Identifiability properties for inverse problems in medical engineering, with observability and optimization issues

We discuss a number of inverse identification problems that arise in medical engineering or in neurosciences for functional or clinical brain analysis purposes, like source recovery or conductivity estimation from boundary data, for electroand magneto-encephalography (EEG/MEG), or in electrical impedance tomograpahy (EIT). Maxwell’s equations under physical assumptions are to the effect that the electrical potential within the head can be modelled as a solution to some partial differential equation (PDE), in spherical or more general 3-dimensional domains [13]. In particular, the quasi-static assumption (time derivatives of the electromagnetic fields are neglected), the PDE is an elliptic Poisson equation for a variable conductivity that only involves the space variable. For the EEG application, on which we mainly focus, available boundary data are furnished by values of the current flux and the electrical potential (measured by electrodes, see figure 1) on the scalp. From such partial and overdetermined boundary measurements of the current flux and the potential (which may be viewed as input and output of the system), the aim is to identify and to reconstruct: non–measured boundary data (EEG cortical mapping step, a Cauchy transmission problem), unknown current sources supported within the brain (EEG and MEG, singularities of the potential), that correspond to the primary cerebral current, or unknown conductivity coefficients (EIT). These questions can be stated as identification or observation issues for infinite dimensional systems, of which the electrical potential should be viewed as the state. We consider the two first ones, that are deconvolution issues (as in