Dynamic Shape Analysis Through the Offset-Normal Distribution

Abstract

Statistical analysis of dynamic shapes is a problem with significant challenges due to the difficulty in providing a description of the shape changes over time, across subjects and over groups of subjects.

Recent attempts to study the shape change in time are based on the Procrustes tangent coordinates or spherical splines in Kendall shape spaces (Kent et al. (2001) Functional models of growth for landmark data. In: Proceedings in functional and spatial data analysis. University Press, Leeds, pp 109–115; Kume et al. Biometrika 94:513–528, 2007; Fishbaugh et al. (2012) Analysis of longitudinal shape variability via subject specific growth modeling. In: Ayache N, Delingette H, Golland P, Mori K (eds) Medical image computing and computer-assisted intervention – MICCAI 2012. Lecture notes in computer science, vol 7510. Springer, Berlin, Heidelberg, pp 731–738; Hinkle et al. (2012) International anthropometric study of facial morphology in various ethnic groups/races. In: Computer Vision - ECCV. Lecture Notes in Computer Science, vol 7574. pp 1–14; Fontanella et al. (2013) A functional spatio-temporal model for geometric shape analysis. In: Torelli N, Pesarin F, Bar-Hen A (eds) Advances in theoretical and applied statistics. Springer, Berlin, pp 75–86).

This chapter deals with the statistical analysis of a temporal sequence of landmark data using the exact distribution theory for the shape of planar correlated Gaussian configurations. Specifically, we extend the theory introduced in the second chapter to a dynamic framework and discuss the use of the offset-normal distribution for the description of time-varying shapes.

Modeling the temporal correlation structure of the dynamic process is a complex task, in general. For two time points, Mardia and Walder (Biometrika 81:185–196, 1994) have shown that the density function of the offset-normal distribution has a rather complicated form and have discussed the difficulty of extending their results to t > 2. In the final part of the chapter we show that, in principle, it is possible to calculate the closed form expression of the offset-normal distribution for a general t, though its calculation can be computationally expensive.

This chapter is organized as follows. In Sect. 3.1 we describe the offset-normal shape distribution in a dynamic context. In Sect. 3.2 we introduce the EM algorithm for general spatio-temporal covariance matrices while Sect. 3.3 describes the necessary adjustments of the general update rules under separability assumptions of the spatio-temporal covariance structure. A discussion of the computational difficulties concerning the performance of the algorithm is also provided. Following Kume and Welling (J Comput Graph Stat 19:702–723, 2010), in Sect. 3.4 we discuss the case in which the temporal dynamics of the shape variables are only modeled through a polynomial regression which captures the large-scale temporal variability of the process. The fit of this regression model to happiness and surprise data are shown in Sect. 3.4.1. For the same expressions, we also consider the problem of matching symmetry and provide some comments in Sect. 3.5. Finally, Sect. 3.6 concludes the chapter by discussing the use of mixture models for classification purposes in a dynamic setting.