| 201. |
- Nordebo, Sven, et al.
(författare)
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Fisher information analysis and preconditioning in electrical impedance tomography
- 2010
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Ingår i: Journal of Physics: Conference Series 224 (2010), 1742-6588. - : IOP Publishing. - 1742-6596 .- 1742-6588. ; , s. 012057-
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Konferensbidrag (refereegranskat)abstract
- 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 consistent 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.
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| 202. |
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| 203. |
- Nordebo, Sven, et al.
(författare)
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Fisher information analysis for two-dimensional microwave tomography
- 2007
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Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 23:3, s. 859-877
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, a Fisher information analysis is employed to establish some important physical performance bounds in microwave tomography. As a canonical problem, the two-dimensional electromagnetic inverse problem of imaging a cylinder with isotropic dielectric losses is considered. A fixed resolution is analysed by introducing a finite basis, i.e., pixels representing the material properties. The corresponding Cramer-Rao bound for estimating the pixel values is computed based on a calculation of the sensitivity field which is obtained by differentiating the observed field with respect to the estimated parameter. An optimum trade-off between the accuracy and the resolution is defined based on the Cramer-Rao bound, and its application to assess a practical resolution limit in the inverse problem is discussed. Numerical examples are included to illustrate how the Fisher information analysis can be used to investigate the significance of measurement distance, operating frequency and losses in the canonical tomography set-up.
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| 204. |
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| 205. |
- Nordebo, Sven, et al.
(författare)
-
Fisher information analysis in electrical impedance tomography
- 2012
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Ingår i: Geophysical Research Abstract. ; , s. 2012-3553-1-
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Konferensbidrag (refereegranskat)abstract
- 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.
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| 206. |
- Nordebo, Sven, et al.
(författare)
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Fisher information analysis in electrical impedance tomography
- 2013
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Ingår i: Journal of Geophysics and Engineering. - : Institute of Physics (IOP). - 1742-2132 .- 1742-2140. ; 10:6, s. Article ID: 064008-
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 207. |
- Nordebo, Sven, et al.
(författare)
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Fisher information analysis in microwave tomography
- 2009
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Ingår i: The Third International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP09). - 9781424451807 ; , s. 217-220
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Konferensbidrag (refereegranskat)
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| 208. |
- Nordebo, Sven, et al.
(författare)
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Fisher information for inverse problems and trace class operators
- 2012
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Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 53:12
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Tidskriftsartikel (refereegranskat)abstract
- This paper provides a mathematical framework for Fisher information analysis forinverse problems based on Gaussian noise on infinite-dimensional Hilbert space. The covariance operator for the Gaussian noise is assumed to be trace class, andthe Jacobian of the forward operator Hilbert-Schmidt. We show that the appropriatespace for defining the Fisher information is given by the Cameron-Martin space. This is mainly because the range space of the covariance operator always is strictlysmaller than the Hilbert space. For the Fisher information to be well-defined, it is furthermore required that the range space of the Jacobian is contained in the Cameron-Martin space. In order for this condition to hold and for the Fisher information tobe trace class, a sufficient condition is formulated based on the singular values ofthe Jacobian as well as of the eigenvalues of the covariance operator, together withsome regularity assumptions regarding their relative rate of convergence. An explicit example is given regarding an electromagnetic inverse source problem with “external”spherically isotropic noise, as well as “internal” additive uncorrelated noise.
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| 209. |
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| 210. |
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