Identifiability Properties for Inverse Problems in EEG Data Processing Medical Engineering with Observability and Optimi
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Identifiability Properties for Inverse Problems in EEG Data Processing Medical Engineering with Observability and Optimization Issues Juliette Leblond
Received: 26 September 2013 / Accepted: 28 April 2014 © Springer Science+Business Media Dordrecht 2014
Abstract We consider inverse problems of source identification in electroencephalography, modelled by elliptic partial differential equations. Being given boundary data that consist in values of the current flux and of the electric potential on the scalp, the aim is to reconstruct unknown current sources supported within the brain. For spherical layered models of the head, and after a preliminary data transmission step, such inverse source problems are tackled using best rational approximation techniques on planar sections. Both theoretical and constructive aspects are described, while numerical illustrations are provided. Keywords Inverse boundary value problems · Elliptic partial differential equation · Medical imaging · EEG (electroencephalography) · Observability · Optimization Mathematics Subject Classification 30E10 · 31A25 · 31B20 · 35J05
1 Introduction We discuss some inverse identification problems that arise in medical engineering or in neurosciences for functional and clinical brain analysis purposes. We focus on source recovery issues from boundary data in electroencephalography (EEG). Maxwell’s equations are to the effect that the electric potential within the head can be modelled as a solution to some partial differential equation (PDE), in spherical or more general 3-dimensional domains [14]. In particular, with the quasi-static assumption (time derivatives of the electromagnetic fields are neglected), the EEG problem is modelled by an elliptic Poisson–Laplace PDE that only involves the space variable. Boundary data are furnished by a number of pointwise values of the electric potential on the scalp (measured by electrodes on a part of the scalp, see Fig. 1), together with the (vanishing) current flux. From such partial and overdetermined boundary measurements of the current flux and the potential, the aim is to identify and to reconstruct:
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J. Leblond ( ) Team APICS, INRIA Sophia Antipolis, BP 93, 06902 Sophia Antipolis Cedex, France e-mail: [email protected]
J. Leblond
– non-measured boundary data (a Cauchy transmission problem, cortical mapping step), – unknown current sources supported within the brain (singularities of the potential), that correspond to the primary cerebral current. These questions can be rephrased as identification or observation issues for infinite dimensional systems, where the given boundary measurements (flux and potential) coincide with the input and output of the system, of which the electric potential and the current flux inside the head should be viewed as the state. We consider below these inverse potential problems [13]. Related considerations in magnetoencephalography (MEG) will be briefly discussed in conclusion, with others from electric impedance tomography (EIT). Observe further that similar decon
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