Resumen de Approximation by spline functions on Powell-Sabin triangulations
Salah Eddargani
Smooth splines on triangulations are the subject of many applications in various fields, among them approximation theory, computer-aided geometric design, entertainment industry, etc. Smooth spline spaces with a lower degree are the classical choice, which is extremely difficult to achieve in arbitrary triangulations. An alternative is to use macro elements of lower degree that split each triangle into a number of macro-triangles. In particular, Powell-Sabin (PS-) split which divides each triangle into six macro-triangles.
In this thesis, we deal with the approximation by quartic PS-splines. Namely, we start by solving a Hermite interpolation problem in the space of C^1 quartic PS-splines and providing several local quasi-interpolation schemes reproducing quartic polynomials and not requiring the resolution of any linear system. The provided schemes are constructed with the help of Marsden's identity. Then, we address the geometric characterization of Powell-Sabin triangulations allowing the construction of bivariate quartic splines of class C^2.
Quasi-interpolation in a space of sextic PS-splines are also considered. These spline functions are C^2 continuous on the whole domain but fourth-order regularity is required at vertices and C^3 smoothness conditions are imposed across the edges of the refined triangulation and also at the interior point chosen to define the refinement. An algorithm is proposed to define the Powell-Sabin triangles with small area and diameter needed to construct a normalized basis. Quasi-interpolation operators which reproduce sextic polynomials are constructed after deriving Marsden's identity from a more explicit version of the control polynomials introduced some years ago in the literature.
Examining the applicability of PS-splines the numerical quadratures, we proved that any Gaussian quadrature formula exact on the space of quadratic polynomials defined on a triangle T endowed with a specific PS-refinement integrates also the functions in the space of C^1 quadratic PS-splines defined on T. This extends the existing results in the literature, where the inner split point Z chosen to define the split had to lie on a very specific subset of the T. Now Z can be freely chosen inside T.
When dealing with Digital Elevation Models in engineering, the construction of normalized basis functions can be extremely expensive and memory demanding when treating big data. To avoid this problem, we provide quasi-interpolation schemes defined on a uniform triangulation of type-1 endowed with a PS-split. The spline schemes are generated by setting their Bézier ordinates to suitable combinations of the given data values.
Inspiring from bivariate PS-splines theory, we define a family of univariate many knot spline spaces of arbitrary degree defined on an initial partition that is refined by adding a point in each sub-interval. For an arbitrary smoothness r, splines of degrees 2r and 2r + 1 are considered by imposing additional regularity when necessary. For an arbitrary degree, a B-spline-like basis is constructed by using the Bernstein-Bézier representation. Blossoming is then used to establish a Marsden's identity from which several quasi-interpolation operators having optimal approximation orders are defined.
Finally, we address the approximation by C^2 cubic splines via two approaches. In the first one, we discuss the construction of C^2 cubic spline quasi-interpolation schemes defined on a refined partition. These schemes are reduced in terms of the degree of freedom compared to those existing in the literature. Namely, we provide a recipe for reducing the degree of freedom by imposing super-smoothing conditions while preserving full smoothness and cubic precision. In addition, we provide subdivision rules by means of blossoming. The derived rules are designed to express the B-spline coefficients associated with a finer partition from those associated with the former one. While in the second approach, we construct a novel normalized B-spline-like representation for C^2 continuous cubic spline space defined on an initial partition refined by inserting two new points inside each sub-interval. Thus, we derive several families of super-convergent quasi-interpolation operators.
