Geometric Modelling

The main research in the group will continue to be in (i) splines and mesh-based modelling, and (ii) applied algebraic geometry, with activity also in optimization and linear algebra. However, we do want to adjust our research plan slightly by focusing on (1) a particular application area and (2) development of algorithms suited for implementation on the GPU (Graphics Processing Unit) and other accelerator units.

We will continue to pursue broad research in the area of splines and mesh-based modelling. The main goal is to create and improve methods, and extend mathematical theory for curve and surface representations. We will seek to improve modelling, interpretation, and visualization of structured and unstructured 3D data sets, and time-varying 4D data sets, such as in medical imaging and large simulation grids.

Topics of particular interest will be continuation of work on transfinite interpolation and barycentric coordinates; further research on the close link between a spline and its control polygon, and more generally multiresolution representations; investigation of new surface formats; as well as a general interest in the development of discrete (numerical) differential geometry.

During the first three years of the CMA, the main activity concerning algebraic geometry and geometric modelling centred around the EU project GAIA II, which ended in 2005. Now SINTEF and CMA, together with eight European academic and industrial partners, have succeeded in obtaining the EU FP7 ITN project ShApes, Geometry and Algebra (SAGA), which will start in September 2008 and last for four years, thus permitting the continuation of work started in the GAIA project as well as new initiatives.

The SAGA project will focus on four research areas: 1) Change of representations, 2) Geometric computing - algebraic tools, 3) Algebraic geometry for CAD applications, 4) Practical industrial problems. Two examples of topics within these areas that will be studied by CMA geometers are the following.

Algebraic spline geometry: Approximation of larger regions of a CAD-model using multivariate algebraic spline surfaces, will keep the polynomial degree lower, and allow for flexible approximations. The aim is to develop a new theory, “algebraic spline geometry,” by replacing polynomial functions with algebraic spline functions. This could lead to the development and understanding of certain aspects of classical real (semi-)algebraic geometry so that it can extend to the theory of multivariate algebraic splines. CAD applications: For applications of algebraic geometry in CAD, the theory of classical projective algebraic geometry of curves and surfaces needs to be extended and developed in the real, affine, and bounded cases. Some aspects are: enumerative geometry of real curves and surfaces; singularity theory: existence and description of real surface singularities; the theory of polar and dual varieties, related to Sturm-Habicht methods for determining the topology of real curves and surfaces, and applied to finding efficient ways of computing points on the components of real algebraic varieties; the theory of moduli spaces of varieties and of parameterized varieties and the study of the semi algebraicstratification of these moduli spaces; the design of catalogues of surfaces (parametric or not) that can be used in CAD.

In optimization and linear algebra the focus will continue to be on combinatorial and geometrical problems in matrix theory, as well as optimization problems motivated by applications in fields like networks, transportation, and image analysis. These problems are studied both with a theoretical focus and a constructive/algorithmic focus where the goal is to develop efficient numerical/combinatorial algorithms for specific problem classes.

An application area of particular interest will be Mathematical methods in image processing, with special emphasis on 3D medical image processing. This is an active field driven not least by developments in medical imaging (ultrasound, x-ray, CT, MR, PET). There are obvious links to geometric modelling, but also to partial differential equations, optimization/linear algebra and other fields. This area should therefore be a good basis for cross-disciplinary work. A particularly interesting topic is the so-called image fusion problem. A trend in modern surgery is for the surgeon to be guided not by direct view of the operation area, but by an indirect view from a video camera. By combining the video stream with data from other imaging techniques, the surgeon can be given a much better view, where information that is hidden from the video camera (e.g., a tumour) can become visible. This image fusion problem is extremely challenging both mathematically and computationally as it should ultimately be solved in (close to) real time. The research will be based on cooperation with the Interventional Centre at the National Hospital in Oslo, in particular with the technical director Eigil Samset who is an adjunct professor at the Department of Informatics and CMA. The expertise of Xue Cheng Tai (PDE-group) on PDE-based methods in imag processing, and Geir Dahl in optimization, will be important in this context. It should also be pointed out that 3D image processing is closely related to processing of scanned 3D data, which is a central topic in geometric modelling

There is already activity geared towards exploiting the GPU for computational purposes in cooperation with SINTEF (CMA partner). Staff members from both the Geometry and PDE groups at CMA are involved and we view this as a good framework for cross-disciplinary work. The GPU is just one of several low-cost but high-performance accelerator units that have been developed for computer gaming and other entertainment purposes; such accelerators may also beutilized for scientific computing. At the same time the performance of traditional CPUs has reached a level where further gains are difficult without excessive demands on energy. Because of the momentum behind the entertainment industry, we anticipate that the development of high-performance accelerators will continue and lead to new types of supercomputers with an unprecedented price/performance ratio. We therefore expect an increasing focus on parallel algorithms for these kinds of hardware in all the major activities in geometry.