The Experts below are selected from a list of 183552 Experts worldwide ranked by ideXlab platform
Giorgio Picci - One of the best experts on this subject based on the ideXlab platform.
-
on the multivariate circulant rational covariance Extension Problem
Conference on Decision and Control, 2013Co-Authors: Anders Lindquist, Chiara Masiero, Giorgio PicciAbstract: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.
-
The Circulant Rational Covariance Extension Problem: The Complete Solution
IEEE Transactions on Automatic Control, 2013Co-Authors: Anders G. Lindquist, Giorgio PicciAbstract:The rational covariance Extension Problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion Problem to construct an infinite-dimensional positive-definite Toeplitz matrix the northwest corner of which is given. The circulant rational covariance Extension Problem considered in this paper is a modification of this Problem to partial stochastic realization of periodic stationary process, which are better represented on the discrete unit circle rather than on the discrete real line . The corresponding matrix completion Problem then amounts to completing a finite-dimensional Toeplitz matrix that is circulant. Another important motivation for this Problem is that it provides a natural approximation, involving only computations based on the fast Fourier transform, for the ordinary rational covariance Extension Problem, potentially leading to an efficient numerical procedure for the latter. The circulant rational covariance Extension Problem is an inverse Problem with infinitely many solutions in general, each corresponding to a bilateral ARMA representation of the underlying periodic process. In this paper, we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them. A procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented.
-
CDC - On the multivariate circulant rational covariance Extension Problem
52nd IEEE Conference on Decision and Control, 2013Co-Authors: Anders Lindquist, Chiara Masiero, Giorgio PicciAbstract: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.
-
The Circulant Rational Covariance Extension Problem: The Complete Solution
arXiv: Optimization and Control, 2012Co-Authors: Anders Lindquist, Giorgio PicciAbstract:The rational covariance Extension Problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion Problem to construct an infinite-dimensional positive-definite Toeplitz matrix the north-west corner of which is given. The circulant rational covariance Extension Problem considered in this paper is a modification of this Problem to partial stochastic realization of reciprocal and periodic stationary process, which are better represented on the discrete unit circle $\mathbb{Z}_{2N}$ rather than on the discrete real line $\mathbb{Z}$. The corresponding matrix completion Problem then amounts to completing a finite-dimensional Toeplitz matrix that is circulant. Another important motivation for this Problem is that it provides a natural approximation, involving only computations based on the fast Fourier transform, for the ordinary rational covariance Extension Problem, potentially leading to an efficient numerical procedure for the latter. The circulant rational covariance Extension Problem is an inverse Problem with infinitely many solutions in general, each corresponding to a bilateral ARMA representation of the underlying periodic (reciprocal) process. In this paper we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them. A procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented.
-
a maximum entropy solution of the covariance Extension Problem for reciprocal processes
IEEE Transactions on Automatic Control, 2011Co-Authors: Francesca Paola Carli, Augusto Ferrante, Michele Pavon, Giorgio PicciAbstract:Stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of Markov random fields restricted to one dimension. Nonstationary reciprocal processes have been extensively studied in the past especially by Jamison et al. The specialization of the nonstationary theory to the stationary case, however, does not seem to have been pursued in sufficient depth in the literature. Stationary reciprocal processes (and reciprocal stochastic models) are potentially useful for describing signals which naturally live in a finite region of the time (or space) line. Estimation or identification of these models starting from observed data seems still to be an open Problem which can lead to many interesting applications in signal and image processing. In this paper, we discuss a class of reciprocal processes which is the acausal analog of auto-regressive (AR) processes, familiar in control and signal processing. We show that maximum likelihood identification of these processes leads to a covariance Extension Problem for block-circulant covariance matrices. This generalizes the famous covariance band Extension Problem for stationary processes on the integer line. As in the usual stationary setting on the integer line, the covariance Extension Problem turns out to be a basic conceptual and practical step in solving the identification Problem. We show that the maximum entropy principle leads to a complete solution of the Problem.
Anders Lindquist - One of the best experts on this subject based on the ideXlab platform.
-
on the multivariate circulant rational covariance Extension Problem
Conference on Decision and Control, 2013Co-Authors: Anders Lindquist, Chiara Masiero, Giorgio PicciAbstract: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.
-
CDC - On the multivariate circulant rational covariance Extension Problem
52nd IEEE Conference on Decision and Control, 2013Co-Authors: Anders Lindquist, Chiara Masiero, Giorgio PicciAbstract: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.
-
The Circulant Rational Covariance Extension Problem: The Complete Solution
arXiv: Optimization and Control, 2012Co-Authors: Anders Lindquist, Giorgio PicciAbstract:The rational covariance Extension Problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion Problem to construct an infinite-dimensional positive-definite Toeplitz matrix the north-west corner of which is given. The circulant rational covariance Extension Problem considered in this paper is a modification of this Problem to partial stochastic realization of reciprocal and periodic stationary process, which are better represented on the discrete unit circle $\mathbb{Z}_{2N}$ rather than on the discrete real line $\mathbb{Z}$. The corresponding matrix completion Problem then amounts to completing a finite-dimensional Toeplitz matrix that is circulant. Another important motivation for this Problem is that it provides a natural approximation, involving only computations based on the fast Fourier transform, for the ordinary rational covariance Extension Problem, potentially leading to an efficient numerical procedure for the latter. The circulant rational covariance Extension Problem is an inverse Problem with infinitely many solutions in general, each corresponding to a bilateral ARMA representation of the underlying periodic (reciprocal) process. In this paper we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them. A procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented.
-
a convex optimization approach to the rational covariance Extension Problem
Siam Journal on Control and Optimization, 1999Co-Authors: Christopher I Byrnes, Sergei V Gusev, Anders LindquistAbstract:In this paper we present a convex optimization Problem for solving the rational covariance Extension Problem. Given a partial covariance sequence and the desired zeros of the modeling filter, the poles are uniquely determined from the unique minimum of the corresponding optimization Problem. In this way we obtain an algorithm for solving the covariance Extension Problem, as well as a constructive proof of Georgiou's seminal existence result and his conjecture, a stronger version of which we have resolved in [Byrnes et al., IEEE Trans. Automat. Control, AC-40 (1995), pp. 1841--1857].
-
On the well-posedness of the rational covariance Extension Problem
Current and Future Directions in Applied Mathematics, 1997Co-Authors: Christopher I Byrnes, Henry Landau, Anders LindquistAbstract:In this paper, we give a new proof of the solution of the rational covariance Extension Problem, an interpolation Problem with historical roots in potential theory, and with recent application in speech synthesis, spectral estimation, stochastic systems theory, and systems identification. The heart of this Problem is to parameterize, in useful systems theoretical terms, all rational, (strictly) positive real functions having a specified window of Laurent coefficients and a bounded degree. In the early 1980’s, Georgiou used degree theory to show, for any fixed “Laurent window”, that to each Schur polynomial there exists, in an intuitive systems-theoretic manner, a solution of the rational covariance Extension Problem. He also conjectured that this solution would be unique, so that the space of Schur polynomials would parameterize the solution set in a very useful form. In a recent paper, this Problem was solved as a corollary to a theorem concerning the global geometry of rational, positive real functions. This corollary also asserts that the solutions are analytic functions of the Schur polynomials.
Giuseppe Liotta - One of the best experts on this subject based on the ideXlab platform.
-
The Partial Visibility Representation Extension Problem
Algorithmica, 2018Co-Authors: Steven Chaplick, Grzegorz Guśpiel, Grzegorz Gutowski, Tomasz Krawczyk, Giuseppe LiottaAbstract:For a graph G , a function $$\psi $$ ψ is called a bar visibility representation of G when for each vertex $$v \in V(G)$$ v ∈ V ( G ) , $$\psi (v)$$ ψ ( v ) is a horizontal line segment ( bar ) and $$uv \in E(G)$$ u v ∈ E ( G ) if and only if there is an unobstructed, vertical, $$\varepsilon $$ ε -wide line of sight between $$\psi (u)$$ ψ ( u ) and $$\psi (v)$$ ψ ( v ) . Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph G , a bar visibility representation of G , additionally, puts the bar $$\psi (u)$$ ψ ( u ) strictly below the bar $$\psi (v)$$ ψ ( v ) for each directed edge ( u , v ) of G . We study a generalization of the recognition Problem where a function $$\psi '$$ ψ ′ defined on a subset $$V'$$ V ′ of V ( G ) is given and the question is whether there is a bar visibility representation $$\psi $$ ψ of G with $$\psi (v) = \psi '(v)$$ ψ ( v ) = ψ ′ ( v ) for every $$v \in V'$$ v ∈ V ′ . We show that for undirected graphs this Problem, and other closely related Problems, is $$\mathsf {NP}$$ NP -complete, but for certain cases involving directed graphs it is solvable in polynomial time.
-
the partial visibility representation Extension Problem
Graph Drawing, 2016Co-Authors: Steven Chaplick, Grzegorz Guśpiel, Grzegorz Gutowski, Tomasz Krawczyk, Giuseppe LiottaAbstract:For a graph G, a function \(\psi \) is called a bar visibility representation of G when for each vertex \(v \in V(G)\), \(\psi (v)\) is a horizontal line segment (bar) and \(uv \in E(G)\) iff there is an unobstructed, vertical, \(\varepsilon \)-wide line of sight between \(\psi (u)\) and \(\psi (v)\). Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph G, a bar visibility representation \(\psi \) of G, additionally, for each directed edge (u, v) of G, puts the bar \(\psi (u)\) strictly below the bar \(\psi (v)\). We study a generalization of the recognition Problem where a function \(\psi '\) defined on a subset \(V'\) of V(G) is given and the question is whether there is a bar visibility representation \(\psi \) of G with \(\psi |V' = \psi '\). We show that for undirected graphs this Problem together with closely related Problems are \(\mathsf {NP}\)-complete, but for certain cases involving directed graphs it is solvable in polynomial time.
-
the partial visibility representation Extension Problem
arXiv: Computational Geometry, 2015Co-Authors: Steven Chaplick, Grzegorz Guśpiel, Grzegorz Gutowski, Tomasz Krawczyk, Giuseppe LiottaAbstract:For a graph $G$, a function $\psi$ is called a \emph{bar visibility representation} of $G$ when for each vertex $v \in V(G)$, $\psi(v)$ is a horizontal line segment (\emph{bar}) and $uv \in E(G)$ iff there is an unobstructed, vertical, $\varepsilon$-wide line of sight between $\psi(u)$ and $\psi(v)$. Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph $G$, a bar visibility representation $\psi$ of $G$, additionally, puts the bar $\psi(u)$ strictly below the bar $\psi(v)$ for each directed edge $(u,v)$ of $G$. We study a generalization of the recognition Problem where a function $\psi'$ defined on a subset $V'$ of $V(G)$ is given and the question is whether there is a bar visibility representation $\psi$ of $G$ with $\psi(v) = \psi'(v)$ for every $v \in V'$. We show that for undirected graphs this Problem together with closely related Problems are \NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time.
Francesca Paola Carli - One of the best experts on this subject based on the ideXlab platform.
-
a maximum entropy solution of the covariance Extension Problem for reciprocal processes
IEEE Transactions on Automatic Control, 2011Co-Authors: Francesca Paola Carli, Augusto Ferrante, Michele Pavon, Giorgio PicciAbstract:Stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of Markov random fields restricted to one dimension. Nonstationary reciprocal processes have been extensively studied in the past especially by Jamison et al. The specialization of the nonstationary theory to the stationary case, however, does not seem to have been pursued in sufficient depth in the literature. Stationary reciprocal processes (and reciprocal stochastic models) are potentially useful for describing signals which naturally live in a finite region of the time (or space) line. Estimation or identification of these models starting from observed data seems still to be an open Problem which can lead to many interesting applications in signal and image processing. In this paper, we discuss a class of reciprocal processes which is the acausal analog of auto-regressive (AR) processes, familiar in control and signal processing. We show that maximum likelihood identification of these processes leads to a covariance Extension Problem for block-circulant covariance matrices. This generalizes the famous covariance band Extension Problem for stationary processes on the integer line. As in the usual stationary setting on the integer line, the covariance Extension Problem turns out to be a basic conceptual and practical step in solving the identification Problem. We show that the maximum entropy principle leads to a complete solution of the Problem.
-
a maximum entropy solution of the covariance Extension Problem for reciprocal processes
arXiv: Optimization and Control, 2011Co-Authors: Francesca Paola Carli, Augusto Ferrante, Michele Pavon, Giorgio PicciAbstract:Stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of Markov random fields restricted to one dimension. Non stationary reciprocal processes have been extensively studied in the past especially by Jamison, Krener, Levy and co-workers. The specialization of the non-stationary theory to the stationary case, however, does not seem to have been pursued in sufficient depth in the literature. Stationary reciprocal processes (and reciprocal stochastic models) are potentially useful for describing signals which naturally live in a finite region of the time (or space) line. Estimation or identification of these models starting from observed data seems still to be an open Problem which can lead to many interesting applications in signal and image processing. In this paper, we discuss a class of reciprocal processes which is the acausal analog of auto-regressive (AR) processes, familiar in control and signal processing. We show that maximum likelihood identification of these processes leads to a covariance Extension Problem for block-circulant covariance matrices. This generalizes the famous covariance band Extension Problem for stationary processes on the integer line. As in the usual stationary setting on the integer line, the covariance Extension Problem turns out to be a basic conceptual and practical step in solving the identification Problem. We show that the maximum entropy principle leads to a complete solution of the Problem.
Steven Chaplick - One of the best experts on this subject based on the ideXlab platform.
-
The Partial Visibility Representation Extension Problem
Algorithmica, 2018Co-Authors: Steven Chaplick, Grzegorz Guśpiel, Grzegorz Gutowski, Tomasz Krawczyk, Giuseppe LiottaAbstract:For a graph G , a function $$\psi $$ ψ is called a bar visibility representation of G when for each vertex $$v \in V(G)$$ v ∈ V ( G ) , $$\psi (v)$$ ψ ( v ) is a horizontal line segment ( bar ) and $$uv \in E(G)$$ u v ∈ E ( G ) if and only if there is an unobstructed, vertical, $$\varepsilon $$ ε -wide line of sight between $$\psi (u)$$ ψ ( u ) and $$\psi (v)$$ ψ ( v ) . Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph G , a bar visibility representation of G , additionally, puts the bar $$\psi (u)$$ ψ ( u ) strictly below the bar $$\psi (v)$$ ψ ( v ) for each directed edge ( u , v ) of G . We study a generalization of the recognition Problem where a function $$\psi '$$ ψ ′ defined on a subset $$V'$$ V ′ of V ( G ) is given and the question is whether there is a bar visibility representation $$\psi $$ ψ of G with $$\psi (v) = \psi '(v)$$ ψ ( v ) = ψ ′ ( v ) for every $$v \in V'$$ v ∈ V ′ . We show that for undirected graphs this Problem, and other closely related Problems, is $$\mathsf {NP}$$ NP -complete, but for certain cases involving directed graphs it is solvable in polynomial time.
-
the partial visibility representation Extension Problem
Graph Drawing, 2016Co-Authors: Steven Chaplick, Grzegorz Guśpiel, Grzegorz Gutowski, Tomasz Krawczyk, Giuseppe LiottaAbstract:For a graph G, a function \(\psi \) is called a bar visibility representation of G when for each vertex \(v \in V(G)\), \(\psi (v)\) is a horizontal line segment (bar) and \(uv \in E(G)\) iff there is an unobstructed, vertical, \(\varepsilon \)-wide line of sight between \(\psi (u)\) and \(\psi (v)\). Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph G, a bar visibility representation \(\psi \) of G, additionally, for each directed edge (u, v) of G, puts the bar \(\psi (u)\) strictly below the bar \(\psi (v)\). We study a generalization of the recognition Problem where a function \(\psi '\) defined on a subset \(V'\) of V(G) is given and the question is whether there is a bar visibility representation \(\psi \) of G with \(\psi |V' = \psi '\). We show that for undirected graphs this Problem together with closely related Problems are \(\mathsf {NP}\)-complete, but for certain cases involving directed graphs it is solvable in polynomial time.
-
the partial visibility representation Extension Problem
arXiv: Computational Geometry, 2015Co-Authors: Steven Chaplick, Grzegorz Guśpiel, Grzegorz Gutowski, Tomasz Krawczyk, Giuseppe LiottaAbstract:For a graph $G$, a function $\psi$ is called a \emph{bar visibility representation} of $G$ when for each vertex $v \in V(G)$, $\psi(v)$ is a horizontal line segment (\emph{bar}) and $uv \in E(G)$ iff there is an unobstructed, vertical, $\varepsilon$-wide line of sight between $\psi(u)$ and $\psi(v)$. Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph $G$, a bar visibility representation $\psi$ of $G$, additionally, puts the bar $\psi(u)$ strictly below the bar $\psi(v)$ for each directed edge $(u,v)$ of $G$. We study a generalization of the recognition Problem where a function $\psi'$ defined on a subset $V'$ of $V(G)$ is given and the question is whether there is a bar visibility representation $\psi$ of $G$ with $\psi(v) = \psi'(v)$ for every $v \in V'$. We show that for undirected graphs this Problem together with closely related Problems are \NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time.
