Quasi-interpolation by quadratic -splines on truncated octahedral partitions

Rhein, Markus; Kalbe, Thomas

doi:10.1016/j.cagd.2009.04.002

Cited by 5 publications

(8 citation statements)

References 26 publications

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“…8 Here we summarize the main ideas and the necessary basics needed to understand our visualization pipeline.…”

Section: Splines On Truncated Octahedral Partitionsmentioning

confidence: 99%

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Interactive isosurfaces with quadratic C¹splines on truncated octahedral partitions

Marinc¹,

Kalbe²,

Rhein³

et al. 2011

Visualization and Data Analysis 2011

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The reconstruction of a continuous function from discrete data is a basic task in many applications such as the visualization of 3D volumetric data sets. We use a local approximation method for quadratic C 1 splines on uniform tetrahedral partitions to achieve a globally smooth function. The spline is based on a truncated octahedral partition of the volumetric domain, where each truncated octahedron is further split into a fixed number of disjunct tetrahedra. The Bernstein-Bézier coefficients of the piecewise polynomials are directly determined by appropriate combinations of the data values in a local neighborhood. As previously shown, the splines provide an approximation order two for smooth functions as well as their derivatives. We present the first visualizations using these splines and show that they are well-suited for GPU-based, interactive high-quality visualization of isosurfaces from discrete data.

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“…8 Here we summarize the main ideas and the necessary basics needed to understand our visualization pipeline.…”

Section: Splines On Truncated Octahedral Partitionsmentioning

confidence: 99%

“…8 This spline is based on a more complex tetrahedral partition using a truncated octahedral partition of the volumetric domain, see Section 3, and provides a better numerical approximation of the original data compared to the cubic splines on type-6 tetrahedral partitions.…”

Section: Introductionmentioning

confidence: 99%

Interactive isosurfaces with quadratic C¹splines on truncated octahedral partitions

Marinc¹,

Kalbe²,

Rhein³

et al. 2011

Visualization and Data Analysis 2011

View full text Add to dashboard Cite

show abstract

“…Therefore, to cover the whole domain we need one-quarter as many TOs as there are data points in the volumetric grid. The spline approximation scheme 9 is described through a linear operator, which maps the set of discrete data values into the space of quadratic C 1 splines regarding the uniform tetrahedral partition described in the previous section. This operator is given explicitly by concrete formulae for the computation of the spline coefficients in B-form as simple linear combinations of some local data values.…”

Section: The Truncated Octahedral Partition and The Data Meshmentioning

confidence: 99%

“…Basics about the B-form will be given below; further details and references can be found in Rhein and Kalbe. 9 The B-form has several advantages, including stable evaluation of the spline and its derivatives. For smoothness between neighbouring polynomials only local rules based on geometrical conditions have to be considered.…”

Section: Trivariate Bernstein–bézier Splinesmentioning

confidence: 99%

“…We use ray casting on the GPU based on a new trivariate spline, which, for the first time, solves the problem of finding a local approximation method by quadratic polynomials and is globally C 1 continuous. 9 This spline is based on a more complex tetrahedral partition using a truncated octahedral partition of the volumetric domain (see 'Splines on truncated octahedral partitions'), and provides a better numerical approximation of the original data compared with the cubic splines on type-6 tetrahedral partitions.…”

Section: Introductionmentioning

confidence: 99%

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Interactive isosurfaces with quadratic C 1 splines on truncated octahedral partitions

Marinc

Kalbe

Rhein

et al. 2012

Information Visualization

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The reconstruction of a continuous function from discrete data is a basic task in many applications such as the visualization of 3D volumetric data sets. We use a local approximation method for quadratic C 1 splines on uniform tetrahedral partitions to achieve a globally smooth function. The spline is based on a truncated octahedral partition of the volumetric domain, where each truncated octahedron is further split into a fixed number of disjunct tetrahedra. The Bernstein-Bézier coefficients of the piecewise polynomials are directly determined by appropriate combinations of the data values in a local neighbourhood. As previously shown, the splines provide an approximation order two for smooth functions as well as their derivatives. We present the first visualizations using these splines and show that they are well suited for graphics processing unit (GPU)based, interactive, high-quality visualization of isosurfaces from discrete data.

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Multivariate normalized Powell–Sabin B-splines and quasi-interpolants

Speleers

2013

Computer Aided Geometric Design

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We present the construction of a multivariate normalized B-spline basis for the quadratic C 1 -continuous spline space defined over a triangulation in R s (s ≥ 1) with a generalized Powell-Sabin 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 s-simplices that must contain a specific set of points. We also propose a family of quasi-interpolants based on this multivariate Powell-Sabin B-spline representation. Their spline coefficients only depend on a set of local function values. The multivariate quasi-interpolants reproduce quadratic polynomials and have an optimal approximation order.

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Quasi-interpolation by quadratic -splines on truncated octahedral partitions

Cited by 5 publications

References 26 publications

Interactive isosurfaces with quadratic C¹splines on truncated octahedral partitions

Interactive isosurfaces with quadratic C¹splines on truncated octahedral partitions

Interactive isosurfaces with quadratic C 1 splines on truncated octahedral partitions

Multivariate normalized Powell–Sabin B-splines and quasi-interpolants

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Quasi-interpolation by quadratic -splines on truncated octahedral partitions

Cited by 5 publications

References 26 publications

Interactive isosurfaces with quadratic C1splines on truncated octahedral partitions

Interactive isosurfaces with quadratic C1splines on truncated octahedral partitions

Interactive isosurfaces with quadratic C 1 splines on truncated octahedral partitions

Multivariate normalized Powell–Sabin B-splines and quasi-interpolants

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Product

Resources

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Interactive isosurfaces with quadratic C¹splines on truncated octahedral partitions

Interactive isosurfaces with quadratic C¹splines on truncated octahedral partitions