Surface Motions in Random Media

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This research is concerned with the analysis of the time evolution of random surfaces. Special emphasis will be placed on the computer algorithms needed to implement simulations of the mathematical models. The understanding of the motion of interfaces and boundary surfaces is of great importance for many industrial processes (image analysis, crystal growth, thin films, crack nucleation, etc.) The motivation for this research came originally from the denoising and enhancement of still images and videos. For this reason, the efforts will concentrate on the two and three dimensional cases. Very little attention has been given so far to stochastic versions of such problems, even though they are believed to offer a more realistic alternative that deterministic approaches. Many problems in science and engineering involve moving boundaries and surfaces. Examples range from melting and solidification to the automatic detection of edges in videos. The mathematical methods that have been developed for these purposes so far usually do not take randomness into account. This research will begin to incorporate such effects, focussing on applications that are related to the denoising and enhancement of images and videos.