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...

Boundary conditions are still an open question in the field of approximating subdivision since the problem of determining a general construction of the endpoint rules we need when subdividing a B-spline curve/surface with Bézier end conditions has not been solved yet. This consideration prompted us to present an efficient algorithm for the conversion between B-spline bases defined over differen...

Knot insertion is the operation of obtaining a new representation of a B-spline curve by introducing additional knot values to the defining knot vector. The new curve has control points consisting of the original control points and additional new control points corresponding to the number of new knot values. So knot insertions give additional control points which provide extra shape control wit...

Evaluation routines are essential for any application that uses triangular B-spline surfaces. This paper describes an algorithm to efficiently evaluate triangular B-spline surfaces with arbitrary many variables. The novelty of the algorithm is its generality: there is no restriction on the degree of the B-spline surfaces or on the dimension of the domain. Constructing an evaluation graph allows...

The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two–scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spli...

We prove that a compactly supported spline function of degree k satisses the scaling equation (x) = P N n=0 c(n)(mx?n) for some integer m 2, if and only if (x) = P n p(n)B k (x?n) where p(n) are the coeecients of a polynomial P(z) such that the roots of P(z)(z ? 1) k+1 are mapped into themselves by the mapping z ! z m , and B k is the uniform B-spline of degree k. Furthermore, the shifts of for...

We analyze an alternative to classical Zernike fitting based on the cubic B-spline model, and compare the strengths and weaknesses of each representation over a set of different wavefronts that cover a wide range of shape complexity. The results obtained show that a Zernike low-degree polynomial expansion or a cubic B-spline with a low number of breakpoints are the best choices for fitting simp...

Multi-degree Tchebycheffian splines are with pieces drawn from extended (complete) Tchebycheff spaces, which may differ interval to interval, and possibly of different dimensions. These a natural extension multi-degree polynomial splines. Under quite mild assumptions, they can be represented in terms so-called B-spline (MDTB-spline) basis; such basis possesses all the characterizing properties ...