Therese Sjödén

An adjoint field approach to Fisher information based sensitivity analysis in electrical impedance tomography

An adjoint field approach is used to formulate a general numerical framework for Fisher information-based sensitivity analysis in electrical impedance tomography. General expressions are given for the gradients used in standard least-squares optimization, i.e. the Jacobian related to the forward problem, and it is shown that these gradient expressions are compatible with commonly used electrode models such as the shunt model and the complete electrode model. By using the adjoint field formulations together with a variational analysis, it is also shown how the computation of the Fisher information can be integrated with the gradient calculations used for optimization. It is furthermore described how the Fisher information analysis and the related sensitivity map can be used in a preconditioning strategy to obtain a well-balanced parameter sensitivity and improved performance for gradient-based quasi-Newton optimization algorithms in electrical impedance tomography. Numerical simulations as well as reconstructions based on experimental data are used to illustrate the sensitivity analysis and the performance of the improved inversion algorithm in a four-electrode measurement set-up.

A method for under-bark detection of the wood grain angle radial dependence

Abstract Twist in wood, being closely related to spiral grain, may cause serious problems in building structures, furniture and joinery. It is therefore of great interest to sort out, at an early stage in the manufacturing process, trees, logs and boards that have an excess of spiral grain. The spiral grain pattern is described by a helical deviation of the fibre direction in relation to the longitudinal direction of a living tree or a log and seems to be an indicator for other defects such as compression wood. Remote microwave sensing of spiral grain has received a lot of interest during the past two decades. Its development has been impeded by the large variation with moisture content of the microwave properties of wood and by the complexity in modelling the electromagnetic field in a log with spiral grain. A review is presented of a direct method with no requirement for information on moisture content for boards. This procedure has recently been generalized to cylindrical logs and trees having a constant slope of the grain. A further generalization is presented here to allow for the normal spiral grain pattern with radially changing slope of grain in wood under bark. Based on this theory, a measurement procedure is proposed for the detection of wood grain angle with radial dependence, requiring no information on moisture content in the sapwood, which is also applicable to completely or partially frozen wood. A suitable application would be an instrument to use in the forest for measurements on living trees or logs.

Data Fusion for Electromagnetic and Electrical Resistive Tomography Based on Maximum Likelihood

This paper presents a maximum likelihood based approach to data fusion for electromagnetic (EM) and electrical resistive (ER) tomography. The statistical maximum likelihood criterion is closely linked to the additive Fisher information measure, and it facilitates an appropriate weighting of the measurement data which can be useful with multiphysics inverse problems. The Fisher information is particularly useful for inverse problems which can be linearized similar to the Born approximation. In this paper, a proper scalar product is defined for the measurements and a truncated Singular Value Decomposition (SVD) based algorithm is devised which combines the measurement data of the two imaging modalities in a way that is optimal in the sense of maximum likelihood. As a multiphysics problem formulation with applications in geophysics, the problem of tunnel detection based on EM and ER tomography is studied in this paper. To illustrate the connection between the Greens functions, the gradients and the Fisher information, two simple and generic forward models are described in detail regarding two-dimensional EM and ER tomography, respectively.

Fisher information analysis in electrical impedance tomography

This paper provides a quantitative analysis of the optimal accuracy and resolution in electrical impedance tomography (EIT) based on the Cramer–Rao lower bound. The imaging problem is characterized by the forward operator and its Jacobian. The Fisher information operator is defined for a deterministic parameter in a real Hilbert space and a stochastic measurement in a finite-dimensional complex Hilbert space with a Gaussian measure. The connection between the Fisher information and the singular value decomposition (SVD) based on the maximum likelihood (ML) criterion (the ML-based SVD) is established. It is shown that the eigenspaces of the Fisher information provide a suitable basis to quantify the trade-off between the accuracy and the resolution of the (nonlinear) inverse problem. It is also shown that the truncated ML-based pseudo-inverse is a suitable regularization strategy for a linearized problem, which exploits sufficient statistics for estimation within these subspaces. The statistical-based Cramer–Rao lower bound provides a complement to the deterministic upper bounds and the L-curve techniques that are employed with linearized inversion. To this end, electrical impedance tomography provides an interesting example where the eigenvalues of the SVD usually do not exhibit a very sharp cut-off, and a trade-off between the accuracy and the resolution may be of practical importance. A numerical study of a hypothetical EIT problem is described, including a statistical analysis of the model errors due to the linearization.

Fisher information analysis and preconditioning in electrical impedance tomography

In this contribution, it is described how the Fisher information can be computed by using adjoint field techniques, and integrated with the gradient calculations used for optimization in electrical impedance tomography. In particular, the Fisher information can be used as a preconditioner to obtain improved convergence properties and a regularization for quasi-Newton optimization algorithms in electrical impedance tomography. Experimental data have been used to study the possibility of combining a good reconstruction quality with a low system complexity, which is achieved by using a four-electrode measurement technique to avoid the need of high-precision electrode modeling and allowing a very coarse FEM grid. Here, the Fisher information based preconditioning implies a regularization and an algorithm that works well on a coarse FEM grid with very small electrodes modeled as point sources.

Numerical Verification of a Microwave Impedance Model for a Twisted Wooden Cylinder

To determine the grain angle under bark for wooden logs, microwaves are suitable in contrast to a laser system that requires that a part of the bark layer is removed. Such measurements can be used to sort out logs in a sawmill in order to avoid problems in the processing, low quality in finished products and unnecessary costs with sawing, drying and transporting. Since the logs are moving with a high speed, the measurements must be done quickly in real time. To get quick algorithms it has been proposed to model the log with a normal surface impedance rather than as a penetrable cylinder. This paper determines the accuracy in such a modelling by comparing results from the two models in two‐dimensions. The comparison is based on measured values of dielectric data for wood. The conclusion is that there exists a region of moisture content, frequency and azimuthal angle for which the relative error is less than 1% for the longitudinal and less than 10% for the tangential component of the electric field, and st...

Wood grain angle estimation in logs with microwave modelling

We study the detection of the wood grain angle using microwaves in order to avoid twist in finished boards. For such a determination of spiral grain, we exploit the anisotropic dielectric propertie ...