Probabilistic Models for Shapes as Continuous Curves

We develop new shape models by defining a standard shape from which we can explain shape deformation and variability. Currently, planar shapes are modelled using a function space, which is applied to data extracted from images. We regard a shape as a continuous curve and identified on the Wiener measure space whereas previous methods have primarily used sparse sets of landmarks expressed in a Euclidean space. The average of a sample set of shapes is defined using measurable functions which treat the Wiener measure as varying Gaussians. Various types of invariance of our formulation of an average are examined in regard to practical applications of it. The average is examined with relation to a Fréchet mean in order to establish its validity. In contrast to a Fréchet mean, however, the average always exists and is unique in the Wiener space. We show that the average lies within the range of deformations present in the sample set. In addition, a measurement, which we call a quasi-score, is defined in order to evaluate “averages” computed by different shape methods, and to measure the overall deformation in a sample set of shapes. We show that the average defined within our model has the least spread compared with methods based on eigenstructure. We also derive a model to compactly express shape variation which comprises the average generated from our model. Some examples of average shape and deformation are presented using well-known datasets and we compare our model to previous work.