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Sjödén, Therese
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Sjödén, T., Nordebo, S. & Nilsson, B. (2015). 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.

Keywords
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
Sjödén, T. & Nordebo, S. (2015). 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.

Keywords
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
Ekevid, T., Carlqvist, C. & Sjödén, T. (2015). Virtual Articulated Haulers: prediction of vehicle performance and design loads. In: Josep M. Font-Llagunes (Ed.), Multibody Dynamics 2015: Barcelona, June 29 – July 2. Paper presented at ECCOMAS Thematic Conference on Multibody Dynamics June 29 - July 2, 2015, Barcelona, Catalonia, Spain (pp. 1377-1387). Barcelona: International Center for Numerical Methods in Engineering (CIMNE)
Open this publication in new window or tab >>Virtual Articulated Haulers: prediction of vehicle performance and design loads
2015 (English)In: Multibody Dynamics 2015: Barcelona, June 29 – July 2 / [ed] Josep M. Font-Llagunes, Barcelona: International Center for Numerical Methods in Engineering (CIMNE), 2015, p. 1377-1387Conference paper, Published paper (Refereed)
Abstract [en]

Articulated haulers are suited for any kind of transports in difficult terrain. The concept was developed from an agricultural tractor in the 1960’s by Volvo. The main features are the hydraulic steering, the individual front and rear frames connected with a rotational hitch joint, all-wheel drive and large tires. These features give the extraordinary terrain capability. Complete vehicle simulations support development of stability, handling and ride comfort properties during early development of the hauler. Complete vehicle simulations can also be utilized to extract design loads. This paper presents the current status in this field.

The complete vehicle MBS simulation model includes flexible bodies, hydraulic system, tires and powertrain. A refined powertrain model in Simulink is used for the analysis presented. The environment is defined by the road and the driver. A comfort track road is used for comfort evaluations and a digitalized endurance test road for load predictions. The driver model aims to follow the given target path and velocity.

Simulation and measurement signals are compared by means of relevant spectra; load ranges, level crossings and Power Spectral Density. A scalar measure is defined by the equivalent value to carry out comparisons between simulations and test. Fatigue load simulations need an accurate dynamic description of the vehicle, capturing low frequency excitation. The results shows that the load simulation captures the frequency peaks below 15 Hz, although there are some differences in magnitude and peak frequency. When simulated and measured equivalent values are compared most entities are in the range between 1/2 and 2 regarding estimated life. Further improvements are possible but the simulated load quality should also be related to other uncertainties in the fatigue evaluation process.

Place, publisher, year, edition, pages
Barcelona: International Center for Numerical Methods in Engineering (CIMNE), 2015
Keywords
Articulated Hauler, Co-simulation, Structure loads, Multi Body Dynamics
National Category
Vehicle Engineering
Research subject
Technology (byts ev till Engineering), Mechanical Engineering
Identifiers
urn:nbn:se:lnu:diva-45690 (URN)978-84-944244-0-3 (ISBN)
Conference
ECCOMAS Thematic Conference on Multibody Dynamics June 29 - July 2, 2015, Barcelona, Catalonia, Spain
Available from: 2015-08-14 Created: 2015-08-14 Last updated: 2018-05-18Bibliographically approved
Nordebo, S., Gustafsson, M., Nilsson, B., Sjödén, T. & Soldovieri, F. (2013). Fisher information analysis in electrical impedance tomography. Journal of Geophysics and Engineering, 10(6), Article ID: 064008
Open this publication in new window or tab >>Fisher information analysis in electrical impedance tomography
Show others...
2013 (English)In: Journal of Geophysics and Engineering, ISSN 1742-2132, E-ISSN 1742-2140, Vol. 10, no 6, p. Article ID: 064008-Article in journal (Refereed) Published
Abstract [en]

This paper provides a quantitative analysis of the optimal accuracy and resolution in electrical impedance tomography (EIT) based on the Cramér–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 Cramér–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.

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2013
National Category
Electrical Engineering, Electronic Engineering, Information Engineering
Research subject
Physics, Waves and Signals
Identifiers
urn:nbn:se:lnu:diva-27493 (URN)10.1088/1742-2132/10/6/064008 (DOI)000334994600009 ()2-s2.0-84893496491 (Scopus ID)
Available from: 2013-07-09 Created: 2013-07-09 Last updated: 2017-12-06Bibliographically approved
Nordebo, S., Sjödén, T., Gustafsson, M. & Soldovieri, F. (2012). Fisher information analysis in electrical impedance tomography. In: Geophysical Research Abstract: EGU General Assembly 2012. Paper presented at European Geosciences Union General Assembly, Vienna, 22-27 April, 2012 (pp. 2012-3553-1).
Open this publication in new window or tab >>Fisher information analysis in electrical impedance tomography
2012 (English)In: Geophysical Research Abstract: EGU General Assembly 2012, 2012, p. 2012-3553-1-Conference paper, Oral presentation with published abstract (Refereed)
Abstract [en]

In this contribution it is demonstrated how the Cramér-Rao lower bound provides a quantitative analysis of theoptimal accuracy and resolution in inverse imaging, see also Nordebo et al., 2010, 2010b, 2010c. The imagingproblem is characterized by the forward operator and its Jacobian. The Fisher information operator is defined fora deterministic parameter in a real Hilbert space and a stochastic measurement in a finite-dimensional complexHilbert space with Gaussian measure. The connection between the Fisher information and the Singular ValueDecomposition (SVD) based on the Maximum Likelihood (ML) criterion (the ML-based SVD) is established. Itis shown that the eigenspaces of the Fisher information provide a suitable basis to quantify the trade-off betweenthe accuracy and the resolution of the (non-linear) inverse problem. It is also shown that the truncated ML-basedpseudo-inverse is a suitable regularization strategy for a linearized problem, which exploits a sufficient statisticsfor estimation within these subspaces.The statistical-based Cramér-Rao lower bound provides a complement to the deterministic upper bounds and theL-curve techniques that are employed with linearized inversion (Kirsch, 1996; Hansen, 1992, 1998, 2010). To thisend, the Electrical Impedance Tomography (EIT) provides an interesting example where the eigenvalues of theSVD usually do not exhibit a very sharp cut-off, and a trade-off between the accuracy and the resolution may be ofpractical importance. A numerical study of EIT is described, including a statistical analysis of the model errors dueto the linearization. The Fisher information and sensitivity analysis is also used to compare, evaluate, and optimizemeasurement configurations in EIT.

National Category
Physical Sciences
Research subject
Natural Science, Physics
Identifiers
urn:nbn:se:lnu:diva-19207 (URN)
Conference
European Geosciences Union General Assembly, Vienna, 22-27 April, 2012
Available from: 2012-05-31 Created: 2012-05-31 Last updated: 2017-01-10Bibliographically approved
Sjödén, T. (2012). Sensitivity Analysis and Material Parameter Estimation using Electromagnetic Modelling. (Doctoral dissertation). Växjö, Kalmar: Linnaeus University Press
Open this publication in new window or tab >>Sensitivity Analysis and Material Parameter Estimation using Electromagnetic Modelling
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Känslighetsanalys och estimering av materialparametrar med elektromagnetisk modellering
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. p. 111
Series
Linnaeus University Dissertations ; 87/2012
Keywords
sensitivity analysis, material parameter, Fisher information, Cramér-Rao bound, electromagnetic modelling, grain angle, wood, electrical impedance tomography, inverse problems, 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:nbn:se:lnu:diva-18689 (URN)978-91-86983-60-4 (ISBN)
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
Sjödén, T., Nordebo, S. & Nilsson, B. (2012). Sensitivity analysis for inverse problems in Electrical Impedance Tomography. In: : . Paper presented at The 6th International Conference on Inverse Problems: Modeling and Simulation, Antalya, Turkey, 21-26 May, 2012 (pp. 220-221).
Open this publication in new window or tab >>Sensitivity analysis for inverse problems in Electrical Impedance Tomography
2012 (English)Conference paper, Published paper (Refereed)
National Category
Medical Engineering
Identifiers
urn:nbn:se:lnu:diva-19213 (URN)
Conference
The 6th International Conference on Inverse Problems: Modeling and Simulation, Antalya, Turkey, 21-26 May, 2012
Available from: 2012-05-31 Created: 2012-05-31 Last updated: 2017-01-10Bibliographically approved
Nordebo, S., Gustafsson, M., Sjödén, T. & Soldovieri, F. (2011). Data fusion for electromagnetic and electrical resistive tomography based on maximum likelihood. International Journal of Geophysics, 2011, Article ID 617089.
Open this publication in new window or tab >>Data fusion for electromagnetic and electrical resistive tomography based on maximum likelihood
2011 (English)In: International Journal of Geophysics, ISSN 1687-885X, E-ISSN 1687-8868, Vol. 2011, article id 617089Article in journal (Refereed) Published
Abstract [en]

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 Green's 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.

National Category
Electrical Engineering, Electronic Engineering, Information Engineering
Research subject
Physics, Waves and Signals
Identifiers
urn:nbn:se:lnu:diva-16729 (URN)10.1155/2011/617089 (DOI)2-s2.0-84871880554 (Scopus ID)
Available from: 2012-01-11 Created: 2012-01-11 Last updated: 2017-12-08Bibliographically approved
Nordebo, S., Bayford, R., Bengtsson, B., Fhager, A., Gustafsson, M., Hashemzadeh, P., . . . Sjödén, T. (2010). An adjoint field approach to Fisher information based sensitivity analysis in electrical impedance tomography. Inverse Problems, 26(12), Article ID: 125008
Open this publication in new window or tab >>An adjoint field approach to Fisher information based sensitivity analysis in electrical impedance tomography
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2010 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 26, no 12, p. Article ID: 125008-Article in journal (Refereed) Published
Abstract [en]

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.

National Category
Engineering and Technology
Research subject
Physics, Waves and Signals
Identifiers
urn:nbn:se:lnu:diva-27496 (URN)10.1088/0266-5611/26/12/125008 (DOI)2-s2.0-78649784569 (Scopus ID)
Available from: 2013-07-09 Created: 2013-07-09 Last updated: 2017-12-06Bibliographically approved
Sjödén, T., Nilsson, B. & Nordebo, S. (2010). Determination of wood parameters using electromagnetic measurements. In: 5th International Conference on Inverse Problems, Modeling and Simulation, Antalya, Turkey, 24-29 May, 2010.: (pp. 70-71).
Open this publication in new window or tab >>Determination of wood parameters using electromagnetic measurements
2010 (English)In: 5th International Conference on Inverse Problems, Modeling and Simulation, Antalya, Turkey, 24-29 May, 2010., 2010, p. 70-71Conference paper, Published paper (Refereed)
National Category
Physical Sciences
Research subject
Physics, Waves and Signals
Identifiers
urn:nbn:se:lnu:diva-7099 (URN)
Available from: 2010-08-10 Created: 2010-08-10 Last updated: 2017-01-10Bibliographically approved
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