Construction of 3D shape models of femoral articular cartilage using harmonic maps

A previous publication has described a method of pairwise 3D surface correspondence for the automated generation of landmarks on a set of examples from a class of shape [3]. However, that method did not guarantee a diffeomorphic correspondence between examples. This affected the model compactness (the ability of the model to capture shape variation in a small number of parameters) and model specificity (the fact that the model will describe shapes only within the class used for training). In this paper we describe a method of generating the pairwise correspondences using piecewise-linear harmonic maps of the surfaces which is constrained to be diffeomorphic. In particular, we are interested in producing shape models of articular cartilage. In general these models will be close to being planar discs which makes the use of harmonic mapping particularly suitable for our application. An example statistical model built using this new correspondence method is shown for the human femoral articular cartilage; a complex biological shape which demonstrates considerable variation between individuals.