An explicit quasi-interpolation scheme based on C1 quartic splines on type-1 triangulations

Institute for Mathematics, University of Mannheim, 68131 Mannheim, Germany
Computer Aided Geometric Design (Impact Factor: 1.64). 01/2008; 25(1):1-13. DOI: 10.1016/j.cagd.2007.05.006
Source: DBLP


We describe a new scheme based on quartic C1-splines on type-1 triangulations approximating regularly distributed data. The quasi-interpolating splines are directly determined by setting the Bernstein–Bézier coefficients of the splines to appropriate combinations of the given data values. Each polynomial piece of the approximating spline is immediately available from local portions of the data, without using prescribed derivatives at any point of the domain. Moreover, the operator interpolates the given data values at all the vertices of the underlying triangulation. Since the Bernstein–Bézier coefficients of the splines are computed directly, an intermediate step making use of certain locally supported splines spanning the space is not needed. We prove that the splines yield nearly-optimal approximation order for smooth functions. The order is known to be best possible for these spaces. Numerical tests confirm the theoretical behavior and show that the approach leads to functional surfaces of high visual quality.

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    • "Note that the results obtained for the discrete quasi-interpolant are comparable to those reported in [32] for a quasi-interpolation operator defined using a quite different method. The error for a step length h associated with Q d,h must be compared with the errors associated with h/4 in the latter. "
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    ABSTRACT: In this paper, we show how by a very simple modification of bivariate spline discrete quasi-interpolants, we can construct a new class of quasi-interpolants, which have remarkable properties such as high order of regularity and polynomial reproduction. More precisely, given a spline discrete quasi-interpolation operator Q d , which is exact on the space P m of polynomials of total degree at most m, we first propose a general method to determine a new differential quasi-interpolation operator Q D r which is exact on P m+r . Q D r uses the values of the function to be approximated at the points involved in the linear functional defining Q d as well as the partial derivatives up to the order r at the same points. From this result, we then construct and study a first order differential quasi-interpolant based on the C 1 B-spline on the equilateral triangulation with a hexagonal support. When the derivatives are not available or extremely expensive to compute, we approximate them by appropriate finite differences to derive new discrete quasi-interpolant Q d . We estimate with small constants the quasi-interpolation errors f −Q D r [f] and f Q d [f] in the infinity norm. Finally, numerical examples are used to analyze the performance of the method.
    Journal of Computational and Applied Mathematics 11/2013; 252. DOI:10.1016/ · 1.27 Impact Factor
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    • "Therefore, the proposed method must be modified to be able to consider functions defined on bounded regions. Once established, it will be possible to adopt a procedure of graphical representation based on the evaluation at the points of a fine grid constructed by subdividing the triangles forming the triangulation of the region, as done, for example, in [25]. These considerations also can be applied to the results obtained in the following section. "
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    ABSTRACT: We define a class of discrete quasi-interpolants based on bivariate box splines by imposing the exactness on a space of polynomials of total degree, depending on the box spline and minimizing a constant appearing in the leading term of an appropriate quasi-interpolation error estimate. We give some C1 quadratic and C2 quartic examples and compare them with other well-known quasi-interpolants.
    Journal of Computational and Applied Mathematics 02/2009; 224(1-224):250-268. DOI:10.1016/ · 1.27 Impact Factor
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    ABSTRACT: We present a novel GPU-based algorithm for high-quality rendering of bivariate spline surfaces. An essential difference to the known methods for rendering graph surfaces is that we use quartic smooth splines on triangulations rather than triangular meshes. Our rendering approach is direct in the sense that since we do not use an intermediate tessellation but rather compute ray-surface intersections (by solving quartic equations numerically) as well as surface normals (by using Bernstein-Bézier techniques) for Phong illumination on the GPU. Inaccurate shading and artifacts appearing for triangular tesselated surfaces are completely avoided. Level of detail is automatic since all computations are done on a per fragment basis. We compare three different (quasi-) interpolating schemes for uniformly sampled gridded data, which differ in the smoothness and the approximation properties of the splines. The results show that our hardware based renderer leads to visualizations (including texturing, multiple light sources, environment mapping, etc.) of highest quality.
    IEEE Transactions on Visualization and Computer Graphics 09/2008; 14(5):1126-39. DOI:10.1109/TVCG.2008.66 · 2.17 Impact Factor
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