Particle filtering for EEG source localization and constrained state spaces

...........................................................................................................................v List of Figures ..................................................................................................................viii List of Tables ...................................................................................................................x Chapter 1: Introduction ....................................................................................................1 1.1 Motivation, Background, and Problem Statement ...............................................1 1.1.1 The EEG Source Localization Problem. .....................................................4 1.1.2 Particle Filtering in Constrained State-Spaces. ...........................................5 1.2 Research Contributions ........................................................................................6 1.3 Organization .........................................................................................................8 Chapter 2: Literature Review ...........................................................................................10 2.1 Problem Statement ...............................................................................................10 2.2 Optimal State Estimation in Linear Models .........................................................13 2.3 Approximate Solutions in Nonlinear Models ......................................................15 2.3.1 The Extended Kalman Filter. ......................................................................15 2.3.2 The Unscented Kalman Filter. ...................................................................17 2.4 The Particle Filter Framework .............................................................................18 2.4.1 Monte Carlo Sampling. ...............................................................................18 2.4.2 Importance Sampling. .................................................................................19 2.4.3 Sequential Importance Sampling. ..............................................................20 2.4.4 The Particle Filter. .....................................................................................24 Chapter 3: The Constrained Particle Filter ......................................................................29 3.