Polygonal blending splines in application to image processing

The paper proposes a novel method of image representation. The basic idea of the method is to transform color images to continuous parametric surfaces. The proposed technique is based on a class of special basis functions, defined on the polygon grid. Besides a exible and symmetric construction, these basis functions are strictly local and Cd-smooth on the entire domain. Having a number of unique features, the proposed representation can be used in various image processing tasks. The main purpose of this paper is to demonstrate the process of the image transformation and discuss possible applications of the presented technique

Geometrically nonlinear polygonal finite element analysis of functionally graded porous plates

In this study, an efficient polygonal finite element method (PFEM) in combination with quadratic serendipity shape functions is proposed to study nonlinear static and dynamic responses of functionally graded (FG) plates with porosities. Two different porosity types including even and uneven distributions through the plate thickness are considered. The quadratic serendipity shape functions over arbitrary polygonal elements including triangular and quadrilateral ones, which are constructed based on a pairwise product of linear shape functions, are employed to interpolate the bending strains. Meanwhile, the shear strains are defined according to the Wachspress coordinates. By using the Timoshenko's beam to interpolate the assumption of the strain field along the edges of polygonal element, the shear locking phenomenon can be naturally eliminated. Furthermore, the C0–type higher-order shear deformation theory (C0–HSDT), in which two additional variables are included in the displacement field, significantly improves the accuracy of numerical results. The nonlinear equations of static and dynamic problems are solved by Newton–Raphson iterative procedure and by Newmark's integration scheme in association with the Picard methods, respectively. Through various numerical examples in which complex geometries and different boundary conditions are involved, the proposed approach yields more stable and accurate results than those generated using other existing approaches

Projective geometry of Wachspress coordinates

We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress in 1975. The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren in 1996. This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves< are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic

On a class of splines free of Gibbs phenomenon

When interpolating data with certain regularity, spline functions are useful. They are defined as piecewise polynomials that satisfy certain regularity conditions at the joints. In the literature about splines it is possible to find several references that study the apparition of Gibbs phenomenon close to jump discontinuities in the results obtained by spline interpolation. This work is devoted to the construction and analysis of a new nonlinear technique that allows to improve the accuracy of splines near jump discontinuities eliminating the Gibbs phenomenon. The adaption is easily attained through a nonlinear modification of the right hand side of the system of equations of the spline, that contains divided differences. The modification is based on the use of a new limiter specifically designed to attain adaption close to jumps in the function. The new limiter can be seen as a nonlinear weighted mean that has better adaption properties than the linear weighted mean. We will prove that the nonlinear modification introduced in the spline keeps the maximum theoretical accuracy in all the domain except at the intervals that contain a jump discontinuity, where Gibbs oscillations are eliminated. Diffusion is introduced, but this is fine if the discontinuity appears due to a discretization of a high gradient with not enough accuracy. The new technique is introduced for cubic splines, but the theory presented allows to generalize the results very easily to splines of any order. The experiments presented satisfy the theoretical aspects analyzed in the paper.We would like to thank the anonymous referees for their valuable comments, which have helped to significantly improve this work. This work was funded by project 20928/PI/18 (Proyecto financiado por la Comunidad Autónoma de la Región de Murcia a través de la convocatoria de Ayudas a proyectos para el desarrollo de investigación científica y técnica por grupos competitivos, incluida en el Programa Regional de Fomento de la Investigación Científica y Técnica (Plan de Actuación 2018) de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia), by the national research project MTM2015- 64382-P (MINECO/FEDER) and by NSF grant DMS-1719410

Polygonal spline spaces and the numerical solution of the poisson equation

It is known that generalized barycentric coordinates (GBCs) can be used to form Bernstein polynomial-like functions over a polygon with any number of sides. We propose to use these functions to form a space of continuous polygonal splines (piecewisely defined functions) of order d over a partition consisting of polygons which is able to reproduce all polynomials of degree d. Locally supported basis functions for the space are constructed for order d>=2. The construction for d=2 is simpler than the `serendipity' quadratic finite elements that have appeared in the recent literature. The number of basis functions is similar to, but fewer than, those of the virtual element method. We use them for the numerical solution of the Poisson equation on two special types of non-triangular partitions to present a proof of concept for solving PDE's over polygonal partitions. Numerical solutions based on quadrangulations and pentagonal partitions are demonstrated to show the efficiency of these polygonal spline functions. They can lead to a more accurate solution by using fewer degrees of freedom than the traditional continuous polynomial finite element method if the solutions are smooth although assembling the mass and stiffness matrices can take more time