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Sensitivity Analysis and Material Parameter Estimation using Electromagnetic Modelling
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Känslighetsanalys och estimering av materialparametrar med elektromagnetisk modellering (Swedish)
Abstract [en]

Estimating parameters is the problem of finding their values from measurements and modelling. Parameters describe properties of a system; material, for instance, are defined by mechanical, electrical, and chemical parameters. Fisher information is an information measure, giving information about how changes in the parameter effect the estimation. The Fisher information includes the physical model of the problem and the statistical model of noise. The Cramér-Rao bound is the inverse of the Fisher information and gives the best possible variance for any unbiased estimator.

This thesis considers aspects of sensitivity analysis in two applied material parameter estimation problems. Sensitivity analysis with the Fisher information and the Cramér-Rao bound is used as a tool for evaluation of measurement feasibilities, comparison of measurement set-ups, and as a quantitative measure of the trade-off between accuracy and resolution in inverse imaging.

The first application is with estimation of the wood grain angle parameter in trees and logs. The grain angle is the angle between the direction of the wood fibres and the direction of growth; a large grain angle strongly correlates to twist in sawn timber. In the thesis, measurements with microwaves are argued as a fast and robust measurement technique and electromagnetic modelling is applied, exploiting the anisotropic properties of wood. Both two-dimensional and three-dimensional modelling is considered. Mathematical modelling is essential, lowering the complexity and speeding up the computations. According to a sensitivity analysis with the Cramér-Rao bound, estimation of the wood grain angle with microwaves is feasible.

The second application is electrical impedance tomography, where the conductivity of an object is estimated from surface measurements. Electrical impedance tomography has applications in, for example, medical imaging, geological surveillance, and wood evaluation. Different configurations and noise models are evaluated with sensitivity analysis for a two-dimensional electrical impedance tomography problem. The relation between the accuracy and resolution is also analysed using the Fisher information.

To conclude, sensitivity analysis is employed in this thesis, as a method to enhance material parameter estimation. The sensitivity analysis methods are general and applicable also on other parameter estimation problems.

Abstract [sv]

Estimering av parametrar är att finna deras värde utifrån mätningar och modellering. Parametrar beskriver egenskaper hos system och till exempel material kan definieras med mekaniska, elektriska och kemiska parametrar. Fisherinformation är ett informationsmått som ger information om hur ändringar i en parameter påverkar estimeringen. Fisherinformationen ges av en fysikalisk modell av problemet och en statistisk modell av mätbruset. Cramér-Rao-gränsen är inversen av Fisherinformationen och ger den bästa möjliga variansen för alla väntevärdesriktiga estimatorer.Den här avhandlingen behandlar aspekter av känslighetsanalys i två tillämpade estimeringsproblem för materialparametrar. Känslighetsanalys med Fisherinformation och Cramér-Rao-gränsen används som ett redskap för utvärdering av möjligheten att mäta och för jämförelser av mätuppställningar, samt som ett kvantitativt mått på avvägningen mellan noggrannhet och upplösning för inversa bilder.

Den första tillämpningen är estimering av fibervinkeln hos träd och stockar. Fibervinkeln är vinkeln mellan växtriktningen och riktningen hos träfibern och en stor fibervinkel är relaterad till problem med formstabilitet i färdiga brädor. Mikrovågsmätningar av fibervinkeln presenteras som en snabb och robust mätteknik. I avhandlingen beskrivs två- och tredimensionella elektromagnetiska modeller som utnyttjar anisotropin hos trä. Eftersom matematisk modellering minskar komplexiteten och beräkningstiden är det en viktig del i estimeringen. Enligt känslighetsanalys med Cramér-Rao-gränsen är estimering av fibervinkeln hos trä möjlig.

Den andra tillämpningen är elektrisk impedanstomografi, där ledningsförmågan hos objekt bestäms genom mätningar på ytan. Elektrisk impedanstomografi har tillämpningar inom till exempel medicinska bilder, geologisk övervakning och trämätningar. Olika mätkonfigurationer och brusmodeller utvärderas med känslighetsanalys för ett tvådimensionellt exempel på elektrisk impedanstomografi. Relationen mellan noggrannhet och upplösning analyseras med Fisher information.

För att sammanfatta beskrivs känslighetsanalys som en metod för att förbättra estimeringen av materialparametrar. Metoderna för känslighetsanalys är generella och kan tillämpas också på andra estimeringsproblem för parametrar.

Place, publisher, year, edition, pages
Växjö, Kalmar: Linnaeus University Press , 2012. , 111 p.
Series
Linnaeus University Dissertations, 87/2012
Keyword [en]
sensitivity analysis, material parameter, Fisher information, Cramér-Rao bound, electromagnetic modelling, grain angle, wood, electrical impedance tomography, inverse problems
Keyword [sv]
känslighetsanalys, estimering, materialparameter, Fisher- information, Cramér-Rao-gränsen, elektromagnetisk modellering, fibervinkel, trä, elektrisk impedanstomografi, inversa problem
National Category
Other Physics Topics Signal Processing
Research subject
Physics, Waves and Signals
Identifiers
URN: urn:nbn:se:lnu:diva-18689ISBN: 978-91-86983-60-4 (print)OAI: oai:DiVA.org:lnu-18689DiVA: diva2:526956
Public defence
2012-06-05, Vidéumsalen, D1136, Vejdes plats 6, Växjö, 16:45 (English)
Opponent
Supervisors
Available from: 2012-05-16 Created: 2012-05-15 Last updated: 2016-03-11Bibliographically approved
List of papers
1. Microwave modelling and measurements for early detection of spiral grain in wood
Open this publication in new window or tab >>Microwave modelling and measurements for early detection of spiral grain in wood
2005 (English)In: 14th International  Symposium on Nondestructive Testing of Wood, Eberswalde, Germany, May 2005., 2005Conference paper, Published paper (Refereed)
Research subject
Physics, Electrotechnology
Identifiers
urn:nbn:se:vxu:diva-4821 (URN)
Note
Part of urn:nbn:se:vxu:diva-2210Available from: 2008-09-05 Created: 2008-09-05 Last updated: 2017-01-10Bibliographically approved
2. Wood Grain Angle Estimation in Logs with Microwave Modeling
Open this publication in new window or tab >>Wood Grain Angle Estimation in Logs with Microwave Modeling
2006 (English)In: Mathematical Modeling of Wave phenomena: 2nd Conference on Mathematical Modelling of Wave Phenomena, Växjö, Sweden, American Institute of Physics , 2006, 278-285 p.Conference paper, Published paper (Refereed)
Abstract [en]

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 properties of wood affecting the polarization of an electromagnetic field. A cylindrical model is used for calculating the electromagnetic scattering from a wooden log. In the model, the material has anisotropic dielectric properties, and is considered homogeneous. Furthermore, the grain angle is small and modelled without radial dependence.

Based on the model, we present a measurement strategy using a linearly polarized incident plane wave without z‐dependence, together with sensors placed close to the log. The grain angle can be estimated, also when the effect of the cylindrical geometry is substantial, through a comparison between the measured and the calculated field. No information on the log except its radius is required; the moisture content can, e.g., be unknown. We apply the Cramér Rao Lower Bound, to present error estimates for the determined grain angle as well as for the dielectric parameters.

Place, publisher, year, edition, pages
American Institute of Physics, 2006
Keyword
nondestructive testing, spiral grain, remote sensing
National Category
Other Electrical Engineering, Electronic Engineering, Information Engineering
Research subject
Physics, Electrotechnology
Identifiers
urn:nbn:se:vxu:diva-4417 (URN)10.1063/1.2205811 (DOI)0-7354-0325-2 (ISBN)
Conference
2nd Conference on Mathematical Modelling of Wave Phenomena, Växjö, Sweden
Available from: 2007-03-29 Created: 2007-03-29 Last updated: 2017-01-10Bibliographically approved
3. Estimation of Twist in Uniaxial Cylinders WithInverse Electromagnetic Scattering
Open this publication in new window or tab >>Estimation of Twist in Uniaxial Cylinders WithInverse Electromagnetic Scattering
2009 (English)In: IEEE Transactions on Antennas and Propagation, ISSN 0018-926X, E-ISSN 1558-2221, Vol. 57, no 10, 3264-3273 p.Article in journal (Refereed) Published
Abstract [en]

For the purpose of determining the twist of a homogeneous, locally reacting, uniaxial cylinder, an inverse microwave scattering theory is presented. Remote measurements of the spiral grain of trees and logs are the prime application. Based on practical considerations, it is assumed that the transmitting and receiving antennas are collocated, requiring a three-dimensional modelling. A general theory is first developed, followed by an asymptotic analysis assuming that the distance from the antennas to the cylinder is many wavelengths and many cylinder radii. In this way, a substantial reduction of the numerical complexity, to the level of the two-dimensional case, is achieved. The error of the determined twist angle as function of inherent parameters of the problem using a CramEacuter-Rao analysis is given. The results from numerical simulations show that this error is low enough for determining the grain angle. Presented parameter studies of the error can be used for minimizing the errors in a measurement set up, of particular interest for non-sophisticated instruments and non-ideal laboratory conditions, by selecting optimum parameters such as frequency and antenna gain. Finally, it is stated that the model has a great potential for developing efficient algorithms for measuring the twist angle.

Keyword
Error analysis, nondestructive testing, real time systems, remote sensing, wood industry
National Category
Signal Processing Control Engineering
Research subject
Physics, Electrotechnology
Identifiers
urn:nbn:se:vxu:diva-4824 (URN)10.1109/TAP.2009.2028542 (DOI)
Note

Part of urn:nbn:se:vxu:diva-2210

Available from: 2008-09-05 Created: 2008-09-05 Last updated: 2017-12-13Bibliographically approved
4. Numerical verification of a microwave impedance model for a twisted wooden cylinder
Open this publication in new window or tab >>Numerical verification of a microwave impedance model for a twisted wooden cylinder
2009 (English)In: Mathematical modelling of wave phenomena: 3rd Conference on Mathematical Modelling of Wave Phenomena, Växjö, Sweden, 9 – 13 June 2008, Melville, New York: American Institute of Physics (AIP), 2009, 243-252 p.Conference paper, Published paper (Refereed)
Abstract [en]

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 still lower for a technically interesting narrow angular region. The induced model error in the determination of the grain angle is about 5% which is of the same order or less than the error introduced by measurement noise.

 

Place, publisher, year, edition, pages
Melville, New York: American Institute of Physics (AIP), 2009
Series
AIP Conferece Proceedings, ISSN 0094-243X ; 1106
Keyword
modelling, microwaves, wood, cylinder, twist, impedance model
National Category
Other Electrical Engineering, Electronic Engineering, Information Engineering
Research subject
Physics, Electrotechnology
Identifiers
urn:nbn:se:vxu:diva-4908 (URN)10.1063/1.3117100 (DOI)978-0-7354-0643-8 (ISBN)
Conference
3rd Conference on Mathematical Modelling of Wave Phenomena, Växjö, Sweden, 9 – 13 June 2008
Available from: 2009-03-23 Created: 2009-03-23 Last updated: 2017-01-10Bibliographically approved
5. Sensitivity Analysis for Measurement Configuration Evaluation in Electrical Impedance Tomography
Open this publication in new window or tab >>Sensitivity Analysis for Measurement Configuration Evaluation in Electrical Impedance Tomography
2015 (English)Manuscript (preprint) (Other academic)
Abstract [en]

Fisher information and sensitivity analysis can be used to compare, evaluate, and optimize measurement configurations. In this paper, an electrical impedance tomography measurement is considered. Four-electrode measurement pairs are applied on a circular, two-dimensional disk. Adjacent and polar current configurations are compared. Adjoint field techniques are used for gradient calculation. Gradient methods, connected to Fisher information and singular value decomposition, are used for the reconstructions. The impact of different noise models is studied. According to the Fisher information analysis, the adjacent current configurations consistently outperforms the polar in the present formulations. The conclusion is verified by numerical reconstructions based on synthetic data.

Keyword
Fisher information, sensitivity analysis, electrical impedance tomography, current configurations, maximum likelihood
National Category
Signal Processing Other Physics Topics
Research subject
Physics, Waves and Signals
Identifiers
urn:nbn:se:lnu:diva-18687 (URN)
Available from: 2012-05-15 Created: 2012-05-15 Last updated: 2017-01-10Bibliographically approved
6. On the accuracy and resolution in inverse imaging
Open this publication in new window or tab >>On the accuracy and resolution in inverse imaging
2015 (English)Manuscript (preprint) (Other academic)
Abstract [en]

This paper provides a quantitative analysis of the optimal accuracy and resolution in inverse imaging based on the Cram ́er-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 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 (non-linear) inverse problem. It is also shown that the truncated ML-based pseudo-inverse is a suitable regularization strategy for a linearized problem, which exploits a sufficient statistics for estimation within these subspaces.

The statistical-based Cram ́er-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, the Electrical Impedance Tomography (EIT) 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 EIT is described, including a statistical analysis of the model errors due to the linearization.

Keyword
inverse imaging, Maximum Likelihood (ML), Singular Value Decompo- sition (SVD), Electrical Impedance Tomography (EIT), accuracy, resolution, Cram ́er- Rao bound (CRB)
National Category
Signal Processing Other Physics Topics
Research subject
Physics, Waves and Signals
Identifiers
urn:nbn:se:lnu:diva-18688 (URN)
Available from: 2012-05-15 Created: 2012-05-15 Last updated: 2017-01-10Bibliographically approved

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