Aku Seppänen

State estimation with fluid dynamical evolution models in process tomography - an application to impedance tomography

In this paper we consider the reconstruction of rapidly varying objects in process tomography. The evolution of the physical parameters can often be approximated with stochastic convection-diffusion and fluid dynamics models. We use the state estimation approach to obtain the tomographic reconstructions and show how these flow models can be exploited with the actual observation models that by themselves induce ill-posed problems. The state estimation problem can be stated in different ways based on the available temporal information. We concentrate on such cases in which continuous monitoring is essential but a small delay for the reconstructions is allowable. The state estimation problem is solved with the fixed-lag Kalman smoother algorithm. As the boundary observations we use the voltage data of electrical impedance tomography. We also give a numerical illustration of the approach in a case in which we track a bolus that moves rapidly through a pipeline.

Electrical impedance tomography-based sensing skin for quantitative imaging of damage in concrete

This paper outlines the development of a large-area sensing skin for damage detection in concrete structures. The developed sensing skin consists of a thin layer of electrically conductive copper paint that is applied to the surface of the concrete. Cracking of the concrete substrate results in the rupture of the sensing skin, decreasing its electrical conductivity locally. The decrease in conductivity is detected with electrical impedance tomography (EIT) imaging. In previous works, electrically based sensing skins have provided only qualitative information on the damage on the substrate surface. In this paper, we study whether quantitative imaging of the damage is possible. We utilize application-specific models and computational methods in the image reconstruction, including a total variation (TV) prior model for the damage and an approximate correction of the modeling errors caused by the inhomogeneity of the painted sensing skin. The developed damage detection method is tested experimentally by applying the sensing skin to polymeric substrates and a reinforced concrete beam under four-point bending. In all test cases, the EIT-based sensing skin provides quantitative information on cracks and/or other damages on the substrate surface: featuring a very low conductivity in the damage locations, and a reliable indication of the lengths and shapes of the cracks. The results strongly support the applicability of the painted EIT-based sensing skin for damage detection in reinforced concrete elements and other substrates.

State space models in process tomography — approximation of state noise covariance

In this paper we consider process tomography in the case of time-varying objects. Especially, we concentrate on the case in which the indirect observations from the system are obtained via electrical impedance tomographic (EIT) measurements and in which the time-evolution of the target can be described by a stochastic convection-diffusion model. We use the state estimation approach to obtain the tomographic reconstructions. The state estimation problem is solved with the fixed-lag Kalman smoother algorithm that is a feasible approach for continuous observation with an insignificant delay in the reconstructions. In particular we focus on the covariance structures associated with state space model. The covariance structures determine the temporal and spatial regularization properties of the algorithm. It is shown that the adoption of nontrivial covariance structures in the evolution model yields good estimates for the time-varying object in such a situation in which stationary reconstructions are completely useless.

Bayesian Approach to Tree Detection Based on Airborne Laser Scanning Data

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).

Electrical Resistance Tomography for Assessment of Cracks in Concrete

A study on the feasibility of using electrical resistance tomography (ERT) for detecting cracks in concrete is presented in this article. The previous studies have demonstrated that cracks in concrete can be detected with ERT in cylindrical geometries. In this work, the main focus is on investigating the capability of ERT for crack detection and characterization in more realistic geometries in which the sensor array is attached to one planar surface of concrete. The feasibility of ERT in crack detection is tested with: 1) concrete slabs containing plastic plates; and 2) beams with real cracks generated by three-point bending. The results with slabs demonstrate the ability of ERT to distinguish between crack-like defects with different depths and to detect laminar defects. The results with beams verify that the method is also feasible in the case of real cracks.

Posterior covariance related optimal current patterns in electrical impedance tomography

Within the deterministic inversion framework, the optimal current pattern theory of electrical impedance tomography is well developed. This theory focuses on the notion of distinguishability, which amounts to optimizing the current patterns so that the difference of voltage measurements corresponding to two different predetermined conductivity distributions is maximized. However, it is often difficult to specify the two conductivity distributions. Especially in the framework of statistical inversion theory in which prior information is specified in the form of probability distributions, other approaches are needed. In the statistical inversion framework, the mean accuracy of the conductivity estimates can be described by the posterior covariance. In this paper, we propose to optimize the current patterns based on criteria that are functionals of the posterior covariance matrix. This approach uses the linearized likelihood distribution and results in nonlinear optimization problems with nonlinear equality constraints. We show that optimal current patterns can be constructed for such cases in which the distinguishability approach cannot be employed. Also, it is shown that in some cases only a few current patterns are needed in order to exhaust most of the information available in EIT measurements in the sense that conducting further measurements does not considerably decrease the uncertainty related to the estimates.

Simultaneous recovery of admittivity and body shape in electrical impedance tomography: an experimental evaluation

In this paper, the simultaneous retrieval of the exterior boundary shape and the interior admittivity distribution of an examined body in electrical impedance tomography is considered. The reconstruction method is built for the complete electrode model and it is based on the Frechet derivative of the corresponding current-to-voltage map with respect to the body shape. The reconstruction problem is cast into the Bayesian framework, and maximum a posteriori estimates for the admittivity and the boundary geometry are computed. The feasibility of the approach is evaluated by experimental data from water tank measurements. The results demonstrate that the proposed method has potential for handling an unknown body shape in a practical setting.

Fluid dynamical models and state estimation in process tomography: Effect due to inaccuracies in flow fields

In this paper we consider the reconstruction of rapidly varying objects in process tomography. The evolution of the physical parameters is approximated with stochastic convection diffusion and fluid dynamics models. The actual time-varying reconstruction is carried out as a state estimation problem. As the boundary observations we use the voltage data of electrical impedance tomography. We have previously shown that state estimation works well in process tomography in the cases in which the fluid dynamics of the system are modeled correctly. In the real case, however, the velocity field cannot usually be determined accurately. This may be caused, for example, by complex nature of the flow, the turbulence, discretization, etc. In adopting the first proposed approach, it is essential to know how much the inaccuracies in the fluid dynamical model affect the state estimates in process tomography. In this paper we consider the tolerance of the approach with respect to these inaccuracies. We show that the estimation scheme is relatively tolerant to modeling errors in the flow field. Thus relatively reliable estimates can be obtained, for example, in a case in which a laminar flow model is used in turbulent flow conditions. However, the degradation that is due to incorrect flow fields is not insignificant and it is also conjectured that it could be possible that an extension of the proposed method could be used to estimate some flow field parameters.

Simultaneous Reconstruction of Outer Boundary Shape and Admittivity Distribution in Electrical Impedance Tomography

The aim of electrical impedance tomography is to reconstruct the admittivity distribution inside a physical body from boundary measurements of current and voltage. Due to the severe ill-posedness of the underlying inverse problem, the functionality of impedance tomography relies heavily on accurate modelling of the measurement geometry. In particular, almost all reconstruction algorithms require the precise shape of the imaged body as an input. In this work, the need for prior geometric information is relaxed by introducing a Newton-type output least squares algorithm that reconstructs the admittivity distribution and the object shape simultaneously. The method is built in the framework of the complete electrode model and is based on the Frechet derivative of the corresponding current-to-voltage map with respect to the object boundary shape. The functionality of the technique is demonstrated via numerical experiments with simulated measurement data.

Optimal current patterns in dynamical electrical impedance tomography imaging

In this paper, the topic of optimal experiment design in electrical impedance tomography (EIT) is studied. More specifically, we consider determination of optimal current patterns in EIT in cases of time-varying targets. The reconstruction problem associated with EIT imaging is known to be an ill-posed inverse problem. Statistical inversion methods have been shown to be advantageous in many cases in EIT. In Kaipio et al (2004 Inverse Problems 20 919–36), we considered the problem of optimal experiment design in statistical framework and we proposed an approach for determining optimal current patterns in cases of imaging of time-invariant targets. The approach was based on the statistical interpretation of the reconstruction problem and optimal current patterns were obtained by minimizing the trace of an approximate posterior covariance matrix. In this paper, we utilize a similar approach to determining optimal current patterns in cases of time-varying targets. The image reconstruction problem of EIT is formulated as a state estimation problem. As in the time-invariant case, the optimality criterion is based on the posterior covariances but instead of considering one specific time instant we minimize the time-averaged mean posterior variance. It is shown in a numerical study that the uncertainties of the estimates obtained with optimized current patterns are smaller than those obtained with conventional current patterns. In addition, the results indicate that in time-varying problems a single optimized current pattern may be sufficient to achieve good accuracy, i.e., multiple optimized current patterns do not provide substantial further information on the target. We also demonstrate that in some cases the increase in the number of current patterns can even decrease the reliability of the estimates. This is one of the reasons for the topic of optimal current patterns being quite important in the case of imaging time-varying targets.