Background

The importance of partial differential equations (PDEs) for modelling in the natural and applied sciences can hardly be overemphasised. Because exact solutions are rarely possible, it is important to have accurate and efficient numerical methods for computing approximate solutions of PDEs. The closest point method [1, 2] is a recently developed simple numerical technique for solving PDEs on general curves and surfaces. For example, it can be used to solve a pattern-formation PDE on the surface of an animal.

Techniques and Challenges

The main aims of this project are to further develop and analyse the closest point method, including a finite element approach, and to investigate and implement possible applications. We are exploring new approaches to the formulation of the method in order to apply it to more general problems. We are also investigating applications of the closest point method in image processing, applying techniques developed for standard two-dimensional images to data defined on surfaces.

Results

In particular, we are working on inpainting (restoring missing or damaged parts of images) and have implemented two different inpainting algorithms on surface images using the closest point method. This is important, for example, in a three-dimensional surface scanning process where the scanner has missed some data, and there is a need to fill in the missing parts from the surrounding known regions.

The Future

We aim to analyse the consistency and accuracy of the method, and implement more general numerical schemes. Future work includes developing an industrial collaboration.

References

[1] Macdonald C.B., Ruuth S.J.: The implicit Closest Point Method for the numerical solution of partial differential equations on surfaces, SIAM J. Sci. Comput., 31(6):4330-4350, 2009

[2] Ruuth S.J., Merriman B.: A Simple Embedding Method for Solving Partial Differential Equations on Surfaces, J. Comput. Phys., 227(3):1943-1961, 2008