Multiscale, Statistical Anomaly Detection Analysis and Algorithms for Linearized Inverse Scattering Problems

  • Eric L. Miller
  • Alan S. Willsky
Article

Abstract

In this paper we explore the utility of multiscale and statistical techniques for detecting and characterizing the structure of localized anomalies in a medium based upon observations of scattered energy obtained at the boundaries of the region of interest. Wavelet transform techniques are used to provide an efficient and physically meaningful method for modeling the non-anomalous structure of the medium under investigation. We employ decision-theoretic methods both to analyze a variety of difficulties associated with the anomaly detection problem and as the basis for an algorithm to perform anomaly detection and estimation. These methods allow for a quantitative evaluation of the manner in which the performance of the algorithms is impacted by the amplitudes, spatial sizes, and positions of anomalous areas in the overall region of interest. Given the insight provided by this work, we formulate and analyze an algorithm for determining the number, location, and magnitudes associated with a set of anomaly structures. This approach is based upon the use of a Generalized, M-ary Likelihood Ratio Test to successively subdivide the region as a means of localizing anomalous areas in both space and scale. Examples of our multiscale inversion algorithm are presented using the Born approximation of an electrical conductivity problem formulated so as to illustrate many of the features associated with similar detection problems arising in fields such as geophysical prospecting, ultrasonic imaging, and medical imaging.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin, “Wavelets for the fast solution of second-kind integral equations,” SIAM J. on Scient. Comput., vol. 14, no. 1, 1993, pp. 159–184.Google Scholar
  2. 2.
    M. Bertero, C. De Mol, and E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Problems, vol. 1, 1985, pp. 301–330.Google Scholar
  3. 3.
    M. Bertero, C. De Mol, and E. R. Pike, “Linear inverse problems with discrete data. II: Stability and regularisation,” Inverse Problems, vol. 4, 1988, pp. 573–594.Google Scholar
  4. 4.
    G. Beylkin, R. Coifman, and V. Rokhlin, “Fast wavelet transforms and numerical algorithms I,” Communications on Pure and Applied Mathematics, vol. 44, 1991, pp. 141–183.Google Scholar
  5. 5.
    Y. Bresler, J. A. Fessler, and A. Macovski, “A Bayesian approach to reconstruction from incomplete projections of a multiple object 3D domain,” IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 11, no. 8, 1989, pp. 840–858.Google Scholar
  6. 6.
    K. E. Brewer and S. W. Wheatcraft, “Including multi-scale information in the characterization of hydraulic conductivity distributions,” In E. Foufoula-Georgiou and P. Kumar (eds.), Wavelets in Geophysics, vol. 4 of Wavelet Analysis and its Applications, pp. 213–248. Academic Press, 1994.Google Scholar
  7. 7.
    S. R. Brown, “Transport of fluid and electric current through a single fracture,” Journal of Geophysical Research, vol. 94, no. B7, 1989, pp. 9429–9438.Google Scholar
  8. 8.
    W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method,” IEEE Trans. Medical Imaging, vol. 9, no. 2, pp. 218–225, 1990.Google Scholar
  9. 9.
    W. C. Chew, Waves and Fields in Inhomogeneous Media, New York: Van Nostrand Reinhold, 1990.Google Scholar
  10. 10.
    K. C. Chou, S. A. Golden, and A. S. Willsky, “Multiresolution stochastic models, data fusion and wavelet transforms,” Technical Report LIDS-P-2110, MIT Laboratory for Information and Decision Systems, 1992.Google Scholar
  11. 11.
    K. C. Chou, A. S. Willsky, and R. Nikoukhah, “Multiscale recursive estimation, data fusion, and regularization,” IEEE Trans. Automatic Control, vol. 39, no. 3, 1994, pp. 464–478.Google Scholar
  12. 12.
    K. C. Chou, A. S. Willsky, and R. Nikoukhah, “Multiscale systems, Kalman filters, and Riccati equations,” IEEE Trans. Automatic Control, vol. 39, no. 3, 1994, pp. 479–492.Google Scholar
  13. 13.
    D. J. Crossley and O. G. Jensen, “Fractal velocity models in refraction seismology,” In C. H. Scholtz and B. B. Mandelbrot (eds.), Fractals in Geophysics, pp. 61–76. Birkhauser, 1989.Google Scholar
  14. 14.
    I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Communications on Pure and Applied Mathematics, vol. 41, 1988, pp. 909–996.Google Scholar
  15. 15.
    A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. on Geoscience and Remote Sensing, vol. GE-22, no. 1, 1984, pp. 3–13.Google Scholar
  16. 16.
    A. J. Devaney and G. A. Tsihrintzis, “Maximum likelihood estimation of object location in diffraction tomography,” IEEE Trans. ASSP, vol. 39, no. 3, 1991, pp. 672–682.Google Scholar
  17. 17.
    A. J. Devaney and G. A. Tsihrintzis, “Maximum likelihood estimation of object location in diffraction tomography, part II: Strongly scattering objects,” IEEE Trans. ASSP, vol. 39, no. 6, 1991, pp. 1466–1470.Google Scholar
  18. 18.
    D. C. Dobson, “Estimates on resolution and stabilization for the linearized inverse conductivity problem,” Inverse Problems, vol. 8, 1992, pp. 71–81.Google Scholar
  19. 19.
    D. C. Dobson and F. Santosa, “An image-enhancement technique for electrical impedance tomography,” Inverse Problems, vol. 10, 1994, pp. 317–334.Google Scholar
  20. 20.
    D. C. Dobson and F. Santosa, “Resolution and stability analysis of an inverse problem in electrical impedance tomography: Dependence on the input current patterns,” SIAM J. Appl. Math., vol. 54, no. 6, pp. 1542–1560.Google Scholar
  21. 21.
    P. Flandrin, “Wavelet analysis and synthesis of fractional Brownian motion,” IEEE Trans. Information Theory, vol. 38, no. 2, pp. 910–917.Google Scholar
  22. 22.
    D. G. Gisser, D. Isaacson, and J. C. Newell, “Electric current computed tomography and eigenvalues,” SIAM J. Appl. Math., vol. 50, no. 6, pp. 1623–1634.Google Scholar
  23. 23.
    T. M. Habashy, W. C. Chew, and E. Y. Chow, “Simultaneous reconstruction of permittivity and conductivity profiles in a radially inhomogeneous slab,” Radio Science, vol. 21, no. 4, pp. 635–645.Google Scholar
  24. 24.
    T. M. Habashy, E. Y. Chow, and D. G. Dudley, “Profile inversion using the renormalized source-type integral equation approach,” IEEE Transactions on Antennas and Propagation, vol. 38, no. 5, pp. 668–682.Google Scholar
  25. 25.
    T. M. Habashy, R. W. Groom, and B. R. Spies, “Beyond the Born and Rytov approximations: A nonlinear approach to electromagnetic scattering,” Journal of Geophysical Research, vol. 98, no. B2, pp. 1759–1775.Google Scholar
  26. 26.
    R. F. Harrington, Field Computations by Moment Methods, Macmillan Publ. Co., 1968.Google Scholar
  27. 27.
    J. H. Hippler, H. Ermert, and L. von Bernus, “Broadband holography applied to eddy current imaging using signals with multiplied phases,” Journal of Nondestructive Evaluation, vol. 12, no. 3, pp. 153–162.Google Scholar
  28. 28.
    S. L. Horowitz and T. Pavlidis, “Picture segmentation by a tree traversal algorithm,” Journal of the ACM, vol. 23, no. 2, 1976, pp. 368–388.Google Scholar
  29. 29.
    D. Isaacson, “Distinguishability of conductivities by electrical current computed tomography,” IEEE Trans. on Medical Imaging, vol. MI-5, no. 2, 1986, pp. 91–95.Google Scholar
  30. 30.
    D. Isaacson and M. Cheney, “Current problems in impedance imaging,” In D. Colton, R. Ewing, and W. Rundell (eds.), Inverse Problems in Partial Differential Equations, Ch. 9, pp. 141–149. SIAM, 1990.Google Scholar
  31. 31.
    D. Isaacson and M. Cheney, “Effects of measurement precision and finite numbers of electrodes on linear impedance imaging algorithms,” SIAM J. Appl. Math., vol. 51, no. 6, 1991, 1705–1731.Google Scholar
  32. 32.
    D. L. Jaggard, “On fractal electrodynamics,” In H. N. Kritikos and D. L. Jaggard (eds.), Recent Advances in Electromagnetic Theory, pp. 183–224. Springer-Verlag, 1990.Google Scholar
  33. 33.
    J. M. Lees and P. E. Malin, “Tomographic images of pwave velocity variation at Parkfield, California,” Journal of Geophysical Research, vol. 95, no. B13, 1990, pp. 21,793–21,804.Google Scholar
  34. 34.
    V. Liepa, F. Santosa, and M. Vogelius, “Crack determination from boundary measurements—Reconstruction using experimental data,” Journal of Nondestructive Evaluation, vol. 12, no. 3, 1993, pp. 163–174.Google Scholar
  35. 35.
    S. G. Mallat, “A theory of multiresolution signal decomposition: The wavelet representation,” IEEE Trans. PAMI, vol. 11, no. 7, 1989, pp. 674–693.Google Scholar
  36. 36.
    J. M. Beaulieu and M. Goldberg, “Hierarchy in picture segmentation: A stepwise optimization approach,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 2, 1989, pp. 150–163.Google Scholar
  37. 37.
    E. L. Miller, “The application of multiscale and statistical techniques to the solution of inverse problems,” Technical Report LIDS-TH-2258, MIT Laboratory for Information and Decision Systems, Cambridge, 1994.Google Scholar
  38. 38.
    E. L. Miller, “A scale-recursive, statistically-based method for anomaly characterization in images based upon observations of scattered radiation,” In 1995 IEEE International Conference on Image Processing, 1995.Google Scholar
  39. 39.
    E. L. Miller and A. S. Willsky, “A multiscale approach to sensor fusion and the solution of linear inverse problems,” Applied and Computational Harmonic Analysis, vol. 2, 1995, pp. 127–147.Google Scholar
  40. 40.
    E. L. Miller and A. S. Willsky, “Multiscale, statistically-based inversion scheme for the linearized inverse scattering problem,” IEEE Trans. on Geoscience and Remote Sensing, March 1996, vol. 36, no. 2, pp. 346–357.Google Scholar
  41. 41.
    E. L. Miller and A. S. Willsky, “Wavelet-based, stochastic inverse scattering methods using the extended Born approximation,” In Progress in Electromagnetics Research Symposium, 1995. Seattle, Washington.Google Scholar
  42. 42.
    J. Le Moigne and J. C. Tilton, “Refining image segmentation by integration of edge and region data,” IEEE Trans. on Geoscience and Remote Sensing, vol. 33, no. 3, 1995, pp. 605–615.Google Scholar
  43. 43.
    J. E. Molyneux and A. Witten, “Impedance tomography: imaging algorithms for geophysical applications,” Inverse Problems, vol. 10, 1994, pp. 655–667.Google Scholar
  44. 44.
    D. J. Rossi and A. S. Willsky, “Reconstruction from projections based on detection and estimation of objects- parts I and II: Performance analysis and robustness analysis,” IEEE Trans. on ASSP, vol. ASSP-32, no. 4, 1984, pp. 886–906.Google Scholar
  45. 45.
    K. Sauer, J. Sachs, Jr., and C. Klifa, “Bayesian estimation of 3D objects from few radiographs,” IEEE Trans. Nuclear Science, vol. 41, no. 5, 1994, pp. 1780–1790.Google Scholar
  46. 46.
    A. Schatzberg, A. J. Devaney, and A. J. Witten, “Estimating target location from scattered field data,” Signal Processing, vol. 40, 1994, pp. 227–237.Google Scholar
  47. 47.
    A. H. Tewfick and M. Kim, “Correlation structure of the discrete wavelet coefficients of fractional Brownian motion,” IEEE Trans. Information Theory, vol. 38, no. 2, pp. 904–909.Google Scholar
  48. 48.
    C. Torres-Verdin and T. M. Habashy, “Rapid 2.5-D forward modeling and inversion via a new nonlinear scattering approximation,” Radio Science, 1994, pp. 1051–1079.Google Scholar
  49. 49.
    H. L. Van Trees, Detection, Estimation and Modulation Theory: Part I. New York: John Wiley and Sons, 1968.Google Scholar
  50. 50.
    G. W. Wornell, “A Karhuenen-Loeve-like expansion for 1/fprocesses via wavelets,” IEEE Transactions on Information Theory, vol. 36, 1990, pp. 859–861.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Eric L. Miller
    • 1
  • Alan S. Willsky
    • 2
  1. 1.The Communications and Digital Signal Processing Center, Department of Electrical and Computer Engineering, 235 ForsythNortheastern UniversityBoston
  2. 2.Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer ScienceMassachusetts Institute of TechnologyCambridge

Personalised recommendations