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.
Information fusion via multimodal inverse problems and different sensors is addressed using a Fisher information analysis approach. The Fisher information measure is inherently additive, and it facilitates an appropriate weighting of the measurement data that is statistically optimal and can hence be useful with reconstruction algorithms in geophysical sensing. Given that there exists proper knowledge about the sensor noise statistics, correlations and spectral contents, as well as a correct forward model, the Fisher information is a natural measure of information because it is closely linked to the statistical maximum likelihood principle. To illustrate the concept of data correlation based on statistical Fisher information analysis, two simple and generic examples are employed in electrical resistivity and electromagnetic tomography, which are motivated by geophysical applications, such as tunnel detection. The examples demonstrate that a properly weighted data fusion can be of crucial importance for an ill-posed multimodal inverse problem.
This paper presents two variants of a generalized Jordan's lemma with applications in waveguide theory. As a main application is considered an asymptotic analysis for open waveguide structures with circular geometry. In particular, the generalized Jordan's lemma can be used to justify that field components can be calculated as the sum of discrete and non-discrete modes, i.e., as the sum of residues and an integral along the branch-cut defined by the transversal wavenumber of the exterior domain. An explicit example regarding the axial symmetric TM0 modes of a single core transmission line, wire, or optical fibre is included to demonstrate the associated asymptotic behavior for a typical open waveguide structure.
A Fourier representation of the electric field volume integral equation for an open and cylindrically symmetrical waveguide is given in this paper. The waveguide material is assumed to be isotropic, non-magnetic and with an arbitrary radial variation in the relative permittivity. The Fourier representation yields a system of one-dimensional integral equations, one system for each azimuthal index and where the Fourier variable for the longitudinal direction plays the role of a spectral parameter. The integral equation is of the second kind and has a kernel that is generally discontinuous on the diagonal and singular at the origin. In the axial symmetric case, it can readily be shown that the elements of the matrix kernel belong to an L2-space, and hence that the integral operator is compact and the analytic Fredholm theorem is applicable.
Fundamental upper bounds are given for the plasmonic multipole absorption and scattering of a rotationally invariant dielectric sphere embedded in a lossy surrounding medium. A specialized Mie theory is developed for this purpose and when combined with the corresponding generalized optical theorem, an optimization problem is obtained which is explicitly solved by straightforward analysis. In particular, the absorption cross section is a concave quadratic form in the related Mie (scattering) parameters and the convex scattering cross section can be maximized by using a Lagrange multiplier constraining the absorption to be non-negative. For the homogeneous sphere, the Weierstrass preparation theorem is used to establish the existence and the uniqueness of the plasmonic singularities and explicit asymptotic expressions are given for the dipole and the quadrupole. It is shown that the optimal passive material for multipole absorption and scattering of a small homogeneous dielectric sphere embedded in a dispersive medium is given approximately as the complex conjugate and the real part of the corresponding pole positions, respectively. Numerical examples are given to illustrate the theory, including a comparison with the plasmonic dipole and quadrupole resonances obtained in gold, silver, and aluminum nanospheres based on some specific Brendel-Bormann (BB) dielectric models for these metals. Based on these BB models, it is interesting to note that the metal spheres can be tuned to optimal absorption at a particular size at a particular frequency.
This paper discusses and analyzes the quasistatic optimal plasmonic dipole resonance of a small dielectric particle embedded in a lossy surrounding medium. The optimal resonance at any given frequency is defined by the complex valued dielectric constant that maximizes the absorption of the particle under the quasistatic approximation and a passivity constraint. In particular, for an ellipsoid aligned along the exciting field, the optimal material property is given by the complex conjugate of the pole position associated with the polarizability of the particle. In this paper, we employ the classical Mie theory to analyze this approximation for spherical particles in a lossy surrounding medium. It turns out that the quasistatic optimal plasmonic resonance is valid, provided that the electrical size of the particle is sufficiently small at the same time as the external losses are sufficiently large. Hence, it is important to note that this approximation cannot be used for a lossless medium, and which is also obvious, since the quasistatic optimal dipole absorption becomes unbounded for this case. Moreover, it turns out that the optimal normalized absorption cross sectional area of the small dielectric sphere has a very subtle limiting behavior and is, in fact, unbounded even in full dynamics when both the electrical size and the exterior losses tend to zero at the same time. A detailed analysis is carried out to assess the validity of the quasistatic estimation of the optimal resonance, and numerical examples are included to illustrate the asymptotic results.
This paper describes some preliminary results regarding Time-Domain pulse Reflection (TDR) measurements and modeling performed on the Baltic Cable submarine HVDC link between southern Sweden and northern Germany. The measurements were conducted in collaboration between the Linnaeus University, Lund University, Baltic Cable AB and ABB High Voltage Cables AB, and is part of the research project: “Fundamental wave modeling for signal estimation on lossy transmission lines”. Preliminary results on measurements and modeling are included here, as well as a first numerical study regarding the low-frequency dispersion characteristics of power cables. The numerical study shows that the finite conductivity of the cable lead shield has a great impact on the losses at low frequencies (0-1 kHz), and that the low-frequency asymptotics of the propagation constant is consistent with common propagation models based on the skin-effect.
This paper provides an exact asymptotic analysis regarding the low-frequency dispersion characteristics of a multilayered coaxial cable. A layer-recursive description of the dispersion function is derived that is well suited for asymptotic analysis. The recursion is based on two well-behaved (meromorphic) subdeterminants defined by a perfectly electrically conducting (PEC) and a perfectly magnetically conducting termination, respectively. For an open waveguide structure, the dispersion function is a combination of two such functions, and there is only one branch point that is related to the exterior domain. It is shown that if there is one isolating layer and a PEC outer shield, then the classical Weierstrass preparation theorem can be used to prove that the low-frequency behavior of the propagation constant is governed by the square root of the complex frequency, and an exact analytical expression for the dominating term of the asymptotic expansion is derived. It is furthermore shown that the same asymptotic expansion is valid to its lowest order even if the outer shield has finite conductivity and there is an infinite exterior region with finite nonzero conductivity. As a practical application of the theory, a high-voltage direct current (HVDC) power cable is analyzed and a numerical solution to the dispersion relation is validated by comparisons with the asymptotic analysis. The comparison reveals that the low-frequency dispersion characteristics of the power cable is very complicated and a first-order asymptotic approximation is valid only at extremely low frequencies (below 1 Hz). It is noted that the only way to come to this conclusion is to actually perform the asymptotic analysis. Hence, for practical modeling purposes, such as with fault localization, an accurate numerical solution to the dispersion relation is necessary and the asymptotic analysis is useful as a validation tool.
This paper presents a general framework for sensitivity analysis for fewparameter inverse problems using the Fisher information and the Cram´er-Rao bound. In particular, the one-dimensional inverse problem of estimating the dispersive parameters of an inhomogeneous dielectric layer with linear spatial variation is studied. The analysis technique is particularly well-suited for inverse problems using few parameters, and it is anticipated that the framework may be used as a basis for extensive numerical investigations and physical interpretations. The ill-posedness of the inverse problem can be explicitely quantified by using the Fisher information analysis. As an example, the sensitivity analysis is used together with asymptotic theory to show that the inverse problem becomes extremely ill-posed when the linear spatial variation vanishes.
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.
This paper gives a report of an ongoing research to develop parametric boundary integral equations for helical structures and their application in the computation of induced currents and losses in three-phase power cables. The proposed technique is formulated in terms of the Electric Field Integral Equation (EFIE) or the Magnetic Field Integral Equation (MFIE) for a penetrable object together with the appropriate periodic Green's functions and a suitable parameterization of the helical structure. A simple and efficient numerical scheme is proposed for the computation of the impedance matrix in the Method of Moments (MoM) which is based on a multi-resolution 4-D FFT computation followed by polynomial extrapolation. Numerical examples are included demonstrating that the singular integrals have almost linear convergence and hence that linear or quadratic extrapolation can be used to yield accurate results.
The ISTIMES project, funded by the European Commission in the frame of a joint Call "ICT and Security" of the Seventh Framework Programme, is presented and preliminary research results are discussed. The main objective of the ISTIMES project is to design, assess and promote an Information and Communication Technologies (ICT)-based system, exploiting distributed and local sensors, for non-destructive electromagnetic monitoring of critical transport infrastructures. The integration of electromagnetic technologies with new ICT information and telecommunications systems enables remotely controlled monitoring and surveillance and real time data imaging of the critical transport infrastructures. The project exploits different non-invasive imaging technologies based on electromagnetic sensing (optic fiber sensors, Synthetic Aperture Radar satellite platform based, hyperspectral spectroscopy, Infrared thermography, Ground Penetrating Radar-, low-frequency geophysical techniques, Ground based systems for displacement monitoring). In this paper, we show the preliminary results arising from the GPR and infrared thermographic measurements carried out on the Musmeci bridge in Potenza, located in a highly seismic area of the Apennine chain (Southern Italy) and representing one of the test beds of the project.
The thorax models for pre-term babies are developed based on the CT scans from new-borns and their effect on image reconstruction is evaluated in comparison with other available models.
Objective: Electrical impedance tomography (EIT) is a functional imaging technique in which cross-sectional images of structures are reconstructed based on boundary trans-impedance measurements. Continuous functional thorax monitoring using EIT has been extensively researched. Increasing the number of electrodes, number of planes and frame rate may improve clinical decision making. Thus, a limiting factor in high temporal resolution, 3D and fast EIT is the handling of the volume of raw impedance data produced for transmission and its subsequent storage. Owing to the periodicity (i.e. sparsity in frequency domain) of breathing and other physiological variations that may be reflected in EIT boundary measurements, data dimensionality may be reduced efficiently at the time of sampling using compressed sensing techniques. This way, a fewer number of samples may be taken. Approach: Measurements using a 32-electrode, 48-frames-per-second EIT system from 30 neonates were post-processed to simulate random demodulation acquisition method on 2000 frames (each consisting of 544 measurements) for compression ratios (CRs) ranging from 2 to 100. Sparse reconstruction was performed by solving the basis pursuit problem using SPGL1 package. The global impedance data (i.e. sum of all 544 measurements in each frame) was used in the subsequent studies. The signal to noise ratio (SNR) for the entire frequency band (0 Hz-24 Hz) and three local frequency bands were analysed. A breath detection algorithm was applied to traces and the subsequent errorrates were calculated while considering the outcome of the algorithm applied to a down-sampled and linearly interpolated version of the traces as the baseline. Main results: SNR degradation was generally proportional with CR. The mean degradation for 0 Hz-8 Hz (of interest for the target physiological variations) was below similar to 15 dB for all CRs. The error-rates in the outcome of the breath detection algorithm in the case of decompressed traces were lower than those associated with the corresponding down-sampled traces for CR >= 25, corresponding to sub-Nyquist rate for breathing frequency. For instance, the mean error-rate associated with CR = 50 was similar to 60% lower than that of the corresponding down-sampled traces. Significance: To the best of our knowledge, no other study has evaluated the applicability of compressive sensing techniques on raw boundary impedance data in EIT. While further research should be directed at optimising the acquisition and decompression techniques for this application, this contribution serves as the baseline for future efforts.
We show how the best passive approximation to a given target material or structure can be found byconvex optimization. The approach is based on a representation of positive real functions, where some of the parameters can be given physical relevance by comparison to low- and high-frequency asymptotics of the material or structure under study. A number of different optimization problems can be formulated, which generalizes previous approaches using sum rules.
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.
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.
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.
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.
We prove that any linear operator with kernel in a Gelfand-Shilov space is a composition of two operators with kernels in the same Gelfand-Shilov space. We also give links on numerical approximations for such compositions. We apply these composition rules to establish Schatten-von Neumann properties for such operators.
An experimental approach to investigate the forward scattering sum rule for periodic structures is presented. This approach allows an upper bound on the total cross section integrated over a bandwidth from a simple static problem to be found. Based on energy conservation, the optical theorem is used to construct a relation between the total cross section and the forward scattering of periodic structures as well as single scatterers inside a parallel-plate waveguide. Dynamic measurements are performed using a parallel-plate waveguide and a parallel-plate capacitor is utilized to find the static polarizability. Convex optimization is introduced to identify the total cross section in the dynamic measurements and estimate an optimal lower bound on the polarizability for objects. The results show that the interactions between the electromagnetic field and an object over all wavelengths are given by the static polarizability of the object.