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.

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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.

The radiative transfer equation is solved with the finite element method utilizing the streamline diffusion modification. Results of simulations are shown.

Finite Element Approximations for the Radiative Transfer Equation

PROBLEM TO BE SOLVED: To overcome problems confronted in three-dimensional reconstruction of an object occurring when a conventional restructure algorithm is used using thin projection data SOLUTION: This is a method to produce three-dimensional information of an object The three-dimensional information is produced by solving a nonlinear equation of evolution from at least two projection data having different direction In the equation: x presents internal coordinates of the imaged object; t presents a time; ψ presents a level set function indicating the border face of the object; h presents a function made into a positive value; m presents projection data; operator A presents a attenuation model by X-ray radiation; A * presents a reverse projection operator of A; α presents a positive number; P presents a bypass filter; and f presents a cut-off function to return 0 to a non-positive independent variable and return a positive number to a positive independent variable COPYRIGHT: (C)2006,JPO&NCIPI

Three-dimensional medical x-ray radiographing method and arrangement

The project "Multiphase flows in process industry (ProMoni)" 1.1.2001– 30.4.2004 was a research consortium carried out jointly by seven research groups from VTT, University of Jyväskylä, Tampere University of Technology, Åbo Akademi University and University of Kuopio. It included modeling, development and validation of numerical methods as well as development of new experimental techniques for multiphase flows found in process industry. The primary fields of application are in fluidized beds and in various processes found in the paper and pulp industry. This extensive final report of the ProMoni project includes results from experimental and numerical research of bubbling fluidized beds indicating that modelling produces the bed behaviour realistically and the simulations can be used in developing approximative macroscopic two-phase models. A new gas-solid drag correlation model was successfully used within Fluent CFD code to simulate the typical flow structures in a circulating fluidized bed riser. Several new or newly adapted experimental methods and measuring devices for fibre suspension flows were developed and brought in regular use in laboratory or pilot scale environments. Particle Image Velocimetry and Digital Imaging measurement were developed and used in two-phase flows to measure hydrodynamical interaction and morphology of the dispersed phase. Optical and electrical impedance tomographic devices and related image reconstruction algorithms were developed for nonintrusive measurement of consistency profiles and flow velocity in flows of diffusive fluids such as wood fibre suspensions. Multiphase flows in process industry

Multiphase flows in process industry: ProMoni

The replacement of a nonlinear parameter-to-observable mapping with a linear (affine) approximation is often carried out to reduce the computational costs associated with solving large-scale inverse problems governed by partial differential equations (PDEs). In the case of a linear parameter-to-observable mapping with normally distributed additive noise and a Gaussian prior measure on the parameters, the posterior is Gaussian. However, substituting an accurate model for a (possibly well justified) linear surrogate model can give misleading results if the induced model approximation error is not accounted for. To account for the errors, the Bayesian approximation error (BAE) approach can be utilised, in which the first and second order statistics of the errors are computed via sampling. The most common linear approximation is carried out via linear Taylor expansion, which requires the computation of (Fréchet) derivatives of the parameter-to-observable mapping with respect to the parameters of interest. In this paper, we prove that the (approximate) posterior measure obtained by replacing the nonlinear parameter-to-observable mapping with a linear approximation is in fact independent of the choice of the linear approximation when the BAE approach is employed. Thus, somewhat non-intuitively, employing the zero-model as the linear approximation gives the same approximate posterior as any other choice of linear approximations of the parameter-to-observable model. The independence of the linear approximation is demonstrated mathematically and illustrated with two numerical PDE-based problems: an inverse scattering type problem and an inverse conductivity type problem.

On global normal linear approximations for nonlinear Bayesian inverse problems

The stability of time-varying autoregressive (AR) models is an important issue in such applications as time-varying spectrum estimation and electroencephalography simulation and estimation. In some cases, such as time-varying spectrum estimation, the models that exhibit roots near unit moduli are difficult to use. Thus a tighter stability condition such as stability with a positive margin is needed. A time-varying AR model is stable with a positive margin if the moduli of the roots of the time-varying characteristic polynomial are somewhat less than unity for every time instant. Recently, a new method for the estimation of the time-varying AR models was introduced. This method is based on the interpretation of the underdetermined time-varying prediction equations as an ill-posed inverse problem that is solved by Tikhonov regularization. The method is referred to as the deterministic regression smoothness priors (DRSP) scheme. In this paper, a stabilization method in which the DRSP scheme is augmented with nonlinear stability constrainst is proposed. The problem is formulated so that stability with a positive margin can also be achieved. The problem is solved iteratively with an exterior point algorithm. The performance of the algorithm is studied with a simulation. It is shown that the proposed approach is well suited to stable modeling of signals containing narrowband transitions.

Jari P. Kaipio

Papers

Finite Element Approximations for the Radiative Transfer Equation

Three-dimensional medical x-ray radiographing method and arrangement

Multiphase flows in process industry: ProMoni

On global normal linear approximations for nonlinear Bayesian inverse problems

Stabilization of smoothness priors time-varying autoregressive models