Title: Biomedical Image Computing: the development and application of mathematical and computational models Submitted to: The University of Manchesterby James Grahamfor the degree of Doctor of ScienceJune 2016Biomedical images contain a great deal of information that is useful and a great deal that is not. Computational analysis and interpretation of biomedical images involves extraction of some or all of the useful information. The useless information can take the form of unwanted clutter or noise that can obscure the useful information or inhibit the interpretation. Various mathematical and computational processes may be applied to reduce the effects of noise and distracting content. The most successful approaches involve the use of mathematical or computational models that express the properties of the required information. Interpretation of images involves finding objects or structures in the image that match the properties of the model. This dissertation describes the development and application of different models required for the interpretation of a variety of different image types arising from clinical medicine or biomedical research. These include:* neural network models, * Point Distribution Models, and the associated Active Shape Models, which have become part of the research toolkit of many academic and commercial organisations, * models of the appearance of nerve fibres in noisy confocal microscope images,* models of pose changes in carpal bones during wrist motion, A number of different application problem are described, in which variants of these methods have been developed and used: * cytogenetics, * proteomics, * assessing bone quality, * segmentation of magnetic resonance images, * measuring nerve fibres * inferring 3D motion from 2D cinefluoroscopy sequences.The methods and applications represented here encompass the progression of biomedical image analysis from early developments, where computational power became adequate to the challenges posed by biomedical image data, to recent, highly computationally-intensive methods.