Powell-Sabin splines for geometric design and PDE

Researchers

• P. Dierckx
• H. Speleers

and previously also

• A. Bultheel
• M. Jansen
• E. Vanraes
• J. Maes

PS spline wavelets for geometric design

Related research

• Wavelets and image de-noising (discontinued)
• Computer aided geometric design

Description

After signal processing, image and video applications, wavelet theory now explores a next research domain of so called ``digital geometry processing''. This involves the generation of surfaces, for example in computer graphics applications. The multiscale structure which is known for classical wavelets is here a very useful property. It is obvious that on surfaces one has to deal with irregular grids. This research builds on work by Ingrid Daubechies (Princeton Univ.) and Wim Sweldens (Luncent Techn., Bell Labs).

Also from a statistical point of view there is much interest for this evolution because the use of wavelets for statistical applications has been hindered until now by the constraint of regular grids. The methods for surface generation, which necessarily involve irregular point sets, could help in finding new algorithms for nonequispaced data analysis.

Subdivision is in the first place a principle to generate a smooth curve (one dimension) or a surface (two dimensions) from a set of function values in a discrete point set. It is the basis for applications in computer graphics and CAGD. If a surface is given in a number of points, it allows for example to refine the surface and subsequently apply some local modifications. Well known examples are the algorithms of de Casteljau and de Boor for splines and the algorithms of Delauries-Dubuc for interpolating subdivision. A more recent topic is the application of wavelets in geometric design. To model 3D objects, wavelets have special properties that allow local modifications without changing the overall representation of the object. Numerical aspects of this kind of modeling on highly iregular grids are investigated.

In geometric modeling and approximation applications, splines are well established. However, the link between subdivision and multiresolution is relatively new. The space spanned by a spline function can be analysed at different scales. The corresponding wavelets (spline wavelets) inherit the interesting properties of the father spline wavelet and add to this the multiresolution analysis. The current results however are mainly involved with regular grids in one dimension or tensor product versions in two dimensions (think of the well known Cohen-Daubechies-Feauveau wavelets).

PS spline for differential equations

This research has been discontinued.

The existing expertise about (Powell-Sabin) splines is applied to design finite element methods for the preconditioning and solution of partial differential equations.

Two Bramble-Pasciak-Xu-type preconditioners for second resp. fourth order elliptic problems on the surface of the two-sphere are developped. To discretize the second order problem, C⁰ linear elements on the sphere are used, and for the fourth order problem C¹ finite elements of Powell-Sabin type on the sphere are used. The main idea why these BPX preconditioners work depends on this particular choice of basis.

Powell-Sabin splines are also defined on an arbitrary domain with boundary conditions imposed. This is used to solve PDEs by the Galerkin and collocation method. Also important in this context is a local subdivision method for the hierarchical Powell-Sabin spline basis to allow adaptivity in the solution of the PDEs.