Markov chain Monte Carlo for Bayesian inference

The chord length transform (CLT) is a useful tool to analyze fibre structures. Assuming e.g. arandom process of straight fibres then a realization of such a process can be observed in a binaryimage. The CLT maps to each point in the foreground of a binary image and to each direction thelength of the related chord, where the chord is the connecting part of a line in the direction,running through the point. The maximal chord length in each point and the related direction can bedetermined and lead to an estimation of the common volume-weighted distribution of fibre lengthand fibre direction. The marginal distribution of the fibre direction can even be estimated when thefibres are curved and overlap each other. This works in twoand also in three-dimensional case.Typically the CLT is applied to estimate these distributions, and also to segment crossing fibres[Sandau & Ohser, 2007]The application shown here uses the direction to study in vitro cell motility. The considered cellsare tumour cells (glioblastom cells of type U-87 MG) which were cultivated and observed in apetri dish. A series of images of the scene have been taken in constant time intervals by a lightmicroscope equipped with a digital camera [Müller et al.]. The set of images can be consideredafterwards as a movie and the cell motility can be estimated by tracking the cell nuclei image byimage and calculating in this way the mean velocity.To automate this measurement an automatic detection of each cell nucleus is necessary. Becausethe shape of the cell changes and because the optical apperance of the cell nuclei changes duringcell proliferation, it is only possible to automatically find a connecting region overlapping the cellnucleus. If several cells are close together, then one region may contain several cell nuclei.Considering now these areas as the foreground in a binary image, the set of images can beconsidered as a three-dimensional stack showing tubes following the cell movement. The velocityof the cells can be derived from the local direction of the tubes and in this way the motility of thecells can be measured without the precise knowledge of each nucleus centre and the exact trackingof the migration process. ReferencesSandau, K., Ohser, J.:The Chord Length Transform and the Segmentation of Crossing Fibres.Journal of Microscopy 226, 2007: 43-53.Müller, A., Pollet, D., Scholz, M., Taucher-Scholz, G., Topsch, J.: Investigation of substrateandradiation-dependent cell motility using videomicroscopy, GSI Scientific Report 2005, GSI Report2006-1: 381