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In this paper, we show that the estimation of an irrotational vector field employing the longitudinal ray transform in the discrete domain is tractable, despite the fact that this problem cannot be solved in the continuous domain using... more
In this paper, we show that the estimation of an irrotational vector field employing the longitudinal ray transform in the discrete domain is tractable, despite the fact that this problem cannot be solved in the continuous domain using the same formulation. We derive a set of algebraic equations and solve the ill conditioned inverse problem by directly inverting the produced ray projection matrix. In particular, we prove that the problem can be regularized via discretization and we provide an upper bound for the numerical error of the approximation, ensuring the stability of our formulation. We validate our theoretical results by performing simple simulations reconstructing irrotational fields based solely on boundary measurements without the need of any prior information (e.g. transversal integrals).
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The expected position error in many cases is far from feasible to be estimated experimentally using real satellite measurements which makes the model-based position dilution of precision (PDOP) crucial in positioning and navigation... more
The expected position error in many cases is far from feasible to be estimated experimentally using real satellite measurements which makes the model-based position dilution of precision (PDOP) crucial in positioning and navigation applications. In the following sections we derive the relationship between PDOP and position error and we explain that this relationship holds as long as the model for the observation errors represents the true sources of errors. 1 Observation model and position statistics For a ground receiver at location r = (rx, ry, rz) , the positioning error at time t between r and a reference location r0, denoted by ∆r = (∆rx,∆ry,∆rz) , can be expressed through the linear system [2, 3] b = A∆r + ε, (1) where b ∈ R is a vector with the differences between the measured and modelled pseudo-range values, A = [n1, . . . , nS ] T ∈ R where ns is a unit column vector pointing from the modelled (approximated) receiver position to the satellite and S is the total number of v...
In this thesis, three different approaches are developed for the estimation of focal brain activity using EEG measurements. The proposed approaches have been tested and found feasible using simulated data. First, we develop a robust... more
In this thesis, three different approaches are developed for the estimation of focal brain activity using EEG measurements. The proposed approaches have been tested and found feasible using simulated data. First, we develop a robust solver for the recovery of focal dipole sources. The solver uses a weighted dipole strength penalty term (also called weighted L1,2 norm) as prior information in order to ensure that the sources are sparse and focal, and that both the source orientation and depth bias are reduced. The solver is based on the truncated Newton interior point method combined with a logarithmic barrier method for the approximation of the penalty term. In addition, we use a Bayesian framework to derive the depth weights in the prior that are used to reduce the tendency of the solver to favor superficial sources. In the second approach, vector field tomography (VFT) is used for the estimation of underlying electric fields inside the brain from external EEG measurements. The ele...
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The electroencephalography (EEG) source imaging problem is very sensitive to the electrical modelling of the skull of the patient under examination. Unfortunately, the currently available EEG devices and their embedded software do not... more
The electroencephalography (EEG) source imaging problem is very sensitive to the electrical modelling of the skull of the patient under examination. Unfortunately, the currently available EEG devices and their embedded software do not take this into account; instead, it is common to use a literature-based skull conductivity parameter. In this paper, we propose a statistical method based on the Bayesian approximation error approach to compensate for source imaging errors due to the unknown skull conductivity and, simultaneously, to compute a low-order estimate for the actual skull conductivity value. By using simulated EEG data that corresponds to focal source activity, we demonstrate the potential of the method to reconstruct the underlying focal sources and low-order errors induced by the unknown skull conductivity. Subsequently, the estimated errors are used to approximate the skull conductivity. The results indicate clear improvements in the source localization accuracy and feasi...
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We focus on electro-/magnetoencephalography imaging of the neural activity and, in particular, finding a robust estimate for the primary current distribution via the hierarchical Bayesian model (HBM). Our aim is to develop a reasonably... more
We focus on electro-/magnetoencephalography imaging of the neural activity and, in particular, finding a robust estimate for the primary current distribution via the hierarchical Bayesian model (HBM). Our aim is to develop a reasonably fast maximum a posteriori (MAP) estimation technique which would be applicable for both superficial and deep areas without specific a priori knowledge of the number or location of the activity. To enable source distinguishability for any depth, we introduce a randomized multiresolution scanning (RAMUS) approach in which the MAP estimate of the brain activity is varied during the reconstruction process. RAMUS aims to provide a robust and accurate imaging outcome for the whole brain, while maintaining the computational cost on an appropriate level. The inverse gamma (IG) distribution is applied as the primary hyperprior in order to achieve an optimal performance for the deep part of the brain. In this proof-of-the-concept study, we consider the detectio...
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Satellite-based communications, navigation systems and many scientific instruments rely on observations of trans-ionospheric signals. The quality of these signals can be deteriorated by ionospheric scintillation which can have detrimental... more
Satellite-based communications, navigation systems and many scientific instruments rely on observations of trans-ionospheric signals. The quality of these signals can be deteriorated by ionospheric scintillation which can have detrimental effects on the mentioned applications. Therefore, monitoring of ionospheric scintillation and quantifying its effect on the ground are of significant interest. In this work, we develop a methodology which estimates the scintillation-induced ionospheric uncertainties in the sky and translates their impact to the end users on the ground. First, by using the risk concept from decision theory and by exploiting the intensity and duration of scintillation events (as measured by the $$S_4$$S4 index), we estimate ionospheric risk maps that could readily give an initial impression on the effects of scintillation on the satellite-receiver communication. However, to better understand the influence of scintillation on the positioning accuracy on the ground, we...
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Research Interests: Physics and Tomography
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This paper develops mathematical methods for localizing focal sources at different depths based on the non-invasive electro-/magnetoencephalography measurements. In the context of hierarchical Bayesian modelling, we introduce a... more
This paper develops mathematical methods for localizing focal sources at different depths based on the non-invasive electro-/magnetoencephalography measurements. In the context of hierarchical Bayesian modelling, we introduce a conditionally exponential prior (CEP) which extends the concept of the conditionally Gaussian prior (CGP) and has been proposed to be advantageous in reconstructing far-field activity, in particular, when coupled with randomized multiresolution scanning (RAMUS). An approach to obtain the shape and scale parameter of the gamma hyperprior steering the CEP is derived from the physiological a priori knowledge of the brain activity. The core concept of this study is to show that the firstdegree CEP will yield and improve the focality compared to the second-order case. The results of the present numerical experiments suggest that sources reconstructed via a combination of the first-degree CEP and RAMUS achieve an accuracy comparable to the second-degree case while ...
— In electroencephalography (EEG) source imaging, the inverse source estimates are depth biased in such a way that their maxima are often close to the sensors. This depth bias can be quantified by inspecting the statistics (mean and... more
— In electroencephalography (EEG) source imaging, the inverse source estimates are depth biased in such a way that their maxima are often close to the sensors. This depth bias can be quantified by inspecting the statistics (mean and covariance) of these estimates. In this paper, we find weighting factors within a Bayesian framework for the used ℓ 1 /ℓ 2 sparsity prior that the resulting maximum a posterior (MAP) estimates do not favour any particular source location. Due to the lack of an analytical expression for the MAP estimate when this sparsity prior is used, we solve the weights indirectly. First, we calculate the Gaussian prior variances that lead to depth un-biased maximum a posterior (MAP) estimates. Subsequently, we approximate the corresponding weight factors in the sparsity prior based on the solved Gaussian prior variances. Finally, we reconstruct focal source configurations using the sparsity prior with the proposed weights and two other commonly used choices of weights that can be found in literature.