THB-splines: The truncated basis for hierarchical splines

Giannelli, Carlotta; Jüttler, Bert; Speleers, Hendrik

doi:10.1016/j.cagd.2012.03.025

Cited by 407 publications

(397 citation statements)

References 18 publications

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“…However, to allow for adaptive refinement, several extensions of the B-spline model have recently emerged, e.g. analysis-suitable T-splines, 40,5 hierarchical splines, 45,26,32 or LR-splines 18,30 ; see also Refs. 31 and 28 for a comparison of (1.5)…”

Section: Adaptivity In Isogeometric Analysismentioning

confidence: 99%

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Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

Gantner

Haberlik

Praetorius

2017

Math. Models Methods Appl. Sci.

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We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension d ≥ 2. We employ hierarchical B-splines of arbitrary degree and different order of smoothness. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (respectively, the sum of energy error plus data oscillations) with optimal algebraic rates. Numerical experiments underpin the theoretical findings.

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Section: Adaptivity In Isogeometric Analysismentioning

confidence: 99%

“…41 with the help of so-called truncated hierarchical B-splines (THB-splines) introduced in Ref. 26. In this section, we recall their definition and list some basic properties.…”

Section: Truncated Hierarchical B-splinesmentioning

confidence: 99%

Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

Gantner

Haberlik

Praetorius

2017

Math. Models Methods Appl. Sci.

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“…Conditions that ensure linear independence of the resulting B-splines were introduced by (Kraft, 1996). Truncated Hierarchical B-Splines (Giannelli et al, 2012) provide an alternative formulation for Hierarchical Bsplines with improved numerical properties. 2.…”

Section: Locally Refined Splinesmentioning

confidence: 99%

Locally Refined Splines Representation for Geospatial Big Data

Dokken

Skytt

Barrowclough

2015

Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci.

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When viewed from distance, large parts of the topography of landmasses and the bathymetry of the sea and ocean floor can be regarded as a smooth background with local features. Consequently a digital elevation model combining a compact smooth representation of the background with locally added features has the potential of providing a compact and accurate representation for topography and bathymetry. The recent introduction of Locally Refined B-Splines (LR B-splines) allows the granularity of spline representations to be locally adapted to the complexity of the smooth shape approximated. This allows few degrees of freedom to be used in areas with little variation, while adding extra degrees of freedom in areas in need of more modelling flexibility. In the EU fp7 Integrating Project IQmulus we exploit LR B-splines for approximating large point clouds representing bathymetry of the smooth sea and ocean floor. A drastic reduction is demonstrated in the bulk of the data representation compared to the size of input point clouds. The representation is very well suited for exploiting the power of GPUs for visualization as the spline format is transferred to the GPU and the triangulation needed for the visualization is generated on the GPU according to the viewing parameters. The LR Bsplines are interoperable with other elevation model representations such as LIDAR data, raster representations and triangulated irregular networks as these can be used as input to the LR B-spline approximation algorithms. Output to these formats can be generated from the LR B-spline applications according to the resolution criteria required. The spline models are well suited for change detection as new sensor data can efficiently be compared to the compact LR B-spline representation.

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“…This local refinement is not inherent to standard methods as Non-Uniform Rational Basis Splines (NURBS) or classical parameterizations of tensorproduct surfaces. Existing methods to insert points at specific locations were developped in [1,2,3,4,5]. We propose a new generic tensor-product parameterization for surfaces where the degrees of freedom can be locally increased without altering the shape of the surface.…”

Section: Introductionmentioning

confidence: 99%

Local refinement for 3D deformable parametric surfaces

Badoual

Schmitter

Unser

2016

2016 IEEE International Conference on Image Processing (ICIP)

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Biomedical image segmentation is an active field of research where deformable models have proved to be efficient. The geometric representation of such models determines their ability to approximate the shape of interest as well as the speed of convergence of related optimization algorithms. We present a new tensor-product parameterization of surfaces that offers the possibility of local refinement. The goal is to allocate additional degrees of freedom to the surface only where an increase in local detail is required. We introduce the possibility of locally increasing the number of control points by inserting basis functions at specific locations. Our approach is generic and relies on refinable functions which satisfy the refinement relation. We show that the proposed method improves brain segmentation in 3D MRI images.

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THB-splines: The truncated basis for hierarchical splines

Cited by 407 publications

References 18 publications

Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

Locally Refined Splines Representation for Geospatial Big Data

Local refinement for 3D deformable parametric surfaces

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