1,720,996 research outputs found
Semi-regular Dubuc–Deslauriers wavelet tight frames
Full text linkIn this paper, we construct wavelet tight frames with n vanishing moments for Dubuc–Deslauriers 2n-point semi-regular interpolatory subdivision schemes. Our motivation for this construction is its practical use for further regularity analysis of wide classes of semi-regular subdivision. Our constructive tools are local eigenvalue convergence analysis for semi-regular Dubuc–Deslauriers subdivision, the Unitary Extension Principle and the generalization of the Oblique Extension Principle to the irregular setting by Chui, He and Stöckler. This group of authors derives suitable approximation of the inverse Gramian for irregular B-spline subdivision. Our main contribution is the derivation of the appropriate approximation of the inverse Gramian for the semi-regular Dubuc–Deslauriers scaling functions ensuring n vanishing moments of the corresponding framelets
Optimized dual interpolating subdivision schemes
Full text linkThis work investigates the non-stepwise interpolation property of the recently introduced class of dual interpolating subdivision schemes, and the “loss of memory” phenomenon that comes with it. New differences between schemes having an odd and an even dilation factors are highlighted. In particular, dual interpolating schemes having an odd dilation factor are proven to satisfy a 2-step interpolation property, while an even dilation factor corresponds to a completely non-stepwise interpolation process. These facts are exploited to define an optimized non-uniform level dependent implementation of dual interpolating schemes in order to overcome the computational drawback due to the “loss of memory”
Algebraic characterization of planar cubic and quintic Pythagorean-Hodograph B-spline curves
Full text linkWe provide a revised representation of planar cubic and quintic Pythagorean-Hodograph B-spline curves (PH B-splines for short) that offers the following advantages: (i) the clamped and closed cases are mostly treated together; (ii) the closed case is represented by using the minimum possible number of knots thus avoiding useless control points as well as control edges of zero length when the curve is regular. The proposed simplified representation turns out to be extremely useful to provide a unified complex algebraic characterization of clamped and closed planar PH B-splines of degree three and five. This is aimed at distinguishing regular planar cubic and quintic PH B-splines from C1 cubic and C2 quintic B-spline curves in general. As for planar cubic PH B-splines consisting of m pieces, we obtain m complex conditions that, differently from what was known so far, can be used to characterize both the clamped and the closed case. As for planar quintic PH B-splines, the complex conditions are 2m and, unlike what is shown for cubic PH B-splines, they also depend on the knot intervals. This is to be considered a completely new result since no complex algebraic characterization working for any arbitrarily chosen knot partition had ever been provided for either clamped or closed planar quintic PH B-splines. The proposed algebraic characterization is finally exploited to fully identify the preimage of a regular planar quintic PH B-spline resolving all the sign ambiguities that affected the existing results
Planar class A Bézier curves: The case of real eigenvalues
Full text linkWe consider planar, special Bézier curves, i.e., polynomial Bézier curves in the plane whose control polygon is fully identified by the first edge and a 2×2 matrix M. We focus on the case where M has two real eigenvalues and we formulate, in terms of the Schur form of M, necessary and sufficient conditions for a regular, planar special Bézier curve to be a class A curve, i.e., a curve with monotone curvature, for any degree and any choice of the first edge. The result is simple in its formulation and can thus be easily used for both designing class A curves and analyzing given special Bézier curves
CONSTRUCTION AND EVALUATION OF PYTHAGOREAN HODOGRAPH CURVES IN EXPONENTIAL-POLYNOMIAL SPACES
Full text linkIn the past few decades polynomial curves with Pythagorean hodograph (PH curves) have received considerable attention due to their usefulness in various CAD/CAM areas, manufacturing, numerical control machining, and robotics. This work deals with classes of PH curves built upon exponential-polynomial spaces (EPH curves). In particular, for the two most frequently encountered exponential-polynomial spaces, we first provide necessary and sufficient conditions to be satisfied by the control polygon of the Bézier-like curve in order to fulfill the PH property. Then, for such EPH curves, fundamental characteristics like parametric speed or arc length are discussed to show the interesting analogies with their well-known polynomial counterparts. Differences and advantages with respect to ordinary PH curves become commendable when discussing the solutions to application problems like the interpolation of first-order Hermite data. Finally, a new evaluation algorithm for EPH curves is proposed and shown to compare favorably with the celebrated de Casteljau-like algorithm and two recently proposed methods: Wózny and Chudy's algorithm and the dynamic evaluation procedure by Yang and Hong
Dual univariate interpolatory subdivision of every arity: Algebraic characterization and construction
No full textA new class of univariate stationary interpolatory subdivision schemes of dual type is presented. As opposed to classical primal interpolatory schemes, these new schemes have masks with an even number of elements and are not step-wise interpolants. A complete algebraic characterization, which covers every arity, is given in terms of identities of trigonometric polynomials associated to the schemes. This characterization is based on a necessary condition for refinable functions to have prescribed values at the nodes of a uniform lattice, as a consequence of the Poisson summation formula. A strategy for the construction is then showed, alongside meaningful examples for applications that have comparable or even superior properties, in terms of regularity, length of the support and/or polynomial reproduction, with respect to the primal counterparts
On the refinement matrix mask of interpolating Hermite splines
No full textWe propose a new computational approach for constructing the refinement matrix mask of interpolating Hermite splines of any order and with general dilation factor. Our strategy exploits the refinability properties of cardinal B-splines with simple knots and simplifies the constructive procedures proposed so far
An annihilator-based strategy for the automatic detection of exponential polynomial spaces in subdivision
Full text linkExponential polynomials are essential in subdivision for the reconstruction of specific families of curves and surfaces, such as conic sections and quadric surfaces. It is well known that if a linear subdivision scheme is able to reproduce a certain space of exponential polynomials, then it must be level-dependent, with rules depending on the frequencies (and eventual multiplicities) defining the considered space. This work discusses a general strategy that exploits annihilating operators to locally detect those frequencies directly from the given data and therefore to choose the correct subdivision rule to be applied. This is intended as a first step towards the construction of self-adapting subdivision schemes able to locally reproduce exponential polynomials belonging to different spaces. An application of the proposed strategy is shown explicitly on an example involving the classical butterfly interpolatory scheme. This particular example is the generalization of what has been done for the univariate case in Donat and López-Ureña (2019), which inspired this work
A compact algebraic representation of cardinal GB-splines
Full text linkThis work introduces a compact algebraic representation of generalized B-spline basis functions built upon uniform knot partitions (also known as cardinal GB-splines), that stands out for its simplicity with respect to the well-known integral formulation. Moreover, this result clarifies the relationship between cardinal GB-splines and classical polynomial B-splines, as it isolates the polynomial component of a GB-spline from the non-polynomial contribution brought by the two non-monomial generators of the function space
New algebraic and geometric characterizations of planar quintic Pythagorean-hodograph curves
Full text linkThe aim of this work is to provide new characterizations of planar quintic Pythagorean-hodograph curves. The first two are algebraic and consist of two and three equations, respectively, in terms of the edges of the Bézier control polygon as complex numbers. These equations are symmetric with respect to the edge indices and cover curves with generic as well as degenerate control polygons. The last two characterizations are geometric and rely both on just two auxiliary points outside the control polygon. One requires two (possibly degenerate) quadrilaterals to be similar, and the other highlights two families of three similar triangles. All characterizations are a step forward with respect to the state of the art, and they can be linked to the well-established counterparts for planar cubic Pythagorean-hodograph curves. The key ingredient for proving the aforementioned results is a novel general expression for the hodograph of the curve
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