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31
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.
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.
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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Weighted averages on surfaces
 ACM Trans. Graph
, 2013
"... Figure 1: Interactive control for various geometry processing and modeling applications made possible with weighted averages on surfaces. From left to right: texture transfer, decal placement, semiregular remeshing and Laplacian smoothing, splines on surfaces. We consider the problem of generalizing ..."
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Cited by 5 (1 self)
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Figure 1: Interactive control for various geometry processing and modeling applications made possible with weighted averages on surfaces. From left to right: texture transfer, decal placement, semiregular remeshing and Laplacian smoothing, splines on surfaces. We consider the problem of generalizing affine combinations in Euclidean spaces to triangle meshes: computing weighted averages of points on surfaces. We address both the forward problem, namely computing an average of given anchor points on the mesh with given weights, and the inverse problem, which is computing the weights given anchor points and a target point. Solving the forward problem on a mesh enables applications such as splines on surfaces, Laplacian smoothing and remeshing. Combining the forward and inverse problems allows us to define a correspondence mapping between two different meshes based on provided corresponding point pairs, enabling texture transfer, compatible remeshing, morphing and more. Our algorithm solves a single instance of a forward or an inverse problem in a few microseconds. We demonstrate that anchor points in the above applications can be added/removed and moved around on the meshes at interactive framerates, giving the user an immediate result as feedback.
Swept Volumes
"... 3 Given a solid S ⊂ R with a piecewise smooth boundary, we compute an approximation of the boundary surface of the volume which is swept by S under a smooth oneparameter motion. Using knowledge from kinematical and elementary differential geometry, the algorithm computes a set of points plus surfac ..."
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Cited by 4 (0 self)
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3 Given a solid S ⊂ R with a piecewise smooth boundary, we compute an approximation of the boundary surface of the volume which is swept by S under a smooth oneparameter motion. Using knowledge from kinematical and elementary differential geometry, the algorithm computes a set of points plus surface normals from the envelope surface. A study of the evolution speed of the so called characteristic set along the envelope is used to achieve a prescribed sampling density. With a marching algorithm in a grid, the part of the envelope which lies on the boundary of the swept volume is extracted. The final boundary representation of the swept volume is either a triangle mesh, a Bspline surface or a pointset surface.
Smoothness of interpolatory multivariate subdivision in Lie groups
, 2008
"... Nonlinear subdivision schemes that operate on manifolds are of use whenever manifold valued data have to be processed in a multiscale fashion. This paper considers the case where the manifold is a Lie group and the nonlinear subdivision schemes are derived from linear interpolatory ones by the soca ..."
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Cited by 4 (1 self)
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Nonlinear subdivision schemes that operate on manifolds are of use whenever manifold valued data have to be processed in a multiscale fashion. This paper considers the case where the manifold is a Lie group and the nonlinear subdivision schemes are derived from linear interpolatory ones by the socalled logexp analogy. The main result of the paper is that a multivariate interpolatory Lie group valued subdivision scheme derived from a linear scheme is at least as smooth as the linear scheme, where smoothness is understood in terms of Hölder exponents.