Robust Estimation of the Discrete Spectrum of Relaxations for Electromagnetic Induction Responses

The electromagnetic induction response of a target can be accurately modeled by a sum of real exponentials. However, it is difficult to obtain the model parameters from measurements when the number of exponentials in the sum is unknown or the terms are strongly correlated. Traditionally, the time constants and residues are estimated by nonlinear iterative search. In this paper, a constrained linear method of estimating the parameters is formulated by enumerating the relaxation parameter space and imposing a nonnegative constraint on the parameters. The resulting algorithm does not depend on a good initial guess to converge to a solution. By using tests on synthetic data and laboratory measurement of known targets, the proposed method is shown to provide accurate and stable estimates of the model parameters.

[1]  G. W. Hohmann,et al.  4. Electromagnetic Theory for Geophysical Applications , 1987 .

[2]  Leonidas J. Guibas,et al.  A metric for distributions with applications to image databases , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[3]  Leslie M. Collins,et al.  An improved Bayesian decision theoretic approach for land mine detection , 1999, IEEE Trans. Geosci. Remote. Sens..

[4]  Gordon K. Smyth,et al.  A Modified Prony Algorithm for Exponential Function Fitting , 1995, SIAM J. Sci. Comput..

[5]  E. Barsoukov,et al.  Impedance spectroscopy : theory, experiment, and applications , 2005 .

[6]  Leslie M. Collins,et al.  Classification of landmine-like metal targets using wideband electromagnetic induction , 2000, IEEE Trans. Geosci. Remote. Sens..

[7]  S. Havriliak,et al.  A complex plane representation of dielectric and mechanical relaxation processes in some polymers , 1967 .

[8]  Stephen P. Boyd,et al.  Enhancing Sparsity by Reweighted ℓ1 Minimization , 2007, 0711.1612.

[9]  J. Weese,et al.  A nonlinear regularization method for the calculation of relaxation spectra , 1993 .

[10]  S. K. Runcorn,et al.  Interpretation theory in applied geophysics , 1965 .

[11]  Carl E. Baum,et al.  On the Singularity Expansion Method for the Solution of Electromagnetic Interaction Problems , 1971 .

[12]  Leslie M. Collins,et al.  Performance of a four parameter model for modeling landmine signatures in frequency domain wideband electromagnetic induction detection systems , 2007, SPIE Defense + Commercial Sensing.

[13]  Kenneth Holmström,et al.  A review of the parameter estimation problem of fitting positive exponential sums to empirical data , 2002, Appl. Math. Comput..

[14]  Martin B. Hansen,et al.  A Nonparametric Bayesian Approach to Inverse Problems , 2003 .

[15]  Stephen P. Boyd,et al.  An Efficient Method for Compressed Sensing , 2007, 2007 IEEE International Conference on Image Processing.

[16]  E. C. Levy.  Complex-curve fitting , 1959, IRE Transactions on Automatic Control.

[17]  T. Sarkar,et al.  Using the matrix pencil method to estimate the parameters of a sum of complex exponentials , 1995 .

[18]  Carl E. Baum,et al.  On the low-frequency natural response of conducting and permeable targets , 1999, IEEE Trans. Geosci. Remote. Sens..

[19]  R. Cole,et al.  Dielectric Relaxation in Glycerol, Propylene Glycol, and n‐Propanol , 1951 .

[20]  Rick Chartrand,et al.  Exact Reconstruction of Sparse Signals via Nonconvex Minimization , 2007, IEEE Signal Processing Letters.

[21]  Thomas H. Bell,et al.  Discriminating capabilities of multifrequency EMI data , 2000, SPIE Optics + Photonics.

[22]  C. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[23]  P. Hansen.  Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

[24]  Emre Ertin,et al.  On the Relation Between Sparse Sampling and Parametric Estimation , 2009, 2009 IEEE 13th Digital Signal Processing Workshop and 5th IEEE Signal Processing Education Workshop.

[25]  Wotao Yin,et al.  Iteratively reweighted algorithms for compressive sensing , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.

[26]  Waymond R. Scott,et al.  Automated, non-metallic measurement facility for testing and development of electromagnetic induction sensors for landmine detection , 2009, Defense + Commercial Sensing.

[27]  G. C.,et al.  Electricity and Magnetism , 1888, Nature.

[28]  Lloyd S. Riggs,et al.  Identification of metallic mine-like objects using low frequency magnetic fields , 2001, IEEE Trans. Geosci. Remote. Sens..

[29]  Robert D. Nowak,et al.  A bound optimization approach to wavelet-based image deconvolution , 2005, IEEE International Conference on Image Processing 2005.

[30]  W. R. Smythe.  Static and Dynamic Electricity , 1989 .

[31]  A Tikhonov,et al.  Solution of Incorrectly Formulated Problems and the Regularization Method , 1963 .

[32]  Carlo Tomasi,et al.  The Earth Mover’s Distance , 2001 .

[33]  K. Cole,et al.  Dispersion and Absorption in Dielectrics I. Alternating Current Characteristics , 1941 .

[34]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[35]  Yogadhish Das,et al.  Limitations in identifying objects from their time-domain electromagnetic induction response , 2002, SPIE Defense + Commercial Sensing.

[36]  A. A. Kaufman,et al.  The theory of inductive prospecting , 2001 .

[37]  Gary D. Sower.  Eddy Current Responses of Canonical Metallic Targets , 2019, Detection and Identification of Visually Obscured Targets.

[38]  Charles L. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[39]  Thomas H. Bell,et al.  Simple phenomenological models for wideband frequency-domain electromagnetic induction , 2001, IEEE Trans. Geosci. Remote. Sens..

[40]  Waymond R. Scott,et al.  Broadband Array of Electromagnetic Induction Sensors for Detecting Buried Landmines , 2008, IGARSS 2008 - 2008 IEEE International Geoscience and Remote Sensing Symposium.

[41]  E. Tuncer,et al.  On dielectric data analysis. Using the Monte Carlo method to obtain relaxation time distribution and comparing non-linear spectral function fits , 2001 .