804 research outputs found
From approximating to interpolatory non-stationary subdivision schemes with the same generation properties
In this paper we describe a general, computationally feasible strategy to
deduce a family of interpolatory non-stationary subdivision schemes from a
symmetric non-stationary, non-interpolatory one satisfying quite mild
assumptions. To achieve this result we extend our previous work [C.Conti,
L.Gemignani, L.Romani, Linear Algebra Appl. 431 (2009), no. 10, 1971-1987] to
full generality by removing additional assumptions on the input symbols. For
the so obtained interpolatory schemes we prove that they are capable of
reproducing the same exponential polynomial space as the one generated by the
original approximating scheme. Moreover, we specialize the computational
methods for the case of symbols obtained by shifted non-stationary affine
combinations of exponential B-splines, that are at the basis of most
non-stationary subdivision schemes. In this case we find that the associated
family of interpolatory symbols can be determined to satisfy a suitable set of
generalized interpolating conditions at the set of the zeros (with reversed
signs) of the input symbol. Finally, we discuss some computational examples by
showing that the proposed approach can yield novel smooth non-stationary
interpolatory subdivision schemes possessing very interesting reproduction
properties
Polynomial-based non-uniform interpolatory subdivision with features control
Starting from a well-known construction of polynomial-based interpolatory 4-point schemes, in this paper we present
an original affine combination of quadratic polynomial samples that leads to a non-uniform 4-point scheme with edge
parameters. This blending-type formulation is then further generalized to provide a powerful subdivision algorithm
that combines the fairing curve of a non-uniform refinement with the advantages of a shape-controlled interpolation
method and an arbitrary point insertion rule. The result is a non-uniform interpolatory 4-point scheme that is unique
in combining a number of distinctive properties. In fact it generates visually-pleasing limit curves where special
features ranging from cusps and flat edges to point/edge tension effects may be included without creating undesired
undulations. Moreover such a scheme is capable of inserting new points at any positions of existing intervals, so that
the most convenient parameter values may be chosen as well as the intervals for insertion.
Such a fully flexible curve scheme is a fundamental step towards the construction of high-quality interpolatory subdivision surfaces with features control
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