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Abstract. A method for building statistical point distribution models is proposed. The novelty in this paper is the adaption of Markov random field regularization of the correspondence field over the set of shapes. The new approach leads to a generative model that produces highly homogeneous polygonized shapes and improves the capability of reconstruction of the training data. Furthermore, the method leads to an overall reduction in the total variance of the point distribution model. Thus, it finds correspondence between semi-landmarks that are highly correlated in the shape tangent space. The method is demonstrated on a set of human ear canals extracted from 3D-laser scans.

From a set of 31 three-dimensional computed tomography (CT) scans we model the temporal shape and size of the human mandible for analysis, simulation, and prediction purposes. Each anatomical structure is represented using 14851 semi-landmarks, and mapped into Procrustes tangent space. Exploratory subspace analyses are performed leading to linear models of mandible shape evolution in Procrustes space. The traditional variance analysis results in a one-dimensional growth model. However, working in a non-Euclidean metric results in a multimodal model with uncorrelated modes of biological variation related to independent component analysis. The applied non-Euclidean metric is governed by the correlation structure of the estimated noise in the data. The generative models are compared, and evaluated on the basis of a cross validation study. The new non-Euclidean analysis is completely data driven. It not only gives comparable results w.r.t. previous studies of the mean modeling error, but seems to better correlate to growth, and in addition provides the data analyst with alternative hypothesis of plausible shape evolution; hence aiding in the understanding of cranio-facial growth.

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The contribution of this paper is the adaptation of data driven methods for nonEuclidean metric decomposition of tangent space shape coordinates. The basic idea is to extend principal component analysis (PCA) to take into account the noise variance at different landmarks and at different shapes. We show examples where these non-Euclidean metric methods allow for easier interpretation by decomposition into meaningful modes of variation. The extensions to PCA are based on adaptation of maximum autocorrelation factors and the minimum noise fraction transform to shape decomposition. A common basis of the methods applied is the assessment of the annotation noise variance at individual landmarks. These assessments are based on local models or repeated annotations by independent operators. We show that the Molgedey-Schuster independent component analysis is equivalent to the maximum autocorrelation factors. Finally, the different subspace methods are compared using a probabilistic formulation based on their ability to represent the data.