Electrical impedance tomography (EIT) is a non-invasive medical imaging technique in which images of the conductivity in a region of interest in the body are computed from measurements of voltages on electrodes arising from low-frequency, low-amplitude applied currents. Mathematically, the inverse conductivity problem is nonlinear and ill-posed, and the reconstructions have characteristically low spatial resolution. One approach to improve the spatial resolution of EIT images is to include anatomically and physiologically-based prior information in the reconstruction algorithm. Statistical inversion theory provides a means of including prior information from a representative sample population. In this paper, a method is proposed to introduce statistical prior information into the D-bar method based on Schur complement properties. The method presents an improvement of the image obtained by the D-bar method by maximizing the conditional probability density function of an image that is consistent with a prior information and the model, given a D-bar image computed from the voltage measurements. Experimental phantoms show an improved spatial resolution by the use of the proposed method for the D-bar image reconstructions.

This paper recasts the problem of online secondary path modeling in the form of a statistical inverse problem. A statistical and, in particular, a Bayesian approach towards secondary path modeling is developed and the computational issues that emerge from this approach are discussed. All signals and parameters are modeled as random variables and the degree of information concerning them is coded in their probability density functions. An abstract solution is formulated in the form of a probability density function for the secondary path model. For extracting point estimates, common statistical estimation methods are investigated. It is shown that maximum likelihood estimation is not stable; however, Bayesian method of maximum a posteriori gives a reliable solution. An adaptive algorithm is then developed to compute this solution in a computationally efficient manner. This algorithm has three advantages, compared to the traditional secondary path modeling algorithms. First, it does not cause any interference with the main active noise control algorithm. Second, it does not require any additive-noise to be injected into the secondary path. Third, it does not require any off-line initiation. The convergence of the proposed algorithm is analyzed theoretically. The validity of the theoretical results is investigated by using computer simulation. Finally successful integration of the proposed algorithm into a real-time ANC system is reported.

Estimation of optical absorption and scattering of a target is an inverse problem associated with quantitative photoacoustic tomography. Conventionally, the problem is expressed as two folded. First, images of initial pressure distribution created by absorption of a light pulse are formed based on acoustic boundary measurements. Then, the optical properties are determined based on these photoacoustic images. The optical stage of the inverse problem can thus suffer from, for example, artefacts caused by the acoustic stage. These could be caused by imperfections in the acoustic measurement setting, of which an example is a limited view acoustic measurement geometry. In this work, the forward model of quantitative photoacoustic tomography is treated as a coupled acoustic and optical model and the inverse problem is solved by using a Bayesian approach. Spatial distribution of the optical properties of the imaged target are estimated directly from the photoacoustic time series in varying acoustic detection and optical illumination configurations. It is numerically demonstrated, that estimation of optical properties of the imaged target is feasible in limited view acoustic detection setting.

In this paper, we consider a computational method for detecting trees on the basis of airborne laser scanning (ALS) data. In the approach, locations, heights, and crown shapes of trees are tracked automatically by fitting multiple 3-D crown height models to ALS data of a field plot. The estimates are computed with an iterative reconstruction method based on Bayesian inversion paradigm. The formulation allows for utilizing prior information on tree shapes in the estimation. Here, the prior models are written based on field measurement data and allometric models for tree shapes. The feasibility of the approach is tested with ALS and field data from a managed boreal forest. The algorithm found 70.2% of the trees in the area, which is a clear improvement compared to a usual 2.5D crown delineation approach (53.1% of the trees detected).

Quantitative photoacoustic tomography is an emerging imaging technique aimed at estimating chromophore concentrations inside tissues from photoacoustic images, which are formed by combining optical information and ultrasonic propagation. This is a hybrid imaging problem in which the solution of one inverse problem acts as the data for another ill-posed inverse problem. In the optical reconstruction of quantitative photoacoustic tomography, the data is obtained as a solution of an acoustic inverse initial value problem. Thus, both the data and the noise are affected by the method applied to solve the acoustic inverse problem. In this paper, the noise of optical data is modelled as Gaussian distributed with mean and covariance approximated by solving several acoustic inverse initial value problems using acoustic noise samples as data. Furthermore, Bayesian approximation error modelling is applied to compensate for the modelling errors in the optical data caused by the acoustic solver. The results show that modelling of the noise statistics and the approximation errors can improve the optical reconstructions.

In this paper, we propose a novel computational approach to tree detection. The aim is to reconstruct the positions, sizes and crown shapes of trees. This amounts to simultaneous tree localization and estimation of tree shapes by fitting multiple CHMs to ALS data. This estimation problem is written in Bayesian inversion framework. The benefit of the Bayesian approach is that it enables accounting for statistical prior information on the unknown parameters in the estimation. The prior information is explicitely written in the form of a probability distribution that models the unknowns. Here, the prior information is associated with the statistics of tree height, crown height and crown width. We hypothesize that utilizing such information in the estimation improves the detection of trees in dense forests, i.e. cases where the existing reconstruction methods usually fail. The feasibility of the proposed approach is tested with ALS data. The estimates are compared with field measurements. The preliminary results demonstrate that the positions and sizes of the trees can be tracked relatively well, even if tree crowns are interlaced.

Electrical impedance tomography is a highly unstable problem with respect to measurement and modeling errors. This instability is especially severe when absolute imaging is considered. With clinical measurements, accurate knowledge about the body shape is usually not available, and therefore an approximate model domain has to be used in the computational model. It has earlier been shown that large reconstruction artefacts result if the geometry of the model domain is incorrect. In this paper, we adapt the so-called approximation error approach to compensate for the modeling errors caused by inaccurately known body shape. This approach has previously been shown to be applicable to a variety of modeling errors, such as coarse discretization in the numerical approximation of the forward model and domain truncation. We evaluate the approach with a simulated example of thorax imaging, and also with experimental data from a laboratory setting, with absolute imaging considered in both cases. We show that the related modeling errors can be efficiently compensated for by the approximation error approach. We also show that recovery from simultaneous discretization related errors is feasible, allowing the use of computationally efficient reduced order models.

Diagnostic and operational tasks based on dental radiology often require three-dimensional (3-D) information that is not available in a single X-ray projection image. Comprehensive 3-D information about tissues can be obtained by computerized tomography (CT) imaging. However, in dental imaging a conventional CT scan may not be available or practical because of high radiation dose, low-resolution or the cost of the CT scanner equipment. In this paper, we consider a novel type of 3-D imaging modality for dental radiology. We consider situations in which projection images of the teeth are taken from a few sparsely distributed projection directions using the dentist's regular (digital) X-ray equipment and the 3-D X-ray attenuation function is reconstructed. A complication in these experiments is that the reconstruction of the 3-D structure based on a few projection images becomes an ill-posed inverse problem. Bayesian inversion is a well suited framework for reconstruction from such incomplete data. In Bayesian inversion, the ill-posed reconstruction problem is formulated in a well-posed probabilistic form in which a priori information is used to compensate for the incomplete information of the projection data. In this paper we propose a Bayesian method for 3-D reconstruction in dental radiology. The method is partially based on Kolehmainen et al. 2003. The prior model for dental structures consist of a weighted l1 and total variation (TV)-prior together with the positivity prior. The inverse problem is stated as finding the maximum a posteriori (MAP) estimate. To make the 3-D reconstruction computationally feasible, a parallelized version of an optimization algorithm is implemented for a Beowulf cluster computer. The method is tested with projection data from dental specimens and patient data. Tomosynthetic reconstructions are given as reference for the proposed method.

Estimation of current or potential distribution on the cortex is used to obtain information about neural sources from the scalp recorded electroencephalogram. If the active sources in the brain are superficial, the estimated field distribution on the cortex also yields information about the active source configuration. In these cases, these methods can be used as source localization methods. In this study, we concentrate on finite-element-based cortex potential estimation. Usually these methods require surface interpolation of the recorded voltages at the electrodes onto the entire scalp surface. We propose a new computational approach which does not require the use of surface interpolation but does it implicitly and uses only the recorded data at the electrodes. We refer to this method as the systematic approach (SA). We compare the SA with the surface interpolation approach (IA) and show that the SA is able to produce somewhat better accuracy than the IA. However, the main asset is that the sensitivity of the cortical potential maps to the regularization parameter is significantly lower than with the IA.

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.

In this paper we propose a method for the estimation of time-varying autoregressive (TVAR) processes. The approach is essentially to regularize the heavily underdetermined unconstrained prediction equations with a smoothness priors type side constraint. The implementation of nonhomogeneous smoothness properties is straightforward. The method is compared to the usual deterministic regression approach (TVAR) in which the coefficient evolutions are constrained to a subspace. It is shown that the typical transient oscillations of TVAR can be avoided with the proposed method.

A method for the stabilization of stationary and time-varying autoregressive models is presented. The method is based on the hyperstability constrained LS-problem with nonlinear constraints. The problems are solved iteratively with Gauss-Newton type algorithm that sequentially linearizes the constraints. The proposed method is applied to simulated data in the stationary case and to real EEG data in the time-varying case.

The solution of impedance distribution in electrical impedance tomography is a nonlinear inverse problem that requires the use of a regularization method. The generalized Tikhonov regularization methods have been popular in the solution of many inverse problems. The regularization matrices that are usually used with the Tikhonov method are more or less ad hoc and the implicit prior assumptions are, thus, in many cases inappropriate. In this paper, we propose an approach to the construction of the regularization matrix that conforms to the prior assumptions on the impedance distribution. The approach is based on the construction of an approximating subspace for the expected impedance distributions. It is shown by simulations that the reconstructions obtained with the proposed method are better than with two other schemes of the same type when the prior is compatible with the true object. On the other hand, when the prior is incompatible with the true object, the method will still give reasonable estimates.