Almost all EEG-based brain-computer interfaces (BCIs) need some labeled subject-specific data to calibrate a new subject, as neural responses are different across subjects to even the same stimulus. So, a major challenge in developing high-performance and user-friendly BCIs is to cope with such individual differences so that the calibration can be reduced or even completely eliminated. This paper focuses on the latter. More specifically, we consider an offline application scenario, in which we have unlabeled EEG trials from a new subject, and would like to accurately label them by leveraging auxiliary labeled EEG trials from other subjects in the same task. To accommodate the individual differences, we propose a novel unsupervised approach to align the EEG trials from different subjects in the Euclidean space to make them more consistent. It has three desirable properties: 1) the aligned trial lie in the Euclidean space, which can be used by any Euclidean space signal processing and machine learning approach; 2) it can be computed very efficiently; and, 3) it does not need any labeled trials from the new subject. Experiments on motor imagery and event-related potentials demonstrated the effectiveness and efficiency of our approach.
Diffusion tensor magnetic resonance imaging (DT-MRI) is emerging as an important tool in medical image analysis of the brain. However, relatively little work has been done on producing statistics of diffusion tensors. A main difficulty is that the space of diffusion tensors, i.e., the space of symmetric, positive-definite matrices, does not form a vector space. Therefore, standard linear statistical techniques do not apply. We show that the space of diffusion tensors is a type of curved manifold known as a Riemannian symmetric space. We then develop methods for producing statistics, namely averages and modes of variation, in this space. In our previous work we introduced principal geodesic analysis, a generalization of principal component analysis, to compute the modes of variation of data in Lie groups. In this work we expand the method of principal geodesic analysis to symmetric spaces and apply it to the computation of the variability of diffusion tensor data. We expect that these methods will be useful in the registration of diffusion tensor images, the production of statistical atlases from diffusion tensor data, and the quantification of the anatomical variability caused by disease.