On the multivariate circulant rational covariance extension problem

Partial stochastic realization of periodic processes from finite covariance data leads to the circulant rational covariance extension problem and bilateral ARMA models. In this paper we present a convex optimization-based theory for this problem that extends and modifies previous results by Carli, Ferrante, Pavon and Picci on the AR solution, which have been successfully applied to image processing of textures. We expect that our present results will provide an enhancement of these procedures.

[1]  Anders Lindquist,et al.  Cepstral coefficients, covariance lags, and pole-zero models for finite data strings , 2001, IEEE Trans. Signal Process..

[2]  Anders Lindquist,et al.  Interior point solutions of variational problems and global inverse function theorems , 2007 .

[3]  Francesca P. Carli,et al.  On the Covariance Completion Problem Under a Circulant Structure , 2011, IEEE Transactions on Automatic Control.

[4]  Anders Lindquist,et al.  Identifiability and Well-Posedness of Shaping-Filter Parameterizations: A Global Analysis Approach , 2002, SIAM J. Control. Optim..

[5]  P. Enqvist.  Spectral Estimation by Geometric, Topological and Optimization Methods , 2001 .

[6]  Anders Lindquist,et al.  The Circulant Rational Covariance Extension Problem: The Complete Solution , 2012, IEEE Transactions on Automatic Control.

[7]  Francesca P. Carli,et al.  A Maximum Entropy Solution of the Covariance Extension Problem for Reciprocal Processes , 2011, IEEE Transactions on Automatic Control.

[8]  C. Byrnes,et al.  A Convex Optimization Approach to the Rational Covariance Extension Problem , 1999 .

[9]  Tryphon T. Georgiou,et al.  Realization of power spectra from partial covariance sequences , 1987, IEEE Trans. Acoust. Speech Signal Process..

[10]  BERNARD C. LEVY,et al.  Characterization of Stationary Discrete-Time Gaussian Reciprocal Processes over a Finite Interval , 2002, SIAM J. Matrix Anal. Appl..

[11]  Tryphon T. Georgiou,et al.  Solution of the general moment problem via a one-parameter imbedding , 2005, IEEE Transactions on Automatic Control.

[12]  Francesca P. Carli,et al.  Modelling and Simulation of Images by Reciprocal Processes , 2008, Tenth International Conference on Computer Modeling and Simulation (uksim 2008).

[13]  Anders Lindquist,et al.  From Finite Covariance Windows to Modeling Filters: A Convex Optimization Approach , 2001, SIAM Rev..

[14]  C. Byrnes,et al.  On the partial stochastic realization problem , 1997, IEEE Trans. Autom. Control..

[15]  Ruggero Frezza,et al.  Gaussian reciprocal processes and self-adjoint stochastic differential equations of second order , 1991 .

[16]  A. Krener,et al.  Modeling and estimation of discrete-time Gaussian reciprocal processes , 1990 .

[17]  C. Byrnes,et al.  A complete parameterization of all positive rational extensions of a covariance sequence , 1995, IEEE Trans. Autom. Control..

[18]  B. Musicus,et al.  Maximum entropy pole-zero estimation , 1986, ICASSP '86. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[19]  Tryphon T. Georgiou,et al.  A generalized entropy criterion for Nevanlinna-Pick interpolation with degree constraint , 2001, IEEE Trans. Autom. Control..

[20]  Michele Pavon,et al.  On the Geometry of Maximum Entropy Problems , 2011, SIAM Rev..

[21]  R. Kálmán.  Realization of Covariance Sequences , 1982 .

[22]  C. Byrnes,et al.  The Generalized Moment Problem with Complexity Constraint , 2006 .

[23]  Anders Lindquist,et al.  The Moment Problem for Rational Measures: Convexity in the Spirit of Krein , 2009 .

[24]  Anders Lindquist,et al.  Matrix-valued Nevanlinna-Pick interpolation with complexity constraint: an optimization approach , 2003, IEEE Trans. Autom. Control..

[25]  M. Pavon,et al.  Further Results on the Byrnes-Georgiou-Lindquist Generalized Moment Problem , 2007 .

[26]  Tryphon T. Georgiou,et al.  Kullback-Leibler approximation of spectral density functions , 2003, IEEE Trans. Inf. Theory.

[27]  A. Chiuso,et al.  Reciprocal realization and modeling of textured images , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.