Sampsa Pursiainen

Conditionally Gaussian Hypermodels for Cerebral Source Localization

Bayesian modeling and analysis of the magnetoencephalography and electroencephalography modalities provide a flexible framework for introducing prior information complementary to the measured data. This prior information is often qualitative in nature, making the translation of the available information into a computational model a challenging task. We propose a generalized gamma family of hyperpriors which allows the impressed currents to be focal and we advocate a fast and efficient iterative algorithm, the iterative alternating sequential algorithm for computing maximum a posteriori (MAP) estimates. Furthermore, we show that for particular choices of the scalar parameters specifying the hyperprior, the algorithm effectively approximates popular regularization strategies such as the minimum current estimate and the minimum support estimate. The connection between priorconditioning and adaptive regularization methods is also pointed out. The posterior densities are explored by means of a Markov chain Monte Carlo strategy suitable for this family of hypermodels. The computed experiments suggest that the known preference of regularization methods for superficial sources over deep sources is a property of the MAP estimators only, and that estimation of the posterior mean in the hierarchical model is better adapted for localizing deep sources.

Complete electrode model in EEG: relationship and differences to the point electrode model

In electroencephalography (EEG) source analysis, a primary current density generated by the neural activity of the brain is reconstructed from external electrode voltage measurements. This paper focuses on accurate and effective simulations of EEG through the complete electrode model (CEM). The CEM allows for the incorporation of the electrode size, shape and effective contact impedance into the forward simulation. Both neural currents in the brain and shunting currents between the electrodes and the skin can affect the measured voltages in the CEM. The goal of this study was to investigate the CEM by comparing it with the point electrode model (PEM), which is the current standard electrode model for EEG. We used a three-dimensional, realistic and high-resolution finite element head model as the reference computational domain in the comparison. The PEM could be formulated as a limit of the CEM, in which the effective impedance of each electrode goes to infinity and the size tends to zero. Numerical results concerning the forward and inverse errors and electrode voltage strengths with different impedances and electrode sizes are presented. Based on the results obtained, limits for extremely high and low impedance values of the shunting currents are suggested.

Two-stage reconstruction of a circular anomaly in electrical impedance tomography

In the electrical impedance tomography inverse problem, an unknown conductivity distribution in a given object is to be reconstructed from a set of noisy voltage measurements made on the boundary. This paper focuses on the development of effective reconstruction techniques for detection of a circular anomaly from an otherwise constant background. The goal is to investigate applicability of a two-stage reconstruction process in which a region of interest (ROI) containing the anomaly (e.g. a tumour) is determined in the first stage, and the actual reconstruction is found in the second stage by exploring the ROI. Bayesian inversion methods are applied. The conductivity distribution is modelled as a random variable that follows a posterior probability density proportional to the product of a prior density and a likelihood function. The investigated two-stage reconstruction strategy is, however, not fully Bayesian. In the first stage, the ROI is determined using a quasi-Newton optimization algorithm and a smoothness prior, and in the second stage, the reconstruction is found using Markov chain Monte Carlo sampling and an anomaly prior. Performances of white noise and enhanced noise models as well as performances of standard and linearized finite element forward simulations are compared.

A High-order Finite Element Method for Electical Impedance Tomography

Electrical impedance tomography (EIT) is a non-invasive imaging technique where a conductivity distribution in a domain is reconstructed from boundary voltage measurements. The voltage data are generated by injecting currents into the domain. This is an ill-conditioned non-linear inverse problem. Small measurement or forward modeling errors can lead to unbounded fluctuations in the reconstructions. A forward model describes the dependence of the noiseless voltage data on the conductivity distribution. The present work focuses on applying the high-order finite element method (p-FEM) for forward modeling. In the traditional version of the finite element method (h-FEM), the polynomial degree of the element shape functions is relatively low and the discretization error is reduced by increasing the number of elements. In the p-version, in contrast, the polynomial degree is increased and the mesh size is kept constant. In many applications of the finite element method the performance of the p-version is better than that of the h-version. In this work, it is proposed that the p-version provides more efficient tool for EIT forward modeling. Numerical results are presented.

Comparison Study for Whitney (Raviart–Thomas)-Type Source Models in Finite-Element-Method-Based EEG Forward Modeling

This study concentrates on finite-element-method (FEM)-based electroencephalography (EEG) forward simulation in which the electric potential evoked by neural activity in the brain is to be calculated at the surface of the head. The main advantage of the FEM is that it allows realistic modeling of tissue conductivity inhomogeneity. However, it is not straightforward to apply the classical model of a dipolar source with the FEM, due to its strong singularity and the resulting irregularity. The focus of this study is on comparing different methods to cope with this problem. In particular, we evaluate the accuracy of Whitney (Raviart-Thomas)-type dipole-like source currents compared to two reference dipole modeling methods: the St. Venant and partial integration approach. Common to all these methods is that they enable direct approximation of the potential field utilizing linear basis functions. In the present context, Whitney elements are particularly interesting, as they provide a simple means to model a divergence-conforming primary current vector field satisfying the square integrability condition. Our results show that a Whitney-type source model can provide simulation accuracy comparable to the present reference methods. It can lead to superior accuracy under optimized conditions with respect to both source location and orientation in a tetrahedral mesh. For random source orientations, the St. Venant approach turns out to be the method of choice over the interpolated version of the Whitney model. The overall moderate differences obtained suggest that practical aspects, such as the focality, should be prioritized when choosing a source model.

Sparse source travel-time tomography of a laboratory target: Accuracy and robustness of anomaly detection

This study concerned conebeam travel-time tomography. The focus was on a sparse distribution of signal sources that can be necessary in a challenging in situ environment such as in asteroid tomography. The goal was to approximate the minimum number of source positions needed for robust detection of refractive anomalies, e.g., voids within an asteroid or a casting defects in concrete. Experimental ultrasonic data were recorded utilizing as a target a 150 mm plastic cast cube containing three stones with diameter between 22 and 41 mm. A signal frequency of 55 kHz (35 mm wavelength) was used. Source counts from one to six were tested for different placements. Based on our statistical inversion approach and analysis of the results, three or four sources were found to lead to reliable inversion. The source configurations investigated were also ranked according to their performance. Our results can be used, for example, in the planning of planetary missions as well as in material testing.

EEG/MEG forward simulation through h- and p-type finite elements

Electro/Magnetoencephalography (EEG/MEG) is a non-invasive imaging modality, in which a primary current density generated by the neural activity in the brain is to be reconstructed from external electric potential/magnetic field measurements. This work focuses on effective and accurate simulation of the EEG/MEG forward model through the h- and p-versions of the finite element method (h- and p-FEM). The goal is to compare the effectiveness of these two versions in forward simulation. Both h- and p-type forward simulations are described and implemented, and the technical solutions found are discussed. These include, for example, suitable ways to generate a finite element mesh for a real head geometry through the use of different element types. Performances of the two implemented forward simulation types are compared by measuring directly the forward modeling error, as well as by computing reconstructions through a regularized FOCUSS (FOCal Underdetermined System Solver) algorithm. The results obtained suggest that the p-type performs better in terms of the forward modeling error. However, both types perform well in regularized FOCUSS reconstruction.

Coarse-to-fine reconstruction in linear inverse problems with application to limited-angle computerized tomography

Abstract The goal of this paper is to propose and test an iterative coarse-to-fine reconstruction procedure for a certain class of linear inverse problems. This procedure is based on preconditioned iterative regularization through the conjugate gradient (CG) method, through Tikhonov preconditioning, as well as through wavelet low-pass filtering. A quadratic minimization problem associated with a linear inverse problem, can be very problematic if the quadratic form is not diagonal or nearly (block) diagonal. In the present reconstruction strategy, a nearly block diagonal representation of a quadratic form is obtained due to wavelet filtering and preconditioning. In the numerical experiments, the proposed procedure is successfully applied to limited-angle computerized tomography (limited-angle CT). The results of these experiments show that a combined use of wavelet filters and preconditioning can be effective within the present problem class.