Magnetic resonance imaging (MRI) is often used to obtain qualitative images of the human brain. Different kinds of tissues can be recognised from the different levels of contrast. Multi-echo spin-echo T 2 MR measurements can be used to obtain a quantitative map of the brain where different tissues or components can be recognised based on a certain tissue property, the T 2 relaxation time this case. This quantitative approach is also called parameter mapping. Multi-component parameter mapping makes it possible to determine the concentrations of different tissues inside a certain region, this is mainly used for the determination of the myelin water fraction. Myelin is a substance present in low concentrations in the brain and can be related to certain neurodegenerative autoimmune diseases. Magnetic resonance fingerprinting (MRF) provides a new way to such qualitative measurements. An advantage is that MRF is sensitive to both T1 and T2 relaxation times, making it possible to distinguish more types of tissues. The drawback is, however, that more combinations of the parameters are possible, leading to a more difficult problem with longer computation times. In this project a new method to perform a multi-component analysis in MRF and multi-echo spin-echo T2 measurements is developed, which is faster and less sensitive to noise than previous methods. The algorithm that is developed in this project, is called Sparsity Promoting Iterative Joint Non-negative least squares (SPIJN). It finds a small number of
components throughout the region of interest without assumptions about the number of components or their relaxation times by imposing a joint sparsity constraint. This new method is compared to previously published methods in both numerical simulations and in vivo experiments. The multi-component decomposition for brain data results in meaningful structures on first sight, although further research would be required on the meaning of the matched components. Moreover, the algorithm has been used for the calculation of the myelin water fraction from multi-echo spin-echo T2 data. The obtained maps are comparable to state-of-the-art methods, show improvements in some cases and have significantly shorter computation times.

Wavelets have been very popular in the field of image compression and noise reduction. Another interesting application is adaptive mesh refinement. There are many wavelets with various properties, which will have different effects on different applications. There is no consensus on which wavelet is the best option for adaptive mesh refinement. Most commonly used wavelets for adaptive mesh refinement are Donoho's interpolating wavelet and Sweldens wavelet, the latter a lifted version of Donoho's interpolating wavelet. A detailed comparison of both wavelets is done on different data sets. Moreover, different manners of handling the boundaries are tested. An algorithm to construct the meshes using wavelets is tested and optimised.
Donoho's interpolating wavelet with the lower order boundary stencil implementation resulted to be the most accurate, whilst resulting in very high compression compared to the original mesh. Furthermore, changing the adaptivity of the algorithm, which constructs the meshes, turned out to be valuable for fast changing shapes. Lastly, an improvement on the inverse transform during the adaptive mesh refinement had promising results.