Mathematical Methods for Medical Imaging

Computational anatomy is an emergingdiscipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model the organ development across time (growth or aging), to establish their variability, and to correlate this variability information with other functional, genetic or structural information. Geometry, and especially differential geometry is a natural foundation of the shape modeling. Dealing with data as extracted from medical images, makes the necessity of handling statistical modeling. Hence, the mathematical foundation of computational anatomy, seeks to unify statistics, and geometry. The aim, that methods may serve the computational anatomy emphasized the need for numerical methods. D’Arcy W. Thomson (1860–1948) used transformations of the underlying space to equalize the form of hand drawn biological objects. In some examples factoring out an affine transformation, in other examples projective transformations, and in a single example a non-linear transformation containing singularities when transforming a Scarus Sp. into a Pomacanthus. David G. Kendall created, used originally in archeology to argue if historic landmarks was more than accidentally aligned, what is now called Kendall’s shape space, by factoring out similarity transformations of a labeled point sets.