Quaternion PCA and sparse PCA for shape variability

Abstract

Principal component analysis (PCA) is commonly used in the fields of computer graphics and geometry processing for constructing subspaces that represent the variability present in a dataset. Examples of such datasets are configurations of a non-rigid object, poses of a deformable character or snapshots from a simulation. By applying PCA to the dataset, one can generate a lower-dimensional space in which the data samples can be approximated well. Similar to PCA, one can also apply a sparse PCA technique to the geometric data. These methods aim to find components that do not only describe the deformations present in the data well, but are also sparse and localized.

One of the problems related to PCA and sparse PCA is that they do not take the correlation between the x, y and z coordinates of each vertex into account when constructing the subspace. Furthermore, the methods are not invariant to rigid motion (rotations and translations) of the data samples, which means that rigid registration has to often be applied to the samples as a preprocessing step. Ideally, we would like to construct a subspace that is invariant to these rigid motions.

This project investigates whether quaternions can be used to solve these problems. By describing each vertex as a pure quaternion, we show how quaternion PCA can be applied to geometric data and introduce a quaternion method of snapshots for improved computational efficiency. Additionally, we derive multiple quaternion sparse PCA techniques, which are inspired by the Sparse Localized Deformation Components (SPLOCS) method. Experimental results show that the quaternion PCA and sparse PCA methods are able to describe a richer space of deformations using fewer components. Furthermore, we show that quaternion PCA leads to improved rigid motion invariance.