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On the accuracy and resolution in inverse imaging
Linnaeus University, Faculty of Technology, Department of Mathematics.
Linnaeus University, Faculty of Technology, Department of Physics and Electrical Engineering.ORCID iD: 0000-0002-7018-6248
Linnaeus University, Faculty of Technology, Department of Mathematics.
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.

Place, publisher, year, edition, pages
2015.
Keywords [en]
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: urn:nbn:se:lnu:diva-18688OAI: oai:DiVA.org:lnu-18688DiVA, id: diva2:526951
Available from: 2012-05-15 Created: 2012-05-15 Last updated: 2017-01-10Bibliographically approved
In thesis
1. Sensitivity Analysis and Material Parameter Estimation using Electromagnetic Modelling
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

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Sjödén, ThereseNordebo, SvenNilsson, Börje

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