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Interpolating Subdivision for Meshes with Arbitrary Topology
"... Subdivision is a powerful paradigm for the generation of surfaces of arbitrary topology. Given an initial triangular mesh the goal is to produce a smooth and visually pleasing surface whose shape is controlled by the initial mesh. Of particular interest are interpolating schemes since they match the ..."
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Cited by 217 (24 self)
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Subdivision is a powerful paradigm for the generation of surfaces of arbitrary topology. Given an initial triangular mesh the goal is to produce a smooth and visually pleasing surface whose shape is controlled by the initial mesh. Of particular interest are interpolating schemes since they match the original data exactly, and play an important role in fast multiresolution and wavelet techniques. Dyn, Gregory, and Levin introduced the Butterfly scheme, which yields C 1 surfaces in the topologically regular setting. Unfortunately it exhibits undesirable artifacts in the case of an irregular topology. We examine these failures and derive an improved scheme, which retains the simplicity of the Butterfly scheme, is interpolating, and results in smoother surfaces.
Interpolatory Subdivision on Open Quadrilateral Nets with Arbitrary Topology
 Computer Graphics Forum
, 1996
"... A simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented which generates C 1 surfaces in the limit. The scheme satisfies important requirements for practical applications in computer graphics and engineering. These requirements include the necessity to ..."
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Cited by 143 (10 self)
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A simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented which generates C 1 surfaces in the limit. The scheme satisfies important requirements for practical applications in computer graphics and engineering. These requirements include the necessity to generate smooth surfaces with local creases and cusps. The scheme can be applied to open nets in which case it generates boundary curves that allow a C 0 join of several subdivision patches. Due to the local support of the scheme, adaptive refinement strategies can be applied. We present a simple device to preserve the consistency of such adaptively refined nets. Keywords: Curve and surface modeling, Interpolatory subdivision, Adaptive meshrefinement 1 Introduction The problem we address in this paper is the generation of smooth interpolating surfaces of arbitrary topological type in the context of practical applications. Such applications range from the design of freeform surfaces an...
Interpolating Wavelet Transform
, 1992
"... We describe several "wavelet transforms" which characterize smoothness spaces and for which the coefficients are obtained by sampling rather than integration. We use them to reinterpret the empirical wavelet transform, i.e. the common practice of applying pyramid filters to samples of a f ..."
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Cited by 131 (13 self)
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We describe several "wavelet transforms" which characterize smoothness spaces and for which the coefficients are obtained by sampling rather than integration. We use them to reinterpret the empirical wavelet transform, i.e. the common practice of applying pyramid filters to samples of a function.
Sets of matrices all infinite products of which converge. Linear Algebra and its Applications
, 1992
"... An infinite product IIT = lMi of matrices converges (on the right) if limi _ _ M,... Mi exists. A set Z = (Ai: i> l} of n X n matrices is called an RCP set (rightconvergent product set) if all infinite products with each element drawn from Z converge. Such sets of matrices arise in constructing ..."
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Cited by 82 (1 self)
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An infinite product IIT = lMi of matrices converges (on the right) if limi _ _ M,... Mi exists. A set Z = (Ai: i> l} of n X n matrices is called an RCP set (rightconvergent product set) if all infinite products with each element drawn from Z converge. Such sets of matrices arise in constructing selfsimilar objects like von Koch’s snowflake curve, in various interpolation schemes, in constructing wavelets of compact support, and in studying nonhomogeneous Markov chains. This paper gives necessary conditions and also some sufficient conditions for a set X to be an RCP set. These are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices. Necessary and sufficient conditions are given for a finite set Z to be an RCP set having a limit function M,(d) = rIT = lAd,, where d = (d,,., d,,..>, which is a continuous function on the space of all sequences d with the sequence topology. Finite RCP sets of columnstochastic matrices are completely characterized. Some results are given on the problem of algorithmically
A chronology of interpolation: From ancient astronomy to modern signal and image processing
 Proceedings of the IEEE
, 2002
"... This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into histo ..."
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Cited by 75 (0 self)
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This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation. Keywords—Approximation, convolutionbased interpolation, history, image processing, polynomial interpolation, signal processing, splines. “It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it. ” 1 I.
Multiresolution representations using the autocorrelation functions of compactly supported wavelets
 IEEE Trans. Signal Processing
, 1993
"... CT 06520 0 ..."
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Smooth Wavelet Decompositions with Blocky Coefficient Kernels
, 1993
"... We describe bases of smooth wavelets where the coefficients are obtained by integration against (finite combinations of) boxcar kernels rather than against traditional smooth wavelets. Bases of this type were first developed in work of Tchamitchian and of Cohen, Daubechies, and Feauveau. Our approac ..."
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Cited by 55 (12 self)
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We describe bases of smooth wavelets where the coefficients are obtained by integration against (finite combinations of) boxcar kernels rather than against traditional smooth wavelets. Bases of this type were first developed in work of Tchamitchian and of Cohen, Daubechies, and Feauveau. Our approach emphasizes the idea of averageinterpolation  synthesizing a smooth function on the line having prescribed boxcar averages  and the link between averageinterpolation and DubucDeslauriers interpolation. We also emphasize characterizations of smooth functions via their coefficients. We describe boundarycorrected expansions for the interval, which have a simple and revealing form. We use these results to reinterpret the empirical wavelet transform  i.e. finite, discrete wavelet transforms of data arising from boxcar integrators (e.g. CCD devices).
Subdivision schemes in Lp spaces
 Adv. Comput. Math
, 1995
"... Subdivision schemes play an important role in computer graphics and wavelet analysis. In this paper we are mainly concerned with convergence of subdivision schemes in Lp spaces (1 ≤ p ≤ ∞). We characterize the Lpconvergence of a subdivision scheme in terms of the pnorm joint spectral radius of two ..."
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Cited by 53 (21 self)
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Subdivision schemes play an important role in computer graphics and wavelet analysis. In this paper we are mainly concerned with convergence of subdivision schemes in Lp spaces (1 ≤ p ≤ ∞). We characterize the Lpconvergence of a subdivision scheme in terms of the pnorm joint spectral radius of two matrices associated with the corresponding mask. We also discuss various properties of the limit function of a subdivision scheme, such as stability, linear independence, and smoothness.
A Multiresolution Framework for Variational Subdivision
, 1998
"... Subdivision is a powerful paradigm for the generation of curves and surfaces. It is easy to implement, computationally efficient, and useful in a variety of applications because of its intimate connection with multiresolution analysis. An important task in computer graphics and geometric modeling is ..."
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Cited by 43 (0 self)
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Subdivision is a powerful paradigm for the generation of curves and surfaces. It is easy to implement, computationally efficient, and useful in a variety of applications because of its intimate connection with multiresolution analysis. An important task in computer graphics and geometric modeling is the construction of curves that interpolate a given set of points and minimize a fairness functional (variational design). In the context of subdivision, fairing leads to special schemes requiring the solution of a banded linear system at every subdivision step. We present several examples of such schemes including one that reproduces nonuniform interpolating cubic splines. Expressing the construction in terms of certain elementary operations we are able to embed variational subdivision in the lifting framework, a powerful technique to construct wavelet filter banks given a subdivision scheme. This allows us to extend the traditional lifting scheme for FIR filters to a certain class of IIR filters. Consequently we show how to build variationally optimal curves and associated, stable wavelets in a straightforward fashion. The algorithms to perform the corresponding decomposition and reconstruction transformations are easy to implement and efficient enough for interactive applications.