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

Contour Detection of Mammogram Masses Using ChanVese Model and B-Spline Approximation

ChanVese model segmentation has been applied for contour detection of mass region in mammogram in our previous work. Available information of the desired object contour is used, in this paper, for B-spline approximation. The mass region boundary (contour) is thereafter approximated by a B-spline curve. This approach allows synthesizing the shape of the suspected mass appearing in the mammogram. Experimental results show the accurateness of mass region contour in mammograms using B-spline approximation

An Algebraic Hyperbolic Spline Quasi-Interpolation Scheme for Solving Burgers-Fisher Equations

In this work, the results on hyperbolic spline quasi-interpolation are recalled to establish the numerical scheme to obtain approximate solutions of the generalized Burgers-Fisher equation. After introducing the generalized Burgers-Fisher equation and the algebraic hyperbolic spline quasi-interpolation, the numerical scheme is presented. The stability of our scheme is well established and discussed. To verify the accuracy and reliability of the method presented in this work, we select two examples to conduct numerical experiments and compare them with the calculated results in the literature

Approximation de surfaces fermées par des fonctions splines et applications.

Dans ce travail nous nous intéressons au problème d approximation ou d interpolation des données sur des surfaces fermées comme la sphère. Dans le premier chapitre, nous rappelons quelques résultats essentiels sur la théorie des splines sphériques. Nous étudierons ainsi, les polynômes de Bernstein-Bézier sphériques, les fonctions homogènes et les polynômes harmoniques. Nous exposons également quelques résultats techniques que nous utiliserons dans les chapitres 2 et 3. Dans le deuxième chapitre, nous nous intéressons à l étude des interpolants sphériques d Hermite et au calcul récursif de ces interpolants. Cette étude mène à la construction d une base hiérarchique qui a des propriétés intéressantes est qui permet de réduire le nombre de coefficients à calculés. Dans le troisième chapitre, nous proposons une méthode de quasi-interpolation basée essentiellement sur les B-splines sphériques de Powell-Sabin Bi,j , i = 1, , n, j = 1, 2, 3, et qui s adapte parfaitement à notre problème. Dans le quatrième chapitre, nous nous intéressons à la construction d un quasi-interpolant de classe C1 (ou C2) obtenu comme produit tensoriel de deux quasi-interpolants unidimensionnels de classe C1 (ou C2). Nous proposons notamment comme moyen de compression une analyse multirésolution basée sur le produit tensoriel des ondelettes splines algébriques et des ondelettes UAT-splines.In this work we consider the problem of approximation or interpolation of data on closed sphere-like surfaces. In the first chapter, we recall some key results on the theory of spherical splines. We study spherical homogeneous polynomial of Bernstein-Bézier functions and harmonic polynomials. We exhibit also some technical results that we use in chapters 2 and 3. In the second chapter, we study a recursive method for constricting a Hermite spline interpolant. This decomposition leads to the construction of a new and interesting basis of a space of Hermite spherical splines. In the third chapter, we deal with the problem of the construction of quasi-interpolants in the space of quadratic spherical Powell Sabin splines on uniform spherical triangulations of a sphere-like surface S. In the last chapter, we propose an efficient multiresolution method for fitting scattered data functions on a sphere S, using a tensor product method of periodic algebraic trigonometric splines of order 3 and quadratic polynomial splines defined on a rectangular map of S. We describe the decomposition and the reconstruction algorithms corresponding to the polynomial and periodic algebraic trigonometric waveletsPAU-BU Sciences (644452103) / SudocSudocFranceF

Contour Detection of Mammogram Masses Using ChanVese Model and B-Spline Approximation

ChanVese model segmentation has been applied for contour detection of mass region in mammogram in our previous work. Available information of the desired object contour is used, in this paper, for B-spline approximation. The mass region boundary (contour) is thereafter approximated by a B-spline curve. This approach allows synthesizing the shape of the suspected mass appearing in the mammogram. Experimental results show the accurateness of mass region contour in mammograms using B-spline approximation

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

A Reverse Non-Stationary Generalized B-Splines Subdivision Scheme

In this paper, two new families of non-stationary subdivision schemes are introduced. The schemes are constructed from uniform generalized B-splines with multiple knots of orders 3 and 4, respectively. Then, we construct a third-order reverse subdivision framework. For that, we derive a generalized multi-resolution mask based on their third-order subdivision filters. For the reverse of the fourth-order scheme, two methods are used; the first one is based on least-squares formulation and the second one is based on solving a linear optimization problem. Numerical examples are given to show the performance of the new schemes in reproducing different shapes of initial control polygons

<i>C</i>² Cubic Algebraic Hyperbolic Spline Interpolating Scheme by Means of Integral Values

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