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Linear modeling of multidimensional non-gaussian processes using cumulants

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Abstract

Extending the notion of second-order correlations, we define thecumulants of stationary non-Gaussian random fields, and demonstrate their potential for modeling and reconstruction of multidimensional signals and systems. Cumulants and their Fourier transforms calledpolyspectra preserve complete amplitude and phase information of a multidimensional linear process, even when it is corrupted by additive colored Gaussian noise of unknown covariance function. Relying on this property, phase reconstruction algorithms are developed using polyspectra, which can be computed via a 2-D FFT-based algorithm. Additionally, consistent ARMA parameter estimators are derived for identification of linear space-invariant multidimensional models which are driven by unobservable, i.i.d., non-Gaussian random fields. Contrary to autocorrelation based multidimensional modeling approaches, when cumulants are employed, the ARMA model is allowed to be non-minimum phase, asymmetric non-causal or non-separable.

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References

  • Bartelt, H., Lohmann, A.W., and Wirnitzer, B. 1984. Phase and amplitude recovery from bispectra.Applied Optics. 23:3121–29.

    Google Scholar 

  • Brillinger, D.R. 1965. Introduction to Polyspectra.Ann. Math. Statist. 36:1351–74.

    Google Scholar 

  • Brillinger, D.R., and Rosenblatt, M. 1967. Computation and interpretation ofk-th order spectra. InSpectral Analysis of Time Series. Edited by B. Harris. New York: Wiley. 189–232.

    Google Scholar 

  • Capon, J. 1969. High-resolution frequency-wavenumber spectral analysis.Proc. IEEE, 57:1408–18.

    Google Scholar 

  • Chellappa, R., and Kashyap, R.L. 1982. Digital Image Restoration Using Spatial Interaction Models.IEEE Trans. ASSP. 461–472.

  • Dianat, S.A., and Raghuveer, M.R. 1989. Two-dimensional non-minimum phase signal reconstruction.Proc. Workshop on Higher-Order Spectral Analysis. (June) 112–117, Vail, CO.

  • Dudgeon, D.E., and Mersereau, R.M. 1984.Multidimensional digital signal processing. Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • Giannakis, G.B., 1987. Cumulants: A powerful tool in signal processing.Proc. IEEE. 74, (Sept.):1333–4.

    Google Scholar 

  • Giannakis, G.B. 1989. Signal reconstruction from multiple correlations: Frequency- and time-domain approaches.J. Opt. Soc. Amer., (May) 682–697.

  • Giannakis, G.B., and Delopoulos, A. 1989. Non-parametric estimation of autocorrelation and spectra using cumulants and polyspectra. University of Virginia Report No. UVA/192/444b/EE89/113, (June); also submitted toIEEE Trans. ASSP.

  • Giannakis, G.B., and Swami, A. 1987. New Results on State-Space and Input-Output Identification of Non-Gaussian Processes using Cumulants.Proc. SPIE-87, 826,Advanced Algorithms and Architectures for Signal Processing II, (Aug.):199–204, San Diego, CA.

  • Giannakis, G.B., and Swami, A. 1988. On Estimating Non-Causal Non-Minimum Phase ARMA Models of Non-Gaussian Processes.Proc. IV IEEE ASSP Workshop on Spectrum Estimation and Modeling, 187–192, Minneapolis, MN, (Aug.); to appear inIEEE Trans. on ASSP, March 1990.

  • Hall, T.E. and Wilson, S.G. 1989. Stochastic image modeling using cumulants with application to predictive image coding.Proc. Workshop on Higher-Order Spectral Analysis, (June):239–244, Vail, CO.

  • Hall, E., Giannakis, G.B., and Wilson, S.G. 1989. Predictive image coding using cumulant based causal and non-causal models.Proc. of Conf. on Info. Sciences and Systems, Johns Hopkins Univ., Baltimore, (March):413–418.

  • Huber, P.J., Kleiner, B., Gasser, T., and Dumermuth, G. 1971. Statistical methods for investigating phase relations in stationary stochastic processes.IEEE Trans. on Audio and Electroacoustics. AU-19, (March) 78–86.

    Google Scholar 

  • IEEE Trans. Biomed. Engg. 1981. Special Issue on Computerized Medical Imaging, BME-28, Feb.

  • Jain, A.K. 1989.Fundamentals of Digital Image Processing. Prentice-Hall.

  • Kashyap, R.L. 1984. Characterization and Estimation of Two-Dimensional ARMA Models.IEEE Trans. Info Theory. IT-30: 736–45.

    Google Scholar 

  • Kashyap, R.L., and Chellappa, R. 1983. Estimation and choice of neighbors in spatial interaction models of images.IEEE Trans. Info Theory. 60–72.

  • Kashyap, R.L., Chellappa, R., and Khotanzad, A. 1982. Texture classification using features derived from random field models.Pattern Recognition Letters. 43–50.

  • Kung, S.Y., Arun, K.S., and D.V. Bhaskar Rao. 1983. State Space and SVD Based Approximation Methods for the Harmonic Retrieval Problem.J. Opti. Soc. Amer. 73:1799–1811.

    Google Scholar 

  • Lii, K.S., and Helland, K.N. 1981. Cross-bispectrum computation and variance estimation.ACM Trans. Math Software. 7:284–94.

    Google Scholar 

  • Lii, K.S., and Rosenblatt, M. 1982. Deconvolution and estimation of transfer function phase and coefficients for non-Gaussian linear processes.Ann. Stat. 10:1195–1208.

    Google Scholar 

  • Lii, K.S., Helland, K.N., and Rosenblatt, M. 1982. Estimating three-dimensional energy transfer in isotropic turbulence.J. Time Series Analysis. 1.

  • Lohmann, A.W., and Wirnitzer, B., 1984. Triple Correlations.Proc. IEEE. 72:889–901.

    Google Scholar 

  • Lohmann, A.W., Weigelt, G., and Wirnitzer, B. 1983. Speckle Masking in Astronomy: Triple Correlation Theory and Applications.Applied Optics. 22:4028–4037.

    Google Scholar 

  • Matsuoka, T., and Ulrych, T.J. 1984. Phase estimation using the bispectrum.Proc. IEEE. 72:1403–1411.

    Google Scholar 

  • Nikias, C.L., and Raghuveer, M. 1987. Bispectrum Estimation: A Digital Signal Processing Framework.Proc. IEEE. 75:869–891.

    Google Scholar 

  • Northcott, M.J., Ayers, G.R. and Dainty, J.C. 1988. Algorithms for image reconstruction from phase-limited data using triple correlation.J. Opt. Soc. Amer. 5:986–992.

    Google Scholar 

  • Oppenheim, A.V., and Lim, J.S. 1981. The importance of phase in signals.Proc. IEEE, 69:529–41.

    Google Scholar 

  • Oppenheim, A.V., and Schafer, R.W. 1975.Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • Pan, R. and Nikias, C.L. 1988. The complex cepstrum of higher-order cumulants and non-minimum phase system identification.IEEE Trans. on ASSP. 186–205.

  • Porat, B., and Friedlander, B. 1988. Optimal Estimation of MA and ARMA Parameters of Non-Gaussian Processes From Higher-Order Cumulants.Proc. IV IEEE ASSP Workshop on Spectrum Estimation and Modeling. Minneapolis, MN. (Aug.) 208–212.

  • Priestley, M.B. 1981. Spectral Analysis and Time Series.Academic Press. London.

    Google Scholar 

  • Proc. IEEE. 1983. Special Issue on Computerized Tomography. 71 (March).

  • Raghuveer, M.R., and Dianat, S.A. 1988. Detection of non-linear phase coupling in multidimensional stochastic processes.Int. Conf. on Adv. in Contr. and Comm. October. 729–732. Baton Rouge, LA.

  • Raghuveer, M.R., Dianat, S.A., and Sundaramoorthy, G. 1988. Reconstruction of non-minimum phase multidimensional signals using the bispectrum.Proc. III SPIE Conf. on Visual Communication and Image Processing. (Nov.) Cambridge, MA.

  • Robinson, E.A. 1982. Spectral Approach to Geophysical Inversion by Lorentz, Fourier, and Radon Transforms.Proc. IEEE, Special Issue on Spectral Estimation. (Sept.) 1039–1054.

  • Rosenblatt, M. 1985.Stationary Sequences and Random Fields. Boston: Birkhauser.

    Google Scholar 

  • Sadler, B.M. 1989. Shift and rotation invariant object reconstruction using the bispectrum.Proc. Workshop on Higher-Order Spectral Analysis. (June) 106–111, Vail, CO.

  • Sadler, B., and Giannakis, G.B. 1989. Image sequence analysis and reconstruction from the bispectrum.Proc. of Conf. on Info. Sciences and Systems. Johns Hopkins Univ., Baltimore. (March) 242–246.

    Google Scholar 

  • Swami, A. 1988.System Identification Using Cumulants. Ph.D. Dissertation, University of Southern California, Los Angeles, (Oct.).

    Google Scholar 

  • Swami, A., and Giannakis, G.B. 1988. ARMA Modeling and Phase Reconstruction of Multidimensional Non-Gaussian Processes Using Cumulants.Proc. ICASSP-88, New York, NY, (April) 729–732.

  • Swami, A., and Mendel, J.M. 1987. Adaptive System Identification Using Cumulants.Proc. ICASSP-88, (April) 2248–51, New York, NY; alsoUSC-SIPI Report-117, 1987. University of Southern California, Los Angeles, (Dec.).

  • Swami, A. and Mendel, J.M. 1988a. Cumulant Based Approach to the Harmonic Retrieval Problem.Proc. ICASSP-88. New York, NY, (April) 2264–67.

  • Swami, A., and Mendel, J.M. 1988b. ARMA Parameter Estimation Using Only Output Cumulants.Proc. IV IEEE ASSP Workshop on Spectrum Estimation and Modeling. Minneapolis, MN, (Aug.) 193–198.

  • Swami, A., Giannakis, G.B., and Mendel, J.M. 1989. Modeling and Parameter Estimation of Multidimensional Non-Gaussian Processes Using Cumulants.USC-SIPI Report #147, University of Southern California, (Aug.).

  • Tekalp, A.M., and Erdem, A.T. 1989a. Two-dimensional higher-order spectrum factorization with applications in non-Gaussian image modeling.Proc. Workshop on Higher-Order Spectral Analysis. (June) 186–190, Vail, CO.

  • Tekalp, A.M. and Erdem, A.T. 1989b. Higher-order spectrum factorization in one and two dimensions with applications in signal modeling and non-minimum phase system identification.IEEE Trans. on ASSP. (Oct.) (to appear).

  • Tekalp, A.M., Kaufman, H., and Woods, J.W. 1986. Identification of image and blur parameters for the restoration of non-causal filters.IEEE Trans. ASSP. 34:963–972.

    Google Scholar 

  • Tsatsanis, M., and Giannakis, G.B. 1989. Object detection and classification using matched filtering and higher-order statistics.Proc. VI Workshop on Multi-dimensional Signal Processing. (Sept.) 32–33, Monterey, CA.

  • Srinivasa, N., Ramakrishnan, K.R., and Rajgopal, K. 1987. Two-dimensional spectral factorizaton: a Radon transform approach.IEEE J. Oceanic Engrg. 12:90–96.

    Google Scholar 

  • Woods, J.W., 1972. Two-dimensional discrete Markovian fields.IEEE Trans. on Info. Theory. IT-18:101–109.

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering-Systems, Signal and Image Processing Institute, University of Southern California, 90089-0781, Los Angeles, CA

    A. Swami

  2. Department of Electrical Engineering, University of Virginia, 22901, Charlottesville, VA

    G. B. Giannakis

  3. Department of Electrical Engineering-Systems, Signal and Image Processing Institute, University of Southern California, 90089-0781, Los Angeles, CA

    J. M. Mendel

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  1. A. Swami
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  2. G. B. Giannakis
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  3. J. M. Mendel
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Additional information

This work was performed at the University of Southern California, Los Angeles, under National Science Foundation Grant ECS-8602531 and Naval Ocean Systems Center Contract N6601-85-D-0203. The second author was partly supported by Univ. of Virginia Engr. Research Initiation Grant 6-42410, and HDL Contract 5-25227. Parts of this paper were presented at ICASSP-88, New York, NY, April 1988, and at the IV IEEE ASSP Workshop on Spectrum Estimation and Modeling, Minneapolis, MN, August 1988.

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Swami, A., Giannakis, G.B. & Mendel, J.M. Linear modeling of multidimensional non-gaussian processes using cumulants. Multidim Syst Sign Process 1, 11–37 (1990). https://doi.org/10.1007/BF01812204

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