Analysis of Rotational Deformations From Directional Data

Abstract

This article discusses a novel framework to analyze rotational deformations of real three-dimensional objects. The rotational deformations such as twisting or bending have been observed as the major variation in some medical applications, where the features of the deformed three-dimensional objects are directional data. We propose modeling and estimation of the global deformations in terms of generalized rotations of directions. The proposed method can be cast as a generalized small circle fitting on the unit sphere. We also discuss the estimation of descriptors for more complex deformations composed of two simple deformations. The proposed method can be used for a number of different three-dimensional object models. Two analyses of three-dimensional object data are presented in detail: one using skeletal representations in medical image analysis and the other from biomechanical gait analysis of the knee joint. Supplementary materials for this article are available online.

Key Words:

• Axis of rotation
• Directional statistics
• Skeletal model
• Small circle
• Three-dimensional object

ACKNOWLEDGMENTS

The work of J. Schulz was funded by Norwegian Research Council through eVita program grant no 176872/V30. His research was performed as part of Tromsø Telemedicine Laboratory, funded by the Norwegian Research Council 2007-2014, grant no 174934. S. Jung was partially supported by NSF grant DMS-1307178. S. Huckemann gratefully acknowledges support from the Niedersachsen Vorab and DFG HU 1575/4-1. The work of M. Pierrynowski was supported by a travel grant from McMaster University. The authors would like to thank Chong Shao and Wenqi Zhang for their help in generating the ellipsoid data, and Jared Vicory for advice on running Pablo.