On C2 cubic quasi-interpolating splines and their computation by subdivision via blossoming

We discuss the construction of C2 cubic spline quasi-interpolation schemes defined on a refined partition. These schemes are reduced in terms of degrees of freedom compared to those existing in the literature. Namely, we provide a rule for reducing them 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."Maria de Maeztu" Excellence Unit IMAG (University of Granada, Spain) CEX2020-001105-MICIN/AEI/10.13039/501100011033University of Granada University of Granada/CBU

A new approach to deal with C2 cubic splines and its application to super-convergent quasi-interpolation

The authors wish to thank the anonymous referees for their very pertinent and useful comments which helped them to improve the original manuscript. The first and third authors are members of the research group FQM 191 Matematica Aplicada funded by the PAIDI programme of the Junta de Andalucia, Spain. The second author would like to thank the University of Granada, Spain for the financial support for the research stay during which this work was carried out. Funding for open access charge: Universidad de Granada/CBUAIn this paper, we construct a novel normalized B-spline-like representation for C2-continuous cubic spline space defined on an initial partition refined by inserting two new points inside each sub-interval. The basis functions are compactly supported non-negative functions that are geometrically constructed and form a convex partition of unity. With the help of the control polynomial theory introduced herein, a Marsden identity is derived, from which several families of super-convergent quasi-interpolation operators are defined.Junta de AndaluciaUniversity of Granada, SpainUniversidad de Granada/CBU

Horizontal accuracy assessment of a novel algorithm for approximate a surface to a DEM

This study evaluates the horizontal positional accuracy of a new algorithm that defines a surface that approximates DEM data by means of a spline function. This algorithm allows evaluating the surface at any point in its definition domain and allows analytically estimating other parameters of interest, such as slopes, orientations, etc. To evaluate the accuracy achieved with the algorithm, we use a reference DEM 2 m × 2 m (DEMref) from which the derived DEMs are obtained at 4 m × 4 m, 8 m × 8 m and 16 m × 16 m (DEMder). For each DEMder its spline approximant is calculated, which is evaluated at the same points occupied by the DEMref cells, getting a resampled DEM 2x2m (DEMrem). The horizontal accuracy is obtained by computing the area amongs the homologous contour lines derived from DEMref and DEMrem, respectively. It has been observed that the planimetric errors of the proposed algorithm are very small, even in flat areas, where you could expect major differences. Therefore, this algorithm could be used when an evaluation of the horizontal positional accuracy of a DEM product at lower resolution (DEMpro) and a different producing source than the higher resolution DEMref is wanted

Quasi-Interpolation in a Space of C 2 Sextic Splines over Powell–Sabin Triangulations

In this work, we study quasi-interpolation in a space of sextic splines defined over Powell– Sabin triangulations. These spline functions are of class C 2 on the whole domain but fourth-order regularity is required at vertices and C 3 regularity is 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 a 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. Finally, some tests show the good performance of these operators.Erasmus+ International Dimension programme, European CommissionPAIDI programme, Junta de Andalucía, Spai

Aproximación mediante funciones spline sobre triangulaciones de tipo Powell-Sabin

Smooth splines on triangulations are the subject of many applications in various elds, 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 di cult 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 C1 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 C2. Quasi-interpolation in a space of sextic PS-splines has also been considered. These spline functions are C2 continuous on the whole domain but fourth-order regularity is required at vertices and C3 smoothness conditions are imposed across the edges of the re ned triangulation and also at the interior point chosen to de ne the re nement. An algorithm is proposed to de ne 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 have proved that any Gaussian quadrature formula exact on the space of quadratic polynomials de ned on a triangle T endowed with a speci c PS-re nement integrates also the functions in the space of C1 quadratic PS-splines de ned on T. This extends the existing results in the literature, where the inner split point Z chosen to de ne the split had to lie on a very speci c subset of the T. Now Z can be freely chosen inside T. Sometimes, when dealing with Digital Elevation Models in engineering, the construction of normalized basis functions could be extremely expensive regarding time and memory needed, which is caused by the treatment of big data. To avoid this problem, we provide quasiinterpolation schemes de ned on a uniform triangulation of type-1 endowed with a PS-split. The spline schemes are generated by setting their B ezier ordinates to suitable combinations of the given data values. Inspiring from bivariate PS-splines theory, we de ne a family of univariate many knot spline spaces of arbitrary degree de ned on an initial partition that is re ned 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-B ezier representation. Blossoming is then used to establish a Marsden's identity from which several quasi-interpolation operators having optimal approximation orders are de ned. Finally, we address the approximation by C2 cubic splines via two approaches. In the rst one, we discuss the construction of C2 cubic spline quasi-interpolation schemes de ned on a re ned 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 coe cients associated with a ner partition from those associated with the former one. While in the second approach, we construct a novel normalized B-spline-like representation for C2 continuous cubic spline space de ned on an initial partition re ned by inserting two new points inside each sub-interval. Thus, we derive several families of super-convergent quasi-interpolation operators.El objetivo general de esta tesis es la construcción de espacios de funciones spline sobre particiones de Powell-Sabin, tanto en un sentido clásico como en una situación univariada.Tesis Univ. Granada.University of GranadaCNRST Excellence Scholarship Programm

C2 Cubic Algebraic Hyperbolic Spline Interpolating Scheme by Means of Integral Values

The authors wish to thank the anonymous referees for their very pertinent and useful comments, which helped them to improve the original manuscript. The first author would like to thank the Department of Applied Mathematics of the University of Granada for the financial support for the research stay during which this work was carried out. The authors wish to thank the Hassan First University of Settat for the financial aid offered for the final cost of the APC.In this paper, a cubic Hermite spline interpolating scheme reproducing both linear polynomials and hyperbolic functions is considered. The interpolating scheme is mainly defined by means of integral values over the subintervals of a partition of the function to be approximated, rather than the function and its first derivative values. The scheme provided is C2 everywhere and yields optimal order. We provide some numerical tests to illustrate the good performance of the novel approximation scheme.Department of Applied Mathematics of the University of GranadaHassan First University of Setta

A geometric characterization of Powell-Sabin triangulations allowing the construction of C-2 quartic splines

The authors wish to thank the anonymous referees for their very pertinent and useful comments which helped them to improve the original manuscript. The first and third authors are members of the research group FQM 191 Matem?tica Aplicada funded by the PAIDI programme of the Junta de Andaluc?a. The second author would like to thank the University of Granada for the financial support for the research stay during which this work was carried out.The authors wish to thank the anonymous referees for their very pertinent and useful comments which helped them to improve the original manuscript. The first and third authors are members of the research group FQM 191 Matemática Aplicada funded by the PAIDI programme of the Junta de Andalucía. The second author would like to thank the University of Granada for the financial support for the research stay during which this work was carried out.The paper deals with the characterization of Powell-Sabin triangulations allowing the construction of bivariate quartic splines of class C-2. The result is established by relating the triangle and edge split points provided by the refinement of each triangle. For a triangulation fulfilling the characterization obtained, a normalized representation of the splines in the C-2 space is given.Junta de AndaluciaUniversity of Granad