14:00
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.