# THB-splines: The truncated basis for hierarchical splines

@article{Giannelli2012THBsplinesTT, title={THB-splines: The truncated basis for hierarchical splines}, author={Carlotta Giannelli and Bert J{\"u}ttler and Hendrik Speleers}, journal={Comput. Aided Geom. Des.}, year={2012}, volume={29}, pages={485-498} }

The construction of classical hierarchical B-splines can be suitably modified in order to define locally supported basis functions that form a partition of unity. We will show that this property can be obtained by reducing the support of basis functions defined on coarse grids, according to finer levels in the hierarchy of splines. This truncation not only decreases the overlapping of supports related to basis functions arising from different hierarchical levels, but it also improves the…

## Figures, Tables, and Topics from this paper

## 353 Citations

Hierarchical B-splines on regular triangular partitions

- Mathematics, Computer ScienceGraph. Model.
- 2014

This paper introduces the concept of hierarchial bivariate splines over regular triangular partitions and construct basis functions of such spline space that satisfy some nice properties and provides some examples of hierarchicalsplines over triangular partitions in surface fitting and in solving numerical PDEs.

On the linear independence of (truncated) hierarchical subdivision splines

- Computer Science
- 2015

A catalog of safe subdomains is provided that facilitates the design of domain hierarchies with linearly independent (truncated) hierarchical generating systems.

C2interpolation T-splines

- Mathematics
- 2020

Abstract In this work, we construct a class of C 2 interpolation T-spline (called IT-spline) basis functions over regular T-meshes. A new rule for determining the local knot vectors of the IT-spline…

Modified Bases of PHT-Splines

- Mathematics
- 2017

Recently, it was found that during the process of certain refinement of hierarchical T-meshes, some basis functions of PHT-splines decay severely, which is not expected in solving numerical PDEs and…

Coefficient-Based Spline Data Reduction by Hierarchical Spaces

- Mathematics, Computer ScienceMMCS
- 2016

A coefficient–based least squares spline operator is introduced that does not require any pointwise evaluation to approximate (in a lower dimension spline space) a given bivariate B–spline function.

Foundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement

- Mathematics
- 2018

This chapter presents an overview of polynomial spline theory, with special emphasis on the B-spline representation, spline approximation properties, and hierarchical spline refinement. We start with…

Modified T-splines

- Computer Science, MathematicsComput. Aided Geom. Des.
- 2013

The basic idea is to construct a set of basis functions for a given T-mesh that have the following nice properties: non-negativity, linear independence, partition of unity and compact support.

Adaptively refined multilevel spline spaces from generating systems

- Computer Science, MathematicsComput. Aided Geom. Des.
- 2014

A hierarchy of hierarchical B-splines is introduced, which enables us to perform local refinement in the presence of features, and the adaptive multilevel framework is extended to spaces spanned by Zwart-Powell elements, which are special box splines defined on the criss-cross grid.

Modified PHT-splines

- Mathematics, Computer ScienceComput. Aided Geom. Des.
- 2019

Using this approach, polynomial splines defined over the modified hierarchical T-meshes (modified PHT-splines) are introduced to increase the flexibility of PHT, and when applied to surface fitting and isogeometric analysis problems with anisotropic features.

Patchwork B-spline refinement

- Computer Science, MathematicsComput. Aided Des.
- 2017

This work derives the mathematical foundations of multivariate PB-splines, in particular focusing on the construction of a basis that forms a convex partition of unity, and generalizes the concept of truncated hierarchical B-spline to the novel framework.

## References

SHOWING 1-10 OF 23 REFERENCES

Polynomial splines over hierarchical T-meshes

- Mathematics, Computer ScienceGraph. Model.
- 2008

A new type of splines-polynomial splines over hierarchical T-meshes (called PHT-splines) to model geometric objects which have the same important properties as B-spline do, such as nonnegativity, local support and partition of unity.

Quasi-hierarchical Powell-Sabin B-splines

- Mathematics, Computer ScienceComput. Aided Geom. Des.
- 2009

A normalized quasi-hierarchical basis for Hierarchical Powell-Sabin splines is proposed and it is shown that the basis is strongly stable on uniform hierarchical triangulations.

Surface fitting with hierarchical splines

- Mathematics, Computer ScienceTOGS
- 1995

This work considers the fitting of tensor product parametric spline surfaces to gridded data and uses a hierarchical representation for the surface to improve efficiency and adaptively decompose the fitting process into subproblems involving only a portion of the data.

Adaptive isogeometric analysis by local h-refinement with T-splines

- Mathematics
- 2010

Isogeometric analysis based on non-uniform rational B-splines (NURBS) as basis functions preserves the exact geometry but suffers from the drawback of a rectangular grid of control points in the…

Surface modeling with polynomial splines over hierarchical T-meshes

- Computer ScienceThe Visual Computer
- 2007

With the method, spline surfaces can be constructed efficiently and adaptively to fit genus-zero meshes after their spherical parameterization is obtained, where only small sized linear systems of equations are involved.

Multiresolution Surface Reconstruction for Hierarchical B-splines

- Computer ScienceGraphics Interface
- 1998

The resulting hierarchical surface accurately and economically reproduces the original mesh, is free from excessive undulations in the intermediate levels and produces a multiresolution representation suitable for animation and interactive modelling.

Local refinement of analysis-suitable T-splines

- Mathematics
- 2012

We develop a local refinement algorithm for analysis-suitable T-splines which does not produce excessive propagation of control points. We then demonstrate its use as an adaptive framework for…

Interpolating and approximating scattered 3D-data with hierarchical tensor product B-splines

- Mathematics
- 2010

In this note we describe surface reconstruction algorithms based on optimization techniques. The input are scattered 3D-data with specified topology. The surfaces constructed are tensor product…

Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids

- Mathematics
- 2011

Isogeometric analysis has become a powerful alternative to standard finite elements due to its flexibility in handling complex geometries. One of the major drawbacks of NURBS-based isogeometric…

Bases and dimensions of bivariate hierarchical tensor-product splines

- Computer Science, MathematicsJ. Comput. Appl. Math.
- 2013

It is proved that the dimension of bivariate Tensor-product spline spaces of bi-degree with maximum order of smoothness on a multi-cell domain is equal to the number of tensor- product B-spline basis functions, defined by only single knots in both directions.