Abstract
We consider the problem of estimating the harmonics of a noisy 2-D signal. The observed data is modeled as a 2-D sinusoidal signal, with either random or deterministic phases, plus additive Gaussian noise ofunknown covariance. Our method utilizes recently defined higher-order statistics, referred to as “mixed-cumulants”, which permit a formulation that is applicable to both the random and deterministic case. In particular, we first estimate the frequencies in each dimension using an overdetermined Yule-Walker “type” approach. Then, the 1-D frequencies are paired using a matching criterion. To support our theory, we examine the performance of the proposed method via simulations.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
J. M. M. Anderson and G. B. Giannakis. “Image motion estimation algorithms using cumulants,”IEEE Trans. on Image Processing, vol. 4, pp. 346–357, 1995.
J. M. M. Anderson, “2-D Single Record Harmonic Retrieval Using Higher-Order Statistics,”Proc. of 27 th Asilomar Conf. on Signals, Systems, and Computers, Pacific Grove, CA, pp. 441–445, November 1993.
J. M. M. Anderson, G. B. Giannakis, and A. Swami, Harmonic retrieval using higher-order statistics: A deterministic formulation,”IEEE Trans. on Signal Processing, in press.
D. R. Brillinger and M. Rosenblatt, “Asymptotic theory of estimates ofkth order spectra,”Spectral Analysis of Time Series, ed. B. Harris, pp. 153–188, Wiley, 1967.
J. A. Cadzow, “Spectral estimation: An overdetermined rational model equation approach,”Proc. of the IEEE, vol. 70, 1982, pp. 907–939.
J. Capon, “High-resolution frequency-wavenumber spectrum analysis,”Proc. of the IEEE, vol. 57, 1969, pp. 1408–1418.
A. Dandawate and G. B. Giannakis, “Nonparametric polyspectral estimators forkth-order (almost) cyclostationary processes,”IEEE Trans. on Information Theory, January 1994.
A. Dandawate and G. B. Giannakis, “Asymptotic theory of mixed time averages andkth-order cyclicmoment and cumulant statistics,”IEEE Trans. on Information Theory, in press.
A. Dandawate and G. B. Giannakis, “Statistical tests for the presence of cyclostationarity,”IEEE Trans. on Signal Processing, in press.
A. Dandawate, “On consistent and asymptotically normal sample estimators for cyclic moments and cumulants,” Proc. of Intl. Conf. on ASSP, vol. IV, 1993, pp. 504–507, Minneapolis, MN.
A. Dandawate,Exploiting Cyclostationarity and Higher-Order Statistics in Signal Processing, Ph.D. thesis, Dept. of Electrical Engineering, University of Virginia, January 1993.
A. Dandawate and G. B. Giannakis, “Ergodic results for non-stationary processes,”Conf. on Information Sciences and Systems, pp. 976–983, March 1991.
R. De Beer et al., “Processing of two-dimensional nuclear magnetic resonance time domain signals,”Signal Processing, vol. 15, 1988, pp. 293–302.
G. B. Giannakis, “Signal reconstruction from multiple correlations: Frequency- and time-domain approaches,”Journal of the Optical Society of America A, pp. 682–697, May 1989.
R. R. Hansen and R. Chellappa, “Noncausal 2-D spectrum estimation for direction finding,”IEEE Trans. on Information Theory, vol. 36, 1990, pp. 108–125.
Y. Hua, “Estimating two-dimensional frequencies by matrix enhancement and matrix pencil,”Proc. of Intl. Conf. on ASSP, pp. 3073–3076, Toronto, Canada, May 1991.
S. M. Kay and S. T. Marple Jr., “Spectrum analysis — a modern perspective,”Proc. of the IEEE, vol. 69, 1981, pp. 1380–1418.
S. Kay and R. Nekovei, “An efficient two-dimensional frequency estimator,”IEEE Trans. on ASSP, vol. 38, 1990, pp. 1807–1809.
R. Kumaresan and A. K. Shaw, “An exact least squares fitting technique for two-dimensional frequency wavenumber estimation,”Proc. of the IEEE, vol. 74, 1986, pp. 606–607.
R. Kumaresan and D. W. Tufts, “A two-dimensional technique for frequency-wavenumber estimation,”Proc. of the IEEE, vol. 69, 1981, pp. 1515–1517.
S. Y. Kung, K. S. Arun, and D. V. Bhaskar Rao, “State-space and singular-value decomposition-based approximation methods for the harmonic retrieval problem,”J. of the Optical Society of America A, vol. 73, 1983, pp. 1799–1811.
F. Li and R. J. Vaccaro, “On frequency-wavenumber estimation by state-space realization,”IEEE Trans. on Circuits and Systems, vol. 38, 1991, pp. 800–804.
J. S. Lim,Two-Dimensional Signal and Image Processing, Prentice Hall, 1990.
L. Ljung,System Identification: Theory for the User, Prentice Hall, 1987.
S. L. Marple,Digital Spectral Analysis with Applications, Chap. 16, Englewood Cliffs, NJ: Prentice-Hall, 1988.
J. H. McClellan, “Multidimensional spectral estimation,”Proc. of the IEEE, vol. 70, 1982, pp. 1029–1037.
D. V. B. Rao and S. Y. Kung, “A state space approach for the 2-D harmonic retrieval problem,”Proc. of Intl. Conf. on ASSP, pp. 4.10.1–4.10.3, San Diego, CA, April 1984.
A. Swami, “Pitfalls in polyspectra,”Proc. of Intl. Conf. on ASSP, pp. IV97–IV100, Minneapolis, MN, April 1993.
A. Swami and J. M. Mendel, “Cumulant-based approach to the harmonic retrieval problem,”IEEE Trans. on SP, vol. 39, 1991, pp. 1099–1109.
A. Swami, G. B. Giannakis and J. M. Mendel, “Linear Modeling of multidimensional non-Gaussian processes using cumulants,”Multidimensional Systems and Signal Processing, vol. 1, 1990, pp. 11–37.
A. Swami,System Identification Using Cumulants, Ph.D. thesis, University of Southern California, October 1988.
X. Zhang, “Two-dimensional harmonic retrieval and its time-domain analysis technique,”IEEE Trans. on Information Theory, vol. 37, 1991, pp. 1185–1188.
G. Zhou and G. B. Giannakis, “Fourier Series moment polyspectra for mixed processes and coupled harmonics,”IEEE Trans. on Information Theory, December 1993 (submitted); see also “Estimating coupled harmonics in additive and multiplicative noise,”Proc. of 27th Asilomar Conf. on Signals, Systems, and Computers, Pacific Grove, CA, pp. 1250–1254, November 1993.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Anderson, J.M.M., Giannakis, G.B. 2-D harmonic retrieval using higher-order statistics. Multidim Syst Sign Process 6, 313–331 (1995). https://doi.org/10.1007/BF00983558
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00983558