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Intrinsic subdivision with smooth limits for graphics and animation
 ACM TRANS. GRAPH
, 2006
"... This paper demonstrates the definition of subdivision processes in nonlinear geometries such that smoothness of limits can be proved. We deal with curve subdivision in the presence of obstacles, in surfaces, in Riemannian manifolds, and in the Euclidean motion group. We show how to model kinematic s ..."
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Cited by 31 (7 self)
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This paper demonstrates the definition of subdivision processes in nonlinear geometries such that smoothness of limits can be proved. We deal with curve subdivision in the presence of obstacles, in surfaces, in Riemannian manifolds, and in the Euclidean motion group. We show how to model kinematic surfaces and motions in the presence of obstacles via subdivision. As to numerics, we consider the sensitivity of the limit’s smoothness to sloppy computing.
Smoothness analysis of subdivision schemes by proximity
 Constr. Approx
, 2006
"... Abstract. Linear curve subdivision schemes may be perturbed in various ways, e.g. by modifying them such as to work in a manifold, surface, or group. The analysis of such perturbed and often nonlinear schemes “T ” is based on their proximity to the linear schemes “S ” which they are derived from. Th ..."
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Cited by 31 (6 self)
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Abstract. Linear curve subdivision schemes may be perturbed in various ways, e.g. by modifying them such as to work in a manifold, surface, or group. The analysis of such perturbed and often nonlinear schemes “T ” is based on their proximity to the linear schemes “S ” which they are derived from. This paper considers two aspects of this problem: One is to find proximity inequalities which together with C k smoothness of S imply C k smoothness of T. The other is to verify these proximity inequalities for several ways to construct the nonlinear scheme T analogous to the linear scheme S. The first question is treated for general k, whereas the second one is treated only in the case k = 2. The main result of the paper is that convergent geodesic / projection / Lie group analogues of a certain class of factorizable linear schemes have C 2 limit curves. 1.
Stability of nonlinear subdivision and multiscale transforms, Manuscript, submitted for publication
, 2008
"... Extending upon [CDM03] and [AL05], we present a new general sufficient condition for the Lipschitz stability of nonlinear subdivision schemes and multiscale transforms in the univariate case. It covers the special cases (WENO, PPH) considered so far but also implies the stability in some new cases ( ..."
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Cited by 7 (1 self)
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Extending upon [CDM03] and [AL05], we present a new general sufficient condition for the Lipschitz stability of nonlinear subdivision schemes and multiscale transforms in the univariate case. It covers the special cases (WENO, PPH) considered so far but also implies the stability in some new cases (median interpolating transform, powerp schemes, etc.). Although the investigation concentrates on multiscale transforms {v0, d1,..., dJ} 7− → vJ, J ≥ 1, in ℓ∞(Z) given by a stationary recursion of the form vj = Svj−1 + dj, j ≥ 1, involving a nonlinear subdivision operator S acting on ℓ∞(Z), the approach is extendable to other nonlinear multiscale transforms and norms, as well. 1
Introduction to shiftinvariant spaces. Linear independence. Multivariate approximation and applications
, 2001
"... Abstract. Shiftinvariant spaces play an increasingly important role in various areas of mathematical analysis and its applications. They appear ..."
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Cited by 5 (1 self)
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Abstract. Shiftinvariant spaces play an increasingly important role in various areas of mathematical analysis and its applications. They appear
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.
Convergence and C¹ analysis of subdivision schemes on manifolds by proximity
 COMP. AIDED GEOM. DESIGN
, 2004
"... Curve subdivision schemes on manifolds and in Lie groups are constructed from linear subdivision schemes by first representing the rules of affinely invariant linear schemes in terms of repeated affine averages, and then replacing the operation of affine average either by a geodesic average (in the ..."
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Curve subdivision schemes on manifolds and in Lie groups are constructed from linear subdivision schemes by first representing the rules of affinely invariant linear schemes in terms of repeated affine averages, and then replacing the operation of affine average either by a geodesic average (in the Riemannian sense or in a certain Lie group sense), or by projection of the affine averages onto a surface. The analysis of these schemes is based on their proximity to the linear schemes which they are derived from. We verify that a linear scheme S and its analogous nonlinear scheme T satisfy a proximity condition. We further show that the proximity condition implies the convergence of T and continuity of its limit curves, if S has the same property, and if the distances of consecutive points of the initial control polygon are small enough. Moreover, if S satisfies a smoothness condition which is sufficient for its limit curves to be C¹, and if T is convergent, then the curves generated by T are also C¹. Similar analysis of C² smoothness is postponed to a forthcoming paper.
Nonlinear Subdivision Schemes and MultiScale Transforms
"... Over the past 25 years, fast multiscale algorithms such as wavelettype pyramid transforms for hierarchical data representation, multigrid solvers for the numerical solution of operator equations, and subdivision methods in computeraided geometric design lead to tremendous successes in data and g ..."
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Over the past 25 years, fast multiscale algorithms such as wavelettype pyramid transforms for hierarchical data representation, multigrid solvers for the numerical solution of operator equations, and subdivision methods in computeraided geometric design lead to tremendous successes in data and geometry processing, and in