arXiv subjects
Publications and source records attributed to Paweł Woźny.
We define and investigate two families of dual polynomials associated with the Gauss--Legendre polynomials, which have recently found interesting applications in computer graphics. Using the presented results, one can derive representations of the Gauss--Legendre polynomials, construct the dual bases for Lagrange bases and solve certain approximation problems arising, for example, in CAGD.
It is well-known that a $d$-dimensional polynomial Bézier curve of degree $n$ can be subdivided into two segments using the famous de Casteljau algorithm in $O(dn^2)$ time. Can this problem be solved more efficiently? In this paper, we show that it is possible to do this in $O(dn\log{n})$ time using the fast Fourier transform and its inverse. Experiments show that the direct application of the new method performs well only for small values of $n$, as the algorithm is numerically unstable. However, a slightly modified version -- which still has $O(dn\log{n})$ computational complexity -- offers good numerical quality, which is confirmed by numerical experiments conducted in \textsf{Python}. Moreover, the new method has a nice property: if a Bézier curve is extended by an additional control point, the subdivision can be updated in $O(d)$ time. A similar idea can be applied to speed up the subdivision of rational Bézier curves and rectangular Bézier surfaces, as well as to compute the derivatives of Bézier curves more efficiently.
We present new representations of Gauss--Legendre polynomials and their derivatives in the shifted power basis and in bases related to symmetric orthogonal Jacobi polynomials. Using these representations and certain recurrence relations, we propose efficient $O(n^2+dn)$ methods for evaluating a Gauss--Legendre curve of degree $n$ in $\mathbb E^d$. We also propose algorithms for multipoint evaluation with computational complexity $O(Mdn+dn^2)$, where $M$ is the number of evaluation points.
New differential-recurrence relations for B-spline basis functions are given. Using these relations, a recursive method for finding the Bernstein-Bézier coefficients of B-spline basis functions over a single knot span is proposed. The algorithm works for any knot sequence and has an asymptotically optimal computational complexity. Numerical experiments show that the new method gives results which preserve a high number of digits when compared to an approach which uses the well-known de Boor-Cox formula.
New geometric methods for fast evaluation of derivatives of polynomial and rational Bézier curves are proposed. They apply an algorithm for evaluating polynomial or rational Bézier curves, which was recently given by the authors. Numerical tests show that the new approach is more efficient than the methods which use the famous de Casteljau algorithm. The algorithms work well even for high-order derivatives of rational Bézier curves of high degrees.
A new differential-recurrence relation for the B-spline functions of the same degree is proved. From this relation, a recursive method of computing the coefficients of B-spline functions of degree $m$ in the Bernstein-Bézier form is derived. Its complexity is proportional to the number of coefficients in the case of coincident boundary knots. This means that, asymptotically, the algorithm is optimal. In other cases, the complexity is increased by at most $O(m^3)$. When the Bernstein-Bézier coefficients of B-spline basis functions are known, it is possible to compute any B-spline function in linear time with respect to its degree by performing the geometric algorithm proposed recently by the authors. This algorithm scales well when evaluating the B-spline curve at multiple points, e.g., in order to render it, since one only needs to find the coefficients for each knot span once. When evaluating many B-spline curves at multiple points (as is the case when rendering tensor product B-spline surfaces), such approach has lower computational complexity than using the de Boor-Cox algorithm. The numerical tests show that the new method is efficient. The problem of finding the coefficients of the B-spline functions in the power basis can be solved similarly.
Dual Bernstein polynomials find many applications in approximation theory, computational mathematics, numerical analysis and computer-aided geometric design. In this context, one of the main problems is fast and accurate evaluation both of these polynomials and their linear combinations. New simple recurrence relations of low order satisfied by dual Bernstein polynomials are given. In particular, a first-order non-homogeneous recurrence relation linking dual Bernstein and shifted Jacobi orthogonal polynomials has been obtained. When used properly, it allows to propose fast and numerically efficient algorithms for evaluating all $n+1$ dual Bernstein polynomials of degree $n$ with $O(n)$ computational complexity.
A new algorithm for computing a point on a polynomial or rational curve in Bézier form is proposed. The method has a geometric interpretation and uses only convex combinations of control points. The new algorithm's computational complexity is linear with respect to the number of control points and its memory complexity is $O(1)$. Some remarks on similar methods for surfaces in rectangular and triangular Bézier form are also given.
New differential-recurrence properties of dual Bernstein polynomials are given which follow from relations between dual Bernstein and orthogonal Hahn and Jacobi polynomials. Using these results, a fourth-order differential equation satisfied by dual Bernstein polynomials has been constructed. Also, a fourth-order recurrence relation for these polynomials has been obtained; this result may be useful in the efficient solution of some computational problems.
In this paper, we present a new iterative approximate method of solving boundary value problems. The idea is to compute approximate polynomial solutions in the Bernstein form using least squares approximation combined with some properties of dual Bernstein polynomials which guarantee high efficiency of our approach. The method can deal with both linear and nonlinear differential equations. Moreover, not only second order differential equations can be solved but also higher order differential equations. Illustrative examples confirm the versatility of our method.
In the paper, we give methods of construction of dual bases for the B-spline basis and truncated power basis. Explicit formulas for the dual B-spline basis are obtained using the Legendre-like orthogonal basis of the polynomial spline space presented in (Wei et al., Comput.-Aided Des. 45 (2013), 85-92) and a connection between orthogonal and dual bases of any space given in (Lewanowicz and Woźny, J. Approx. Theory 138 (2006), 129-150). Construction of the dual truncated power basis is performed in two phases. We start with explicit formulas for the dual power basis of the space of polynomials. Then, we expand this basis using an iterative algorithm proposed in (Woźny, J. Comput. Appl. Math. 260 (2014), 301-311). As a result, we obtain the dual truncated power basis. We also present some applications of the proposed dual polynomial spline bases and illustrative examples.
In the paper, we show that the transformations between modified Jacobi and Bernstein bases of the constrained space of polynomials of degree at most $n$ can be performed with the complexity $O(n^2)$. As a result, the algorithm of degree reduction of Bézier curves that was first presented in (Bhrawy et al., J. Comput. Appl. Math. 302 (2016), 369--384), and then corrected in (Lu and Xiang, J. Comput. Appl. Math. 315 (2017), 65--69), can be significantly improved, since the necessary transformations are done in those papers with the complexity $O(n^3)$. The comparison of running times shows that our transformations are also faster in practice.
In this paper, we give an efficient algorithm of degree reduction of Bézier curves with box constraints. The idea is to combine the previous iterative approach, that has been presented recently in (P. Gospodarczyk, Comput. Aided Des. 62 (2015), 143--151), with a fast method of construction of dual bases from (P. Woźny, J. Comput. Appl. Math. 260 (2014), 301--311) and a new efficient method of modification of dual bases.
Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining Bézier form of the $L^2$-solution of the problem of best polynomial approximation of Bézier curve or surface. In this connection, the Bézier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost. In this paper, a set of recurrence relations satisfied by the Bézier coefficients of dual bivariate Bernstein polynomials is derived and an efficient algorithm for evaluation of these coefficients is proposed. Applications of this result to some approximation problems of Computer Aided Geometric Design (CAGD) are discussed.
In a recent paper (Appl. Math. Comput. 215, 1622--1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the convergence acceleration theorem in the case of summation of generalized hypergeometric series. Some new theoretical results and illustrative numerical examples are given.
This paper deals with the problem of multi-degree reduction of a composite Bézier curve with the parametric continuity constraints at the endpoints of the segments. We present a novel method which is based on the idea of using constrained dual Bernstein polynomials to compute the control points of the reduced composite curve. In contrast to other methods, ours minimizes the $L_2$-error for the whole composite curve instead of minimizing the $L_2$-errors for each segment separately. As a result, an additional optimization is possible. Examples show that the new method gives much better results than multiple application of the degree reduction of a single Bézier curve.
We propose a novel approach to the problem of polynomial approximation of rational Bézier triangular patches with prescribed boundary control points. The method is very efficient thanks to using recursive properties of the bivariate dual Bernstein polynomials and applying a smart algorithm for evaluating a collection of two-dimensional integrals. Some illustrative examples are given.
In this paper, we present a novel approach to the problem of merging of Bézier curves with respect to the $L_2$-norm. We give illustrative examples to show that the solution of the conventional merging problem may not be suitable for further modification and applications. As in the case of the degree reduction problem, we apply the so-called restricted area approach -- proposed recently in (P. Gospodarczyk, Computer-Aided Design 62 (2015), 143--151) -- to avoid certain defects and make the resulting curve more useful. A method of solving the new problem is based on box-constrained quadratic programming approach.