Scattered Data Interpolation and Applications: A Tutorial and Survey

Franke, Richard; Nielson, Gregory M.

doi:10.1007/978-3-642-76404-2_6

Cited by 120 publications

(63 citation statements)

References 47 publications

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“…We refer to one of many surveys on this topic for further details [10]. Recently, Park et al [25,26] have shown that scattered data reconstruction for large data sets can be achieved at interactive or near-interactive rates when discretizing the approach and resampling over a regular grid.…”

Section: Related Workmentioning

confidence: 99%

Smooth Surface Extraction from Unstructured Point-based Volume Data Using PDEs

Rosenthal

Linsen

2008

IEEE Trans. Visual. Comput. Graphics

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Abstract-Smooth surface extraction using partial differential equations (PDEs) is a well-known and widely used technique for visualizing volume data. Existing approaches operate on gridded data and mainly on regular structured grids. When considering unstructured point-based volume data where sample points do not form regular patterns nor are they connected in any form, one would typically resample the data over a grid prior to applying the known PDE-based methods. We propose an approach that directly extracts smooth surfaces from unstructured point-based volume data without prior resampling or mesh generation. When operating on unstructured data one needs to quickly derive neighborhood information. The respective information is retrieved by partitioning the 3D domain into cells using a kd-tree and operating on its cells. We exploit neighborhood information to estimate gradients and mean curvature at every sample point using a four-dimensional least-squares fitting approach. Gradients and mean curvature are required for applying the chosen PDE-based method that combines hyperbolic advection to an isovalue of a given scalar field and mean curvature flow. Since we are using an explicit time-integration scheme, time steps and neighbor locations are bounded to ensure convergence of the process. To avoid small global time steps, one can use asynchronous local integration. We extract a smooth surface by successively fitting a smooth auxiliary function to the data set. This auxiliary function is initialized as a signed distance function. For each sample and for every time step we compute the respective gradient, the mean curvature, and a stable time step. With these informations the auxiliary function is manipulated using an explicit Euler time integration. The process successively continues with the next sample point in time. If the norm of the auxiliary function gradient in a sample exceeds a given threshold at some time, the auxiliary function is reinitialized to a signed distance function. After convergence of the evolvution, the resulting smooth surface is obtained by extracting the zero isosurface from the auxiliary function using direct isosurface extraction from unstructured point-based volume data and rendering the extracted surface using point-based rendering methods.

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Section: Related Workmentioning

confidence: 99%

Smooth Surface Extraction from Unstructured Point-based Volume Data Using PDEs

Rosenthal

Linsen

2008

IEEE Trans. Visual. Comput. Graphics

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“…Determine cluster C to be split; (1) Create w new clusters C i by splitting cluster C; (2) Determine neighborhood N i of cluster c I ; (3) Locally reclassify data of clusters in N i ; (4)…”

Section: For All Tasks: While Termination Criterion Is Not Met {mentioning

confidence: 99%

Divisive Parallel Clustering for Multiresolution Analysis

Heckel¹,

Hamann

2004

Geometric Modeling for Scientific Visualization

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Abstract. Clustering is a classical data analysis technique that is applied to a wide range of applications in the sciences and engineering. For very large data sets, the performance of a clustering algorithm becomes critical. Although clustering has been thoroughly studied over the last decades, little has been done on utilizing modern multi-processor machines to accelerate the analysis process. We propose a scalable clustering technique that benefits from existing parallel computers and networks of workstations. It supports the creation of multiresolution representations for very large geometric data sets. The output of the clustering process can be used for interactive data exploration, useful for view-dependent rendering, user-guided refinement, and progressive transmission.

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“…This fitting procedure is somewhat complicated by the irregular degrees of the triangulated nodes since it requires that every node have at least five neighbors (a radial ordering for nodes beyond the immediate neighbors is not determined by the triangulation, so we do not search beyond the neighborhood to bring in additional samples as is common in least-squares schemes [13]). Worse, even though a node may have five neighbors they may be positioned parametrically so as to make the full biquadratic fit ill-conditioned.…”

Section: Local Coordinate Fittingmentioning

confidence: 99%

Free-form shape design using triangulated surfaces

Welch

Witkin

1994

Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH '94

222

137

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We present an approach to modeling with truly mutable yet completely controllable free-form surfaces of arbitrary topology. Surfaces may be pinned down at points and along curves, cut up and smoothly welded back together, and faired and reshaped in the large. This style of control is formulated as a constrained shape optimization, with minimization of squared principal curvatures yielding graceful shapes that are free of the parameterization worries accompanying many patch-based approaches. Triangulated point sets are used to approximate these smooth variational surfaces, bridging the gap between patch-based and particle-based representations. Automatic refinement, mesh smoothing, and re-triangulation maintain a good computational mesh as the surface shape evolves, and give sample points and surface features much of the freedom to slide around in the surface that oriented particles enjoy. The resulting surface triangulations are constructed and maintained in real time.

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Scattered Data Interpolation and Applications: A Tutorial and Survey

Cited by 120 publications

References 47 publications

Smooth Surface Extraction from Unstructured Point-based Volume Data Using PDEs

Smooth Surface Extraction from Unstructured Point-based Volume Data Using PDEs

Divisive Parallel Clustering for Multiresolution Analysis

Free-form shape design using triangulated surfaces

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