12 June 2014
Professor Joachim Weickert
Many successful methods in image processing and computer vision involve parabolic and elliptic partial differential equations (PDEs). Thus, there is a growing demand for simple and highly efficient numerical algorithms that work for a broad class of problems. Moreover, these methods should also be well-suited for low-cost parallel hardware such as GPUs. In this talk we show that two of the simplest methods for the numerical analysis of PDEs can lead to remarkably efficient algorithms when they are only slightly modified: To this end, we consider cyclic variants of the explicit finite difference scheme for approximating parabolic problems, and of the Jacobi overrelaxation method for solving systems of linear equations. Although cyclic algorithms have been around in the numerical analysis community for a long time, they have never been very popular for a number of reasons. We argue that most of these reasons have become obsolete and that cyclic methods ideally satisfy the needs of modern image processing applications. Interestingly this transfer of knowledge is not a one-way road from numerical analysis to image analysis: By considering a factorisation of general smoothing filters, we introduce novel, signal processing based ways of deriving cycle parameters. They lead to hitherto unexplored methods with alternative parameter cycles. These methods offer better smoothing properties than classical numerical concepts such as Super Time Stepping and the cyclic Richardson algorithm. We present a number of prototypical applications that demonstrate the wide applicability of our cyclic algorithms. They include isotropic and anisotropic nonlinear diffusion processes, higher dimensional variational problems, and higher order PDEs.
- Computational Mathematics and Applications Seminar