Subdivision Schemes of Sets and the Approximation of Set-Valued Functions in the Symmetric Difference Metric

Sharon

2016

Math. Comp.

Self Cite

A refinement of manifold data is a computational process, which produces a denser set of discrete data from a given one. Such refinements are closely related to multiresolution representations of manifold data by pyramid transforms, and approximation of manifold-valued functions by repeated refinements schemes. Most refinement methods compute each refined element separately, independently of the computations of the other elements. Here we propose a global method which computes all the refined elements simultaneously, using geodesic averages. We analyse repeated refinements schemes based on this global approach, and derive conditions guaranteeing strong convergence.

Section: Optimal Choice Of Parameters In the Three Pyramidmentioning

confidence: 95%

“…Substituting (21) and α m 1 +1 = ρe iθ we get the sufficient condition for contractivity ρ 2 − 2(γ(1 − cos(θ)) − cos(θ))ρ + 1 > 0,…”

Section: A2 Proof Of Theorem 410mentioning

confidence: 98%

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A global approach to the refinement of manifold data

Sharon

2016

Math. Comp.

Self Cite

“…We adapt the 4-point scheme using a geodesic average M t , under the assumption that it is well defined for t in a small neighbourhood of [0, 1]. Note that such an adaptation was already done in [16] for positive definite matrices, and in [17] for sets. The symmetry of the coefficients, (−ω, 1 2 + ω, 1 2 + ω, −ω) implies that the adaptation of (16) is…”

Section: Examples Of Convergent Gim-schemesmentioning

confidence: 99%

Manifold-valued subdivision schemes based on geodesic inductive averaging

Journal of Computational and Applied Mathematics

Sharon

2017

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Subdivision schemes have become an important tool for approximation of manifold-valued functions.In this paper, we describe a construction of manifold-valued subdivision schemes for geodesically complete manifolds. Our construction is based upon the adaptation of linear subdivision schemes using the notion of repeated binary averaging, where as a repeated binary average we propose to use the geodesic inductive mean. We derive conditions on the adapted schemes which guarantee convergence from any initial manifold-valued sequence. The definition and analysis of convergence are intrinsic to the manifold. The adaptation technique and the convergence analysis are demonstrated by several important examples.

“…We use the form suggested in [12] for the refinement rule in (17) written in terms of repeated binary averages as…”

Section: The Modified 4-point Scheme (M4pt)mentioning

confidence: 99%

A weighted binary average of point-normal pairs with application to subdivision schemes

Lipovetsky

Computer Aided Geometric Design

2016

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Subdivision is a well-known and established method for generating smooth curves and surfaces from discrete data by repeated refinements. The typical input for such a process is a mesh of vertices. In this work we propose to refine 2D data consisting of vertices of a polygon and a normal at each vertex. Our core refinement procedure is based on a circle average, which is a new non-linear weighted average of two points and their corresponding normals. The ability to locally approximate curves by the circle average is demonstrated. With this ability, the circle average is a candidate for modifying linear subdivision schemes refining points, to schemes refining point-normal pairs. This is done by replacing the weighted binary arithmetic means in a linear subdivision scheme, expressed in terms of repeated binary averages, by circle averages with the same weights. Here we investigate the modified Lane-Riesenfeld algorithm and the 4-point scheme. For the case that the initial data consists of a control polygon only, a naive method for choosing initial normals is proposed. An example demonstrates the superiority of the above two modified schemes, with the naive choice of initial normals over the corresponding linear schemes, when applied to a control polygon with edges of significantly different lengths.