Range restricted C2 interpolant to scattered data

The construction of a range restricted bivariate C2 interpolant to scattered data is considered in which the interpolant is positive everywhere if the original data are positive. Sufficient conditions are derived on Bezier points in order to ensure that surfaces comprising quintic Bezier triangular patches are always positive and satisfy C2 continuity conditions. The first and second derivatives at the data sites are then calculated (and modified if necessary) to ensure that these conditions are satisfied. Its construction is local and easily extended to include as upper and lower constraints to the interpolating surfaces of the form z = C(x,y) where C is a polynomial of degree less or equal to 5. A number of examples are presented

An improved positivity preserving odd degree-n Said-Ball boundary curves on rectangular grid using partial differential equation

This paper discusses the sufficient conditions for positivity preserving odd degree-n Said-Ball boundary curves defined on a rectangular grid.We derive a sufficient condition on boundary curves of rectangular Said-Ball patches where the lower bound ordinates are adjusted independently.To construct the boundary curves for each rectangular patch, the Said-Ball polynomial solution of fourth order PDE will be considered where its coefficients can be calculated using edge Said-Ball ordinates which fulfill the positivity preserving conditions.Graphical examples are presented using well-known test functions

Constructing new control points for Bézier interpolating polynomials using new geometrical approach

Interpolation is a mathematical technique employed for estimating the value of missing data between data points. This technique assures that the resulting polynomial passes through all data points. One of the most useful interpolating polynomials is the parametric interpolating polynomial. Bézier interpolating curves and surfaces are parametric interpolating polynomials for two-dimensional (2D) and three-dimensional (3D) datasets, respectively, that produce smooth, flexible, and accurate functions. According to the previous studies, the most crucial component in deriving Bézier interpolating polynomials is the construction of control points. However, most of the existing strategies constructed control points that produce partial smooth functions. As a result, the approximate values of the missing data are not accurate. In this study, nine new strategies of geometrical approach for constructing new 2D and 3D Bézier control points are proposed. The obtained control points from each new strategies are substituted in the relevant Bézier curve and surface equations to derive Bézier piecewise and non-piecewise interpolating polynomials which leads to the development of nine new methods. The proposed methods are proven to preserve the stability and smoothness of the generated Bézier interpolating curves and surfaces. In addition, the numerical results show that most of the resulting polynomials are able to approximate the missing values more accurately compared to those derived by the existing methods. The Bézier interpolating surfaces derived by the proposed method with highest accuracy for 3D datasets are then applied to upscale grey and colour images by the factors of two and three. Not only does the proposed method produces higher quality upscaled images, the numerical results also show that it outperforms the existing methods in terms of accuracy. Therefore, this study has successfully proposed new strategies for constructing new 2D and 3D control points for deriving Bézier interpolating polynomials that are capable of approximating the missing values accurately. In terms of application, the derived Bézier interpolating surfaces have a great potential to be employed in image upscaling