t. This paper surveys some of the work our group has done in electrical impedance tomography.

Electrical Impedance Tomography

There has been tremendous advances in our ability to produce images of human brain function. Applications of functional brain imaging extend from improving our understanding of the basic mechanisms of cognitive processes to better characterization of pathologies that impair normal function. Magnetoencephalography (MEG) and electroencephalography (EEG) (MEG/EEG) localize neural electrical activity using noninvasive measurements of external electromagnetic signals. Among the available functional imaging techniques, MEG and EEG uniquely have temporal resolutions below 100 ms. This temporal precision allows us to explore the timing of basic neural processes at the level of cell assemblies. MEG/EEG source localization draws on a wide range of signal processing techniques including digital filtering, three-dimensional image analysis, array signal processing, image modeling and reconstruction, and, blind source separation and phase synchrony estimation. We describe the underlying models currently used in MEG/EEG source estimation and describe the various signal processing steps required to compute these sources. In particular we describe methods for computing the forward fields for known source distributions and parametric and imaging-based approaches to the inverse problem.

Electromagnetic brain mapping

The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

/pdf/inverse-problems-a-bayesian-perspective-1dknxttxjb.pdf

Inverse problems: A Bayesian perspective

Now in its second edition, this accessible text presents a unified Bayesian treatment of state-of-the-art filtering, smoothing, and parameter estimation algorithms for non-linear state space models. The book focuses on discrete-time state space models and carefully introduces fundamental aspects related to optimal filtering and smoothing. In particular, it covers a range of efficient non-linear Gaussian filtering and smoothing algorithms, as well as Monte Carlo-based algorithms. This updated edition features new chapters on constructing state space models of practical systems, the discretization of continuous-time state space models, Gaussian filtering by enabling approximations, posterior linearization filtering, and the corresponding smoothers. Coverage of key topics is expanded, including extended Kalman filtering and smoothing, and parameter estimation. The book's practical, algorithmic approach assumes only modest mathematical prerequisites, suitable for graduate and advanced undergraduate students. Many examples are included, with Matlab and Python code available online, enabling readers to implement algorithms in their own projects.

Bayesian Filtering and Smoothing

A review on continuous wave functional near-infrared spectroscopy and imaging instrumentation and methodology.

Classical electrical impedance tomography is an imaging modality in which the internal impedivity distribution is reconstructed from the known injected currents and voltages measured on the surface of the object. However, in many industrial cases there are a priori known internal structures inside the vessels which could be used as internal electrodes in tomographical imaging. In this paper we consider modelling of certain types of internal structures and using them as internal electrodes in two-dimensional electrical process tomography. We also propose an inversion approach in which directional smoothness of the resistivity distribution can be taken into account. Simulations and laboratory experiments show that, by utilizing internal structural information and using these internal structures as additional electrodes, we can achieve significant improvements in image reconstruction. Improvements are shown for the cases in which the electrodes are placed at the centre of the object and are surrounded by a resistive layer.

https://iopscience.iop.org/article/10.1088/0957-0233/12/8/304/pdf

Modelling of internal structures and electrodes in electrical process tomography

We consider electrical impedance tomography (EIT) imaging of the brain. The brain is surrounded by the poorly conducting skull which has low conductivity compared to the brain. The skull layer causes a partial shielding effect which leads to weak sensitivity for the imaging of the brain tissue. In this paper we propose an approach based on the Bayesian approximation error approach, to enhance the contrast in brain imaging. With this approach, both the (uninteresting) geometry and the conductivity of the skull are embedded in the approximation error statistics, which leads to a computationally efficient algorithm that is able to detect features such as internal haemorrhage with significantly increased sensitivity and specificity. We evaluate the approach with simulations and phantom data.

https://iopscience.iop.org/article/10.1088/0967-3334/37/1/1/pdf

Contrast enhancement in EIT imaging of the brain.

In this paper, a method for the determination of spatially varying thermal conductivity and perfusion coefficients of tissue is proposed. The temperature evolution in tissue is modelled with the Pennes bioheat equation. The main motivation here is a model-based optimal control for ultrasound surgery, in which the tissue properties are needed when the treatment is planned. The overview of the method is as follows. The same ultrasound transducers, which are eventually used for the treatment, are used to inflict small temperature changes in tissue. This temperature evolution is monitored using MR thermal imaging, and the tissue properties are then estimated on the basis of these measurements. Furthermore, an approach to choose transducer excitations for the determination procedure is also considered. The purpose of this paper is to introduce a method and therefore simulations are used to verify the method. Furthermore, computations are accomplished in a 2D spatial domain.

https://iopscience.iop.org/article/10.1088/0031-9155/51/4/017/pdf

Determination of heterogeneous thermal parameters using ultrasound induced heating and MR thermal mapping

The inverse problem of estimating gas temperature distribution based on the measurement of resistances of thin metal filaments spanned across the gas flow is discussed. The estimation is based on the fact that the resistance of the filament material is a function of the temperature. The gas temperature is modelled as a time-dependent stochastic process, and the temperature estimation is performed by using Kalman filters. The proposed method is tested both with numerically-produced synthetic data and with experimental data.

https://iopscience.iop.org/article/10.1088/0266-5611/14/4/003/pdf

Jari P. Kaipio

Papers

Modelling of internal structures and electrodes in electrical process tomography

Contrast enhancement in EIT imaging of the brain.

Determination of heterogeneous thermal parameters using ultrasound induced heating and MR thermal mapping

Dynamical electric wire tomography: a time series approach

A non-homogeneous regularization method for the estimation of narrow aerosol size distributions