Deformation-based nonlinear dimension reduction: Applications to nuclear morphometry

We describe a new approach for elucidating the nonlinear degrees of freedom in a distribution of shapes depicted in digital images. By combining a deformation-based method for measuring distances between two shape configurations together with multidimensional scaling, a method for determining the number of degrees of freedom in a shape distribution is described. In addition, a method for visualizing the most representative modes of variation (underlying shape parameterization) in a nuclei shape distribution is also presented. The novel approach takes into account the nonlinear nature of shape manifolds and is related to the ISOMAP algorithm. We apply the method to the task of analyzing the shape distribution of HeLa cell nuclei and conclude that approximately three parameters are responsible for their shape variation. Excluding differences in size, translation, and orientation, these are: elongation, bending (concavity), and shifts in mass distribution. In addition, results show that, contrary to common intuition, the most likely nuclear shape configuration is not symmetric.