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A local Lagrange interpolation method based on C¹ cubic splines on Freudenthal partitions
, 1997
"... A trivariate Lagrange interpolation method based on C¹ cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation me ..."
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A trivariate Lagrange interpolation method based on C¹ cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation method is local and stable, provides optimal order approximation, and has linear complexity.
Optimal quasiinterpolation by quadratic C1 splines on fourdirectional meshes. Approximation Theory XI: Gatlinburg 2004
, 2005
"... Abstract. We describe a new scheme based on quadratic C1splines on type2 triangulations approximating gridded data. The quasiinterpolating splines are directly determined by setting the BernsteinBézier coefficients of the splines to appropriate combinations of the given data values. In this way ..."
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Abstract. We describe a new scheme based on quadratic C1splines on type2 triangulations approximating gridded data. The quasiinterpolating splines are directly determined by setting the BernsteinBézier coefficients of the splines to appropriate combinations of the given data values. In this way, 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. Since the BernsteinBézier 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 optimal approximation order for smooth functions, where we provide explicit constants in the corresponding error bounds. The aim of this paper is to describe local methods which use quadratic C1splines on the type2 triangulation to approximate data on a
Spline Approximation of General Volumetric Data
, 2004
"... We present an efficient algorithm for approximating huge general volumetric data sets, i.e. the data is given over arbitrarily shaped volumes and consists of up to millions of samples. The method is based on cubic trivariate splines, i.e. piecewise polynomials of total degree three defined w.r.t. un ..."
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We present an efficient algorithm for approximating huge general volumetric data sets, i.e. the data is given over arbitrarily shaped volumes and consists of up to millions of samples. The method is based on cubic trivariate splines, i.e. piecewise polynomials of total degree three defined w.r.t. uniform type6 tetrahedral partitions of the volumetric domain. Similar as in the recent bivariate approximation approaches (cf. [10, 15]), the splines in three variables are automatically determined from the discrete data as a result of a twostep method (see [40]), where local discrete least squares polynomial approximations of varying degrees are extended by using natural conditions, i.e. the continuity and smoothness properties which determine the underlying spline space. The main advantages of this approach with linear algorithmic complexity are as follows: no tetrahedral partition of the volume data is needed, only small linear systems have to be solved, the local variation and distribution of the data is automatically adapted, BernsteinBézier techniques wellknown in Computer Aided Geometric Design (CAGD) can be fully exploited, noisy data are automatically smoothed. Our numerical examples with huge data sets for synthetic data as well as some realworld data confirm the efficiency of the methods, show the high quality of the spline approximation, and illustrate that the rendered isosurfaces inherit a visual smooth appearance from the volume approximating splines.
Fast Visualization by ShearWarp on Quadratic SuperSpline Models Using Wavelet Data Decompositions
"... of shearwarp by using wavelet encoded data. The closeup views are taken from marked areas. Zooming close to the local regions of interest the improved visual quality becomes increasingly evident. Left: Standard model (trilinear interpolation). Center: Approximation by quadratic supersplines. Righ ..."
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of shearwarp by using wavelet encoded data. The closeup views are taken from marked areas. Zooming close to the local regions of interest the improved visual quality becomes increasingly evident. Left: Standard model (trilinear interpolation). Center: Approximation by quadratic supersplines. Right: Approximation by quadratic supersplines with decimated data (14.8 % of the given data). The quadratic supersplines have the potential to reduce noise and leads to almost artifactfree visualizations. We develop the first approach for interactive volume visualization based on a sophisticated rendering method of shearwarp type, wavelet data encoding techniques, and a trivariate spline model, which has been introduced [24] recently. As a first step of our algorithm, we apply standard wavelet expansions [6, 31] to represent and decimate the given gridded threedimensional data. Based on this data encoding, we give a sophisticated version of the shearwarp based volume rendering method [13]. Our new algorithm visits each voxel only once taking advantage of the particular data organization of octrees. In addition, the hierarchies of the data
A C¹ Quadratic Trivariate Macroelement Space Defined Over Arbitrary Tetrahedral Partitions
, 2007
"... In 1988, Worsey and Piper constructed a trivariate macroelement based on C¹ quadratic splines defined over a split of a tetrahedron into 24 subtetrahedra. However, this local element can only be used to construct a corresponding macroelement spline space over tetrahedral partitions that satisfy s ..."
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In 1988, Worsey and Piper constructed a trivariate macroelement based on C¹ quadratic splines defined over a split of a tetrahedron into 24 subtetrahedra. However, this local element can only be used to construct a corresponding macroelement spline space over tetrahedral partitions that satisfy some very restrictive geometric constraints. We show that by further refining their split, it is possible to construct a macroelement also based on C¹ quadratic splines that can be used with arbitrary tetrahedral partitions. The resulting macroelement space is stable and provides full approximation power.
Bsplines and quasiinterpolants
, 2012
"... We present the construction of a multivariate normalized Bspline basis for the quadratic C 1continuous spline space defined over a triangulation in R s (s ≥ 1) with a generalized PowellSabin refinement. The basis functions have a local support, they are nonnegative, and they form a partition of u ..."
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We present the construction of a multivariate normalized Bspline basis for the quadratic C 1continuous spline space defined over a triangulation in R s (s ≥ 1) with a generalized PowellSabin refinement. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction can be interpreted geometrically as the determination of a set of ssimplices that must contain a specific set of points. We also propose a family of quasiinterpolants based on this multivariate PowellSabin Bspline representation. Their spline coefficients only depend on a set of local function values. The multivariate quasiinterpolants reproduce quadratic polynomials and have an optimal approximation order.
AND
, 2005
"... doi:10.1093/imanum/drl014 Local quasiinterpolation by cubic C1 splines on type6 tetrahedral partitions TATYANA SOROKINA† ..."
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doi:10.1093/imanum/drl014 Local quasiinterpolation by cubic C1 splines on type6 tetrahedral partitions TATYANA SOROKINA†