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31
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 properties of lie group subdivision schemes
 ROSSIGNAC / SCREWBENDER: SMOOTHING PIECEWISE HELICAL MOTIONS
, 2006
"... Linear stationary subdivision rules take a sequence of input data and produce ever denser sequences of subdivided data from it. They are employed in multiresolution modeling and have intimate connections with wavelet and more general pyramid transforms. Data which naturally do not live in a vector ..."
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Cited by 26 (8 self)
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Linear stationary subdivision rules take a sequence of input data and produce ever denser sequences of subdivided data from it. They are employed in multiresolution modeling and have intimate connections with wavelet and more general pyramid transforms. Data which naturally do not live in a vector space, but in a nonlinear geometry like a surface, symmetric space, or a Lie group (e.g. motion capture data), require different handling. One way to deal with Lie group valued data has been proposed by D. Donoho [3]: It is to employ a logexponential analogue of a linear subdivision rule. While a comprehensive discussion of applications is given by Ur Rahman et al. in [9], this paper analyzes convergence and smoothness of such subdivision processes and show that the nonlinear schemes essentially have the same properties regarding C¹ and C² smoothness as the linear schemes they are derived from.
Smoothness equivalence properties of manifoldvalued data subdivision schemes based on the projection approach
 SIAM Journal on Numerical Analysis
"... Interpolation of manifoldvalued data is a fundamental problem which has applications in many fields. The linear subdivision method is an efficient and wellstudied method for interpolating or approximating realvalued data in a multiresolution fashion. A natural way to apply a linear subdivision sc ..."
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Cited by 22 (2 self)
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Interpolation of manifoldvalued data is a fundamental problem which has applications in many fields. The linear subdivision method is an efficient and wellstudied method for interpolating or approximating realvalued data in a multiresolution fashion. A natural way to apply a linear subdivision scheme S to interpolate manifoldvalued data is to first embed the manifold at hand to an Euclidean space and construct a projection operator P that maps points from the ambient space to a closest point on the embedded surface, and then consider the nonlinear subdivision operator S: = P ◦ S. When applied to symmetric spaces such
Smoothness analysis of subdivision schemes on regular grids by proximity
, 2006
"... Subdivision is a very powerful way of approximating a continuous object f(x, y) by a sequence ((Slpi,j)i,j∈Z)l∈N of discrete data on finer and finer grids. The rule S, that maps an approximation on a coarse grid, Slp, to the approximation on the next finer grid, Sl+1p, is called subdivision scheme. ..."
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Cited by 17 (8 self)
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Subdivision is a very powerful way of approximating a continuous object f(x, y) by a sequence ((Slpi,j)i,j∈Z)l∈N of discrete data on finer and finer grids. The rule S, that maps an approximation on a coarse grid, Slp, to the approximation on the next finer grid, Sl+1p, is called subdivision scheme. If for a given scheme S every continuous object f(x, y) constructed by S is of Ck smoothness, then S is said to have smoothness order k. Subdivision schemes are well understood if they are linear. However, for various applications the data have values in a manifold which is not a vector space (for example when our data are positions of a moving rigid body). Under these circumstances, subdivsion schemes become nonlinear and much harder to analyze. One way of analyzing such schemes is to relate them to a given linear scheme and establishing a socalled proximity condition between the two schemes, which helps in proving that the two schemes share the same smoothness. The present paper uses this method to show the C1smoothness of a wide class of nonlinear multivariate schemes.
Interpolatory wavelets for manifoldvalued data
, 2009
"... Geometric waveletlike transforms for univariate and multivariate manifoldvalued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of interpolat ..."
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Cited by 15 (6 self)
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Geometric waveletlike transforms for univariate and multivariate manifoldvalued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of interpolatory wavelet transforms, which applies to Riemannian geometry, Lie groups and other geometries, Hölder smoothness of functions is characterized by decay rates of their wavelet coefficients.
Logexponential analogues of univariate subdivision schemes in Lie groups and their smoothness properties
 APPROXIMATION THEORY XII
, 2007
"... The necessity to process data which live in nonlinear geometries (e.g. motion capture data, unit vectors, subspaces, positive definite matrices) has led to some recent developments in nonlinear multiscale representation and subdivision algorithms. The present paper analyzes convergence and C 1 and ..."
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Cited by 9 (1 self)
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The necessity to process data which live in nonlinear geometries (e.g. motion capture data, unit vectors, subspaces, positive definite matrices) has led to some recent developments in nonlinear multiscale representation and subdivision algorithms. The present paper analyzes convergence and C 1 and C 2 smoothness of subdivision schemes which operate in matrix groups or general Lie groups, and which are defined by the socalled logexponential analogy. It is shown that a large class of such schemes has essentially the same smoothness as the linear schemes they are derived from. This work extends previous work on Lie group subdivision schemes – we consider alternative definitions of analogous schemes, arbitrary dilation factors, and symmetry of the nonlinear scheme.
Smoothness equivalence properties of univariate subdivision schemes and their projection analogues
 NUMERISCHE MATHEMATIK
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Smoothness equivalence properties of general manifoldvalued data subdivision schemes. Multiscale Modeling and Simulation
, 2008
"... Based on a vectorbundle formulation, we introduce a new family of nonlinear subdivision schemes for manifoldvalued data. Any such nonlinear subdivision scheme is based on an underlying linear subdivision scheme. We show that if the underlying linear subdivision scheme reproduces Πk, then the nonli ..."
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Cited by 7 (2 self)
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Based on a vectorbundle formulation, we introduce a new family of nonlinear subdivision schemes for manifoldvalued data. Any such nonlinear subdivision scheme is based on an underlying linear subdivision scheme. We show that if the underlying linear subdivision scheme reproduces Πk, then the nonlinear scheme satisfies an order k proximity condition with the linear scheme. We also develop a new “proximity ⇒ smoothness ” theorem, improving the one in [12]. Combining the two results, we can conclude that if the underlying linear scheme is Ck and stable, the nonlinear scheme is also Ck. The family of manifoldvalued data subdivision scheme introduced in this paper includes a variant of the logexp scheme, proposed in [10], as a special case, but not the original logexp scheme when the underlying linear scheme is noninterpolatory. The original logexp scheme uses the same tangent plane for both the odd and the even rules, while the variant uses two different, judiciously chosen, tangent planes. We also present computational experiments that indicate that the original smoothness equivalence conjecture posted in [10] is unlikely to be true. Our result also generalizes the recent results in [17, 16, 5, 6]. It uses only the intrinsic smoothness structure of the manifold and (hence) does not rely on any embedding or Lie group or symmetric space or Riemannian structure. In particular, concepts such as geodesics, log and exp maps, or projection from ambient space play no explicit role in the theorem. Also, the underlying linear scheme needs not be interpolatory.
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