The famous de Boor conjecture states that the condition of the polynomial B-spline collocation matrix at the knot averages is bounded independently of the knot sequence, i.e., it depends only on the spline degree. For highly nonuniform knot meshes, like geometric meshes, the conjecture is known to be false. As an effort towards finding an answer for uniform meshes, we investigate the spectral c...

We present an iterative multiresolution algorithm for the translational and rotational alignment of digital images. An image is represented by an interpolating spline. Coarser versions of this continuous image model are obtained by using spline approximations at various scales (polynomial spline pyramid). We use a coarse-to-fine updating strategy to compute the alignment parameters iteratively,...

In this paper, we discuss the problem of instability in the dimension of a spline space over a T-mesh. For bivariate spline spaces S (5, 5, 3, 3) and S (4, 4, 2, 2), the instability in the dimension is shown over certain types of T-meshes. This result could be considered as an attempt to answer the question of how large the polynomial degree (m,m) should be relative to the smoothness (r, r) to ...

Let 2 be a polyhedral complex embedded in the euclidean space Ed and Sr(2), r ≥ 0, denote the set of all Cr-splines on 2. Then Sr(2) is an R-module where R = E[x1, . . . , xd] is the ring of polynomials in several variables. In this paper we state and prove the existence of an algorithm to write down a free basis for the above R-module in terms of obvious linear forms defining common faces of m...

The major problem when building shape models by minimising the description length (DL) is the computationally demanding optimisation in a highdimensional space. To speed up the convergence when dealing with discretised contours, Ericsson and Åström [6] have shown how to approximate and how to exploit the gradient of the DL. We derive the gradient of the DL for differentiable training sets. Addi...

For a function f 2 L p ?1; 1], 0 < p < 1, with nitely many sign changes, we construct a sequence of polynomials P n 2 n which are copositive with f and such that kf ? P n k p C! ' (f; (n + 1) ?1) p , where ! ' (f; t) p denotes the Ditzian-Totik modulus of continuity in L p metric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, then ! ...

New upper and lower bounds for the L2 and Vo norms of derivatives of the error in polynomial spline interpolation are derived. These results improve corresponding results of Ahlberg, Nilson, and Walsh, cf. [1], and Schultz and Varga, cf. [5].

This paper explores numerical solutions of optimal control problems using B–Spline curves. It is aimed to give a general framework on how to use B–Splines to formulate optimal control problems and to solve them numerically using Nonlinear Trajectory Generation software package. Effects of the selection of the B–Spline parameters, such as, number of intervals, smoothness, piecewise polynomial or...

Multivariate spline functions are smooth piecewise polynomial functions over triangulations consisting of n-simplices in the Euclidean space IR. A straightforward method for using these spline functions to fit given scattered data and numerically solve elliptic partial differential equations is presented . This method does not require constructing macro-elements or locally supported basis funct...