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.

A method for the single trial estimation of the evoked potentials is presented. The method is based on the regularized least squares scheme. An approach to single trial estimation of multichannel evoked potentials is derived by applying additional information of the spatial correlation of the channels to the estimation. Amplitude estimates obtained with these methods are compared. The performance of the methods is evaluated using real data of target responses in the P300 test. The multi channel approach is shown to give realistic and comparable information about the amplitude differences of the P300 peak between different channels.

http://ieeexplore.ieee.org/iel5/7218/19495/00901512.pdf

Comparison of amplitude estimates in the single trial estimation of evoked potentials

Medical X-ray imaging method and an arrangement

A method for the stabilization of time-varying autoregressive models is proposed. The method is based on the smoothness priors regularized prediction equations with nonlinear stability constraints. The problem is solved iteratively with a Gauss-Newton type algorithm in which the original constrained problem is approximated with a sequence of unconstrained problems by using appropriate penalty functions. The performance of the algorithm is studied with a simulation.

Stabilization of smoothness priors deterministic regression TVAR models

This paper considers a statistical method for damage identification of simply-supported elastic beams from static data. The problem is cast as an inverse problem and analyzed in Bayesian inversion ...

Bayesian damage identification of simply supported beams from elastostatic data

The estimation of poroelastic material parameters based on ultrasound measurements is considered. The acoustical characterisation of poroelastic materials based on various measurements is typically carried out by minimising a cost functional of model residuals, such as the least squares functional. With a limited number of unknown parameters, least squares type approaches can provide both reliable parameter and error estimates. With an increasing number of parameters, both the least squares parameter estimates and, in particular, the error estimates often become unreliable. In this paper, the estimation of the material parameters of an inhomogeneous poroelastic (Biot) plate in the Bayesian framework for inverse problems is considered. Reflection and transmission measurements are performed and 11 poroelastic parameters, as well as 4 measurement setup-related nuisance parameters, are estimated. A Markov chain Monte Carlo algorithm is employed for the computational inference to assess the actual uncertainty of the estimated parameters. The results suggest that the proposed approach for poroelastic material characterisation can reveal the heterogeneities in the object, and yield reliable parameter and uncertainty estimates.

Jari P. Kaipio

Papers

Comparison of amplitude estimates in the single trial estimation of evoked potentials

Medical X-ray imaging method and an arrangement

Stabilization of smoothness priors deterministic regression TVAR models

Bayesian damage identification of simply supported beams from elastostatic data

Estimating the material parameters of an inhomogeneous poroelastic plate from ultrasonic measurements in water