Mathematical Imaging Tools in Cancer Research - From Mitosis Analysis to Sparse Regularisation
This dissertation deals with customised image analysis tools in cancer research. In the field of biomedical sciences, mathematical imaging has become crucial in order to account for advancements in technical equipment and data storage by sound mathematical methods that can process and analyse imaging data in an automated way. This thesis contributes to the development of such mathematically sound imaging models in four ways: (i) automated cell segmentation and tracking. In cancer drug development, time-lapse light microscopy experiments are conducted for performance validation. The aim is to monitor behaviour of cells in cultures that have previously been treated with chemotherapy drugs, since atypical duration and outcome of mitosis, the process of cell division, can be an indicator of successfully working drugs. As an imaging modality we focus on phase contrast microscopy, hence avoiding phototoxicity and influence on cell behaviour. As a drawback, the common halo- and shade-off effect impede image analysis. We present a novel workflow uniting both automated mitotic cell detection with the Hough transform and subsequent cell tracking by a tailor-made level-set method in order to obtain statistics on length of mitosis and cell fates. The proposed image analysis pipeline is deployed in a MATLAB software package called MitosisAnalyser. For the detection of mitotic cells we use the circular Hough transform. This concept is investigated further in the framework of image regularisation in the general context of imaging inverse problems, in which circular objects should be enhanced, (ii) exploiting sparsity of first-order derivatives in combination with the linear circular Hough transform operation. Furthermore, (iii) we present a new unified higher-order derivative-type regularisation functional enforcing sparsity of a vector field related to an image to be reconstructed using curl, divergence and shear operators. The model is able to interpolate between well-known regularisers such as total generalised variation and infimal convolution total variation. Finally, (iv) we demonstrate how we can learn sparsity promoting parametrised regularisers via quotient minimisation, which can be motivated by generalised Eigenproblems. Learning approaches have recently become very popular in the field of inverse problems. However, the majority aims at fitting models to favourable training data, whereas we incorporate knowledge about both fit and misfit data. We present results resembling behaviour of well-established derivative-based sparse regularisers, introduce novel families of non-derivative-based regularisers and extend this framework to classification problems.