularity criteria in terms of the refinement mask are established and applied to the examples to estimate their smoothness. AMS Subject Classification: Primary 45A05, 65D05, 65D15, 26B05 Secondary 41A15, 41A63 Keywords: interpolation, subdivision schemes, interpolatory subdivision schemes, box splines, wavelets y Research supported in part by NSERC Canada under Grant # A7687 1 1. Introduction and Metho...
Characterization of smoothness of multivariate refinable functions in Sobolev spaces
 Trans. Amer. Math. Soc
, 1999
"... Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spect ..."
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Cited by 45 (4 self)
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Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace. Several examples are provided to illustrate the general theory. 1.
Linear and Cubic Box Splines for the Body Centered Cubic Lattice
 In Proceedings of the IEEE Conference on Visualization
, 2004
"... In this paper we derive piecewise linear and piecewise cubic box spline reconstruction filters for data sampled on the body centered cubic (BCC) lattice. We analytically derive a time domain representation of these reconstruction filters and using the Fourier sliceprojection theorem we derive their ..."
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Cited by 41 (8 self)
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In this paper we derive piecewise linear and piecewise cubic box spline reconstruction filters for data sampled on the body centered cubic (BCC) lattice. We analytically derive a time domain representation of these reconstruction filters and using the Fourier sliceprojection theorem we derive their frequency responses. The quality of these filters, when used in reconstructing BCC sampled volumetric data, is discussed and is demonstrated with a raycaster. Moreover, to demonstrate the superiority of the BCC sampling, the resulting reconstructions are compared with those produced from similar filters applied to data sampled on the Cartesian lattice.
Compactly supported tight affine spline frames
 Math. Comp
, 1998
"... Abstract. The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in L2(Rd) from box splines. The wavelets obtained are smooth piecewisepolynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The ..."
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Cited by 39 (8 self)
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Abstract. The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in L2(Rd) from box splines. The wavelets obtained are smooth piecewisepolynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of “mother wavelets”, however, increases with the increase of the required smoothness. Two bivariate constructions, of potential practical value, are highlighted. In both, the wavelets are derived from fourdirection mesh box splines that are