770 research outputs found
A global approach to the refinement of manifold data
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.Comment: arXiv admin note: text overlap with arXiv:1407.836
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Uniform subdivision algorithms for curves and surfaces
A convergence analysis for studying the continuity and differentiability of limit curves generated by uniform subdivision algorithms is presented. The analysis is based on the study of corresponding difference and divided difference algorithms. The alternative process of "integrating" the algorithms is considered. A specific example of a 4-point interpolatory curve algorithm is described and its generalization to a surface algorithm defined over a subdivision of a regular triangular partition is illustrated
Analysis of uniform binary subdivision schemes for curve design
The paper analyses the convergence of sequences of control polygons produced by a binary subdivision scheme of the form
.0,1,2,...kz,ikj,ifjbm0j1k12ifjam0j1k2if=∈+Σ==++Σ==+
The convergence of the control polygons to a Cu curve is analysed in terms
of the convergence to zero of a derived scheme for the differences - . The analysis of the smoothness of the limit curve is reduced to kif
the convergence analysis of "differentiated" schemes which correspond to
divided differences of {/i ∈Z} with respect to the diadic parameteriz- kif
ation = i/2kitk . The inverse process of "integration" provides schemes
with limit curves having additional orders of smoothness
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