Component analysis of complex-valued data for machine learning and computer vision tasks

Abstract

This thesis studies component analysis techniques when complex-valued data arise. Until recently, the usual way to apply these techniques to complex-valued data has been either by splitting the complex-valued data into their real and imaginary parts and afterwards use the techniques with these real data or by applying directly the techniques in the complex domain with some mild assumptions. We have focused in bridging the gap between the two alternatives and shown how to work natively with complex data. To this end, we introduce the use of the quite recently proposed widely linear model on component analysis and we compare it with the classical approaches. Applications and experiments for these methods have been performed for the case of computer vision problems such as face reconstruction, face recognition, expression recognition and video tracking.

In order to decipher the whole approach in the complex domain, we present an introduction to the theory of complex calculus, where the concepts of widely linear transformations, augmented matrix algebra, Wirtinger and complex-valued matrix derivatives are illustrated showing how all these principles can be used. Also, we conduct an overview of the recent advances in the field of augmented statistics and widely linear modeling.

The theory of complex-valued kernels and its usage on Principal Component Analysis (PCA) is analysed in depth and the widely linear version of PCA is presented as well as applications of the proposed method. Furthermore, we examine shape representation in real and complex domain and we compare alternative representations for computer vision tasks. Finally, we have worked towards the unification of component analysis methods in a least-square framework and we look for similarities and differences between the circular and widely linear model.