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Didier Henrion - One of the best experts on this subject based on the ideXlab platform.

  • stable radial distortion calibration by Polynomial Matrix inequalities programming
    arXiv: Optimization and Control, 2014
    Co-Authors: Jan Heller, Didier Henrion, Tomas Pajdla
    Abstract:

    Polynomial and rational functions are the number one choice when it comes to modeling of radial distortion of lenses. However, several extrapolation and numerical issues may arise while using these functions that have not been covered by the literature much so far. In this paper, we identify these problems and show how to deal with them by enforcing nonnegativity of certain Polynomials. Further, we show how to model these nonnegativities using Polynomial Matrix inequalities (PMI) and how to estimate the radial distortion parameters subject to PMI constraints using semidefinite programming (SDP). Finally, we suggest several approaches on how to incorporate the proposed method into the overall camera calibration procedure.

  • convergent relaxations of Polynomial Matrix inequalities and static output feedback
    IEEE Transactions on Automatic Control, 2006
    Co-Authors: Didier Henrion, Jean B Lasserre
    Abstract:

    Using a moment interpretation of recent results on sum-of-squares decompositions of nonnegative Polynomial matrices, we propose a hierarchy of convex linear Matrix inequality (LMI) relaxations to solve nonconvex Polynomial Matrix inequality (PMI) optimization problems, including bilinear Matrix inequality (BMI) problems. This hierarchy of LMI relaxations generates a monotone sequence of lower bounds that converges to the global optimum. Results from the theory of moments are used to detect whether the global optimum is reached at a given LMI relaxation, and if so, to extract global minimizers that satisfy the PMI. The approach is successfully applied to PMIs arising from static output feedback design problems.

  • brief an lmi condition for robust stability of Polynomial Matrix polytopes
    Automatica, 2001
    Co-Authors: Didier Henrion, Denis Arzelier, Dimitry Peaucelle, Michael Sebek
    Abstract:

    A sufficient LMI condition is proposed for checking robust stability of a polytope of Polynomial matrices. It hinges upon two recent results: a new approach to Polynomial Matrix stability analysis and a new robust stability condition for convex polytopic uncertainty. Numerical experiments illustrate that the condition narrows significantly the unavoidable gap between conservative tractable quadratic stability results and exact NP-hard robust stability results.

  • An Efficient Numerical Method for the Discrete-time Symmetric Matrix Polynomial Equation
    1997
    Co-Authors: Didier Henrion, Michael Sebek
    Abstract:

    A numerical procedure is proposed to solve a Matrix Polynomial equation frequently encountered in discrete-time control and signal processing. The algorithm is based on a simple rewriting of the original equation in terms of a reduced Sylvester Matrix. In contrast to previously published methods, it does not make use of elementary Polynomial operations. Moreover and most notably, it is numerically reliable. Basic examples borrowed from control and signal processing literature are aimed at illustrating the simplicity and efficiency of this new numerical method. 1 Introduction We consider the discrete-time symmetric Matrix Polynomial equation A 0 (d \Gamma1 )X(d) + X 0 (d \Gamma1 )A(d) = B(d) (1) where A(d) is a given Polynomial Matrix with real coefficients, the superscript 0 denotes Matrix transpose, B(d) is a given two-sided para-Hermitian Polynomial Matrix with real coefficients such that B(d) = B l (d \Gamma1 ) + B r (d) B l (d) = B 0 r (d) (2) and X(d) is an n-by-..

  • An Efficient Numerical Method For The Discrete Time Symmetric Matrix Polynomial Equation
    1996
    Co-Authors: Didier Henrion
    Abstract:

    A novel numerical procedure is proposed to solve the discrete time symmetric Matrix Polynomial equation A 0 (d \Gamma1 )X(d) + X 0 (d \Gamma1 )A(d) = B(d) frequently encountered in control and signal processing. In contrast to previously published methods, it does not make use of elementary Polynomial operations. The algorithm is based on a simple rewriting of the original equation in terms of reduced Sylvester resultant matrices. It handles all critical cases and namely, is numerically reliable. Some basic examples are provided to illustrate the simplicity and efficiency of the numerical method. 1 Introduction Discrete time symmetric Matrix Polynomial equation of the form A 0 (d \Gamma1 )X(d) + X 0 (d \Gamma1 )A(d) = B(d) (1) where A(d) is a given Polynomial Matrix, the superscript 0 denotes Matrix transpose, B(d) is a given "two-sided" para-Hermitian Polynomial Matrix such that B(d) = B l (d \Gamma1 ) + B r (d) B l (d) = B 0 r (d) (2) and X(d) is an n-by-n ..

Michael Sebek - One of the best experts on this subject based on the ideXlab platform.

  • brief an lmi condition for robust stability of Polynomial Matrix polytopes
    Automatica, 2001
    Co-Authors: Didier Henrion, Denis Arzelier, Dimitry Peaucelle, Michael Sebek
    Abstract:

    A sufficient LMI condition is proposed for checking robust stability of a polytope of Polynomial matrices. It hinges upon two recent results: a new approach to Polynomial Matrix stability analysis and a new robust stability condition for convex polytopic uncertainty. Numerical experiments illustrate that the condition narrows significantly the unavoidable gap between conservative tractable quadratic stability results and exact NP-hard robust stability results.

  • An Efficient Numerical Method for the Discrete-time Symmetric Matrix Polynomial Equation
    1997
    Co-Authors: Didier Henrion, Michael Sebek
    Abstract:

    A numerical procedure is proposed to solve a Matrix Polynomial equation frequently encountered in discrete-time control and signal processing. The algorithm is based on a simple rewriting of the original equation in terms of a reduced Sylvester Matrix. In contrast to previously published methods, it does not make use of elementary Polynomial operations. Moreover and most notably, it is numerically reliable. Basic examples borrowed from control and signal processing literature are aimed at illustrating the simplicity and efficiency of this new numerical method. 1 Introduction We consider the discrete-time symmetric Matrix Polynomial equation A 0 (d \Gamma1 )X(d) + X 0 (d \Gamma1 )A(d) = B(d) (1) where A(d) is a given Polynomial Matrix with real coefficients, the superscript 0 denotes Matrix transpose, B(d) is a given two-sided para-Hermitian Polynomial Matrix with real coefficients such that B(d) = B l (d \Gamma1 ) + B r (d) B l (d) = B 0 r (d) (2) and X(d) is an n-by-..

John G. Mcwhirter - One of the best experts on this subject based on the ideXlab platform.

  • Relevance of Polynomial Matrix decompositions to broadband blind signal separation
    Signal Processing, 2017
    Co-Authors: Soydan Redif, Stephan Weiss, John G. Mcwhirter
    Abstract:

    The Polynomial Matrix EVD (PEVD) is an extension of the conventional eigenvalue decomposition (EVD) to Polynomial matrices. The purpose of this article is to provide a review of the theoretical foundations of the PEVD and to highlight practical applications in the area of broadband blind source separation (BSS). Based on basic definitions of Polynomial Matrix terminology such as parahermitian and paraunitary matrices, strong decorrelation and spectral majorisation, the PEVD and its theoretical foundations will be briefly outlined. The paper then focuses on the applicability of the PEVD and broadband subspace techniques — enabled by the diagonalisation and spectral majorisation capabilities of PEVD algorithms — to define broadband BSS solutions that generalise well-known narrowband techniques based on the EVD. This is achieved through the analysis of new results from three exemplar broadband BSS applications — underwater acoustics, radar clutter suppression, and domain-weighted broadband beamforming — and their comparison with classical broadband methods.

  • Multichannel spectral factorization algorithm using Polynomial Matrix eigenvalue decomposition
    2015 49th Asilomar Conference on Signals Systems and Computers, 2015
    Co-Authors: Zeliang Wang, John G. Mcwhirter, Stephan Weiss
    Abstract:

    In this paper, we present a new multichannel spectral factorization algorithm which can be utilized to calculate the approximate spectral factor of any para-Hermitian Polynomial Matrix. The proposed algorithm is based on an iterative method for Polynomial Matrix eigenvalue decomposition (PEVD). By using the PEVD algorithm, the multichannel spectral factorization problem is simply broken down to a set of single channel problems which can be solved by means of existing one-dimensional spectral factorization algorithms. In effect, it transforms the multichannel spectral factorization problem into one which is much easier to solve.

  • sequential Matrix diagonalization algorithms for Polynomial evd of parahermitian matrices
    IEEE Transactions on Signal Processing, 2015
    Co-Authors: Soydan Redif, Stephan Weiss, John G. Mcwhirter
    Abstract:

    For parahermitian Polynomial matrices, which can be used, for example, to characterize space-time covariance in broadband array processing, the conventional eigenvalue decomposition (EVD) can be generalized to a Polynomial Matrix EVD (PEVD). In this paper, a new iterative PEVD algorithm based on sequential Matrix diagonalization (SMD) is introduced. At every step the SMD algorithm shifts the dominant column or row of the Polynomial Matrix to the zero lag position and eliminates the resulting instantaneous correlation. A proof of convergence is provided, and it is demonstrated that SMD establishes diagonalization faster and with lower order operations than existing PEVD algorithms.

  • maximum energy sequential Matrix diagonalisation for parahermitian matrices
    Asilomar Conference on Signals Systems and Computers, 2014
    Co-Authors: Jamie Corr, John G. Mcwhirter, Stephan Weiss, Keith Thompson, Ian K. Proudler
    Abstract:

    Sequential Matrix diagonalisation (SMD) refers to a family of algorithms to iteratively approximate a Polynomial Matrix eigenvalue decomposition. Key is to transfer as much energy as possible from off-diagonal elements to the diagonal per iteration, which has led to fast converging SMD versions involving judicious shifts within the Polynomial Matrix. Through an exhaustive search, this paper determines the optimum shift in terms of energy transfer. Though costly to implement, this scheme yields an important benchmark to which limited search strategies can be compared. In simulations, multiple-shift SMD algorithms can perform within 10% of the optimum energy transfer per iteration step.

  • an approximate Polynomial Matrix eigenvalue decomposition algorithm for para hermitian matrices
    International Symposium on Signal Processing and Information Technology, 2011
    Co-Authors: Soydan Redif, Stephan Weiss, John G. Mcwhirter
    Abstract:

    In this paper, we propose an algorithm for computing an approximate Polynomial Matrix eigenvalue decomposition (PEVD). The PEVD of a para-Hermitian Matrix yields a factorisation into a Polynomial Matrix product consisting of a spectrally majorised diagonal Matrix that is pre- and post-multiplied by paraunitary (PU) matrices. All current PEVD algorithms, such as the second order sequential best rotation (SBR2) algorithm, perform a factorisation whereby diagonalisation and spectral majorisation are only achieved in approximation. The purpose of this paper is to present a new iterative approach which constitutes a “Householder-like” version of SBR2 and is akin to Tkacenko's approximate EVD (AEVD); however, unlike the AEVD, the proposed method carries out the diagonalisation successively by applying arbitrary-degree, finite impulse response PU matrices. We show an application of our algorithm to the design of signal-adapted PU filter banks for subband coding. Simulation results for the proposed approach show very close agreement with the behaviour of the infinite order principal component filter banks and demonstrate its superiority compared to state-of-the-art algorithms in terms of strong decorrelation and spectral majorisation.

Stephan Weiss - One of the best experts on this subject based on the ideXlab platform.

  • Relevance of Polynomial Matrix decompositions to broadband blind signal separation
    Signal Processing, 2017
    Co-Authors: Soydan Redif, Stephan Weiss, John G. Mcwhirter
    Abstract:

    The Polynomial Matrix EVD (PEVD) is an extension of the conventional eigenvalue decomposition (EVD) to Polynomial matrices. The purpose of this article is to provide a review of the theoretical foundations of the PEVD and to highlight practical applications in the area of broadband blind source separation (BSS). Based on basic definitions of Polynomial Matrix terminology such as parahermitian and paraunitary matrices, strong decorrelation and spectral majorisation, the PEVD and its theoretical foundations will be briefly outlined. The paper then focuses on the applicability of the PEVD and broadband subspace techniques — enabled by the diagonalisation and spectral majorisation capabilities of PEVD algorithms — to define broadband BSS solutions that generalise well-known narrowband techniques based on the EVD. This is achieved through the analysis of new results from three exemplar broadband BSS applications — underwater acoustics, radar clutter suppression, and domain-weighted broadband beamforming — and their comparison with classical broadband methods.

  • investigation of a Polynomial Matrix generalised evd for multi channel wiener filtering
    Asilomar Conference on Signals Systems and Computers, 2016
    Co-Authors: Jamie Corr, Soydan Redif, Stephan Weiss, Ian K. Proudler, Jennifer Pestana, Marc Moonen
    Abstract:

    State-of-the-art narrowband noise cancellation techniques utilise the generalised eigenvalue decomposition (GEVD) for multi-channel Wiener filtering, which can be applied to independent frequency bins in order to achieve broadband processing. Here we investigate the extension of the GEVD to broadband, Polynomial matrices, akin to strategies that have already been developed by McWhirter et. al on the Polynomial Matrix eigenvalue decomposition (PEVD). In our approach we extend the Cholesky method for calculating the scalar GEVD to Polynomial matrices. In this paper we outline our Cholesky-like approach, which utilises recently developed techniques for Polynomial Matrix spectral factorisation and Polynomial Matrix inversion.

  • multiple shift second order sequential best rotation algorithm for Polynomial Matrix evd
    European Signal Processing Conference, 2015
    Co-Authors: Zeliang Wang, John Mcwhirter, Jamie Corr, Stephan Weiss
    Abstract:

    In this paper, we present an improved version of the second order sequential best rotation algorithm (SBR2) for Polynomial Matrix eigenvalue decomposition of para-Hermitian matrices. The improved algorithm is entitled multiple shift SBR2 (MS-SBR2) which is developed based on the original SBR2 algorithm. It can achieve faster convergence than the original SBR2 algorithm by means of transferring more off-diagonal energy onto the diagonal at each iteration. Its convergence is proved and also demonstrated by means of a numerical example. Furthermore, simulation results are included to compare its convergence characteristics and computational complexity with the original SBR2, sequential Matrix diagonal-ization (SMD) and multiple shift maximum element SMD algorithms.

  • Multichannel spectral factorization algorithm using Polynomial Matrix eigenvalue decomposition
    2015 49th Asilomar Conference on Signals Systems and Computers, 2015
    Co-Authors: Zeliang Wang, John G. Mcwhirter, Stephan Weiss
    Abstract:

    In this paper, we present a new multichannel spectral factorization algorithm which can be utilized to calculate the approximate spectral factor of any para-Hermitian Polynomial Matrix. The proposed algorithm is based on an iterative method for Polynomial Matrix eigenvalue decomposition (PEVD). By using the PEVD algorithm, the multichannel spectral factorization problem is simply broken down to a set of single channel problems which can be solved by means of existing one-dimensional spectral factorization algorithms. In effect, it transforms the multichannel spectral factorization problem into one which is much easier to solve.

  • sequential Matrix diagonalization algorithms for Polynomial evd of parahermitian matrices
    IEEE Transactions on Signal Processing, 2015
    Co-Authors: Soydan Redif, Stephan Weiss, John G. Mcwhirter
    Abstract:

    For parahermitian Polynomial matrices, which can be used, for example, to characterize space-time covariance in broadband array processing, the conventional eigenvalue decomposition (EVD) can be generalized to a Polynomial Matrix EVD (PEVD). In this paper, a new iterative PEVD algorithm based on sequential Matrix diagonalization (SMD) is introduced. At every step the SMD algorithm shifts the dominant column or row of the Polynomial Matrix to the zero lag position and eliminates the resulting instantaneous correlation. A proof of convergence is provided, and it is demonstrated that SMD establishes diagonalization faster and with lower order operations than existing PEVD algorithms.

Gilles Villard - One of the best experts on this subject based on the ideXlab platform.

  • computing the rank and a small nullspace basis of a Polynomial Matrix
    International Symposium on Symbolic and Algebraic Computation, 2005
    Co-Authors: Arne Storjohann, Gilles Villard
    Abstract:

    We reduce the problem of computing the rank and a null-space basis of a univariate Polynomial Matrix to Polynomial Matrix multiplication. For an input n x n Matrix of degree, d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension, n and degree, d. If the latter multiplication is done in MM(n,d)= O~(nωd operations, with ω the exponent of Matrix multiplication over K, then the algorithm uses O~MM(n,d) operations in, K. For m x n matrices of rank r and degree d, the cost expression is O(nmr ω-2d). The soft-O notation O~ indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of Matrix Hensel high-order lifting and Matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a K[x]-module.

  • Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix
    2005
    Co-Authors: Arne Storjohann, Gilles Villard
    Abstract:

    We reduce the problem of computing the rank and a nullspace basis of a univariate Polynomial Matrix to Polynomial Matrix multiplication. For an input n x n Matrix of degree d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension n and degree d. If the latter multiplication is done in MM(n,d)=softO(n^omega d) operations, with omega the exponent of Matrix multiplication over K, then the algorithm uses softO(MM(n,d)) operations in K. The softO notation indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of Matrix Hensel high-order lifting and Matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a K[x]-module

  • On the Complexity of Polynomial Matrix Computations
    2003
    Co-Authors: Pascal Giorgi, Claude-pierre Jeannerod, Gilles Villard
    Abstract:

    We study the link between the complexity of Polynomial Matrix multiplication and the complexity of solving other basic linear algebra problems on Polynomial matrices. By Polynomial matrices we mean n x n matrices of degree d over K[x] where K is a commutative field. Under the straight-line program model we show that multiplication is reducible to the problem of computing the coefficient of degree d of the determinant. Conversely, we propose algorithms for minimal approximant computation and column reduction that are based on Polynomial Matrix multiplication; for the determinant, the straight-line program we give also relies on Matrix product over K[x] and provides an alternative to Storjohann's determinant algorithm. We further show that all these problems can be solved in particular in O~(n^w d) operations in K. Here the "soft Oh'' notation O~ indicates some missing log(nd) factors and w is the exponent of Matrix multiplication overK.