Mathematical methods in medical image processing

Medical imaging problems, such as magnetic resonance imaging, can typically be modeled as inverse problems. A novel methodological approach which was already proven to be highly effective and widely applicable is based on the assumption that most real-life images are intrinsically of low-dimensional nature. This sparsity property can be revealed by representation systems from the area of applied harmonic analysis such as wavelets or shearlets. The inverse problem itself is then solved by sparse regularization, which in certain situations is referred to as compressed sensing. This chapter shall serve as an introduction to and a survey of mathematical methods for medical imaging problems with a specific focus on sparsity-based methods. The effectiveness of the presented methods is demonstrated with a small case study from sparse parallel magnetic resonance imaging.