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133
Dynamic Textures
, 2002
"... Dynamic textures are sequences of images of moving scenes that exhibit certain stationarity properties in time; these include seawaves, smoke, foliage, whirlwind etc. We present a novel characterization of dynamic textures that poses the problems of modeling, learning, recognizing and synthesizing ..."
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Cited by 377 (18 self)
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Dynamic textures are sequences of images of moving scenes that exhibit certain stationarity properties in time; these include seawaves, smoke, foliage, whirlwind etc. We present a novel characterization of dynamic textures that poses the problems of modeling, learning, recognizing and synthesizing dynamic textures on a firm analytical footing. We borrow tools from system identification to capture the "essence" of dynamic textures; we do so by learning (i.e. identifying) models that are optimal in the sense of maximum likelihood or minimum prediction error variance. For the special case of secondorder stationary processes, we identify the model suboptimally in closedform. Once learned, a model has predictive power and can be used for extrapolating synthetic sequences to infinite length with negligible computational cost. We present experimental evidence that, within our framework, even lowdimensional models can capture very complex visual phenomena.
Krylov Projection Methods For Model Reduction
, 1997
"... This dissertation focuses on efficiently forming reducedorder models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov p ..."
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Cited by 213 (3 self)
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This dissertation focuses on efficiently forming reducedorder models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reducedorder models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczosbased methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also developed to form a complete modelreduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multipleinput multipleoutput systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees.
An Implementation of the LookAhead Lanczos Algorithm for NonHermitian Matrices Part I
, 1991
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Multiresolution markov models for signal and image processing
 Proceedings of the IEEE
, 2002
"... This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coheren ..."
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Cited by 153 (17 self)
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This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for selfsimilar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden
Iterative Solution of Linear Systems
 Acta Numerica
, 1992
"... this paper is as follows. In Section 2, we present some background material on general Krylov subspace methods, of which CGtype algorithms are a special case. We recall the outstanding properties of CG and discuss the issue of optimal extensions of CG to nonHermitian matrices. We also review GMRES ..."
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Cited by 130 (8 self)
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this paper is as follows. In Section 2, we present some background material on general Krylov subspace methods, of which CGtype algorithms are a special case. We recall the outstanding properties of CG and discuss the issue of optimal extensions of CG to nonHermitian matrices. We also review GMRES and related methods, as well as CGlike algorithms for the special case of Hermitian indefinite linear systems. Finally, we briefly discuss the basic idea of preconditioning. In Section 3, we turn to Lanczosbased iterative methods for general nonHermitian linear systems. First, we consider the nonsymmetric Lanczos process, with particular emphasis on the possible breakdowns and potential instabilities in the classical algorithm. Then we describe recent advances in understanding these problems and overcoming them by using lookahead techniques. Moreover, we describe the quasiminimal residual algorithm (QMR) proposed by Freund and Nachtigal (1990), which uses the lookahead Lanczos process to obtain quasioptimal approximate solutions. Next, a survey of transposefree Lanczosbased methods is given. We conclude this section with comments on other related work and some historical remarks. In Section 4, we elaborate on CGNR and CGNE and we point out situations where these approaches are optimal. The general class of Krylov subspace methods also contains parameterdependent algorithms that, unlike CGtype schemes, require explicit information on the spectrum of the coefficient matrix. In Section 5, we discuss recent insights in obtaining appropriate spectral information for parameterdependent Krylov subspace methods. After that, 4 R.W. Freund, G.H. Golub and N.M. Nachtigal
Krylov Subspace Techniques for ReducedOrder Modeling of Nonlinear Dynamical Systems
 Appl. Numer. Math
, 2002
"... Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of largescale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bilinearization method, which extends Kry ..."
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Cited by 93 (5 self)
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Means of applying Krylov subspace techniques for adaptively extracting accurate reducedorder models of largescale nonlinear dynamical systems is a relatively open problem. There has been much current interest in developing such techniques. We focus on a bilinearization method, which extends Krylov subspace techniques for linear systems. In this approach, the nonlinear system is first approximated by a bilinear system through Carleman bilinearization. Then a reducedorder bilinear system is constructed in such a way that it matches certain number of multimoments corresponding to the first few kernels of the VolterraWiener representation of the bilinear system. It is shown that the twosided Krylov subspace technique matches significant more number of multimoments than the corresponding oneside technique.
A survey of model reduction methods for largescale systems
 Contemporary Mathematics
, 2001
"... An overview of model reduction methods and a comparison of the resulting algorithms is presented. These approaches are divided into two broad categories, namely SVD based and moment matching based methods. It turns out that the approximation error in the former case behaves better globally in freque ..."
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Cited by 91 (9 self)
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An overview of model reduction methods and a comparison of the resulting algorithms is presented. These approaches are divided into two broad categories, namely SVD based and moment matching based methods. It turns out that the approximation error in the former case behaves better globally in frequency while in the latter case the local behavior is better. 1 Introduction and problem statement Direct numerical simulation of dynamical systems has been an extremely successful means for studying complex physical phenomena. However, as more detail is included, the dimensionality of such simulations may increase to unmanageable levels of storage and computational requirements. One approach to overcoming this is through model reduction. The goal is to produce a low dimensional system that has
Asymptotic waveform evaluation via a Lanczos method
 Appl. Math. Lett
, 1994
"... AbstractIn this paper we show that the twosided Lanczos procedure combined with implicit restarts, offers significant advantages over Pad6 approximations used typically for model reduction in circuit simulation. ..."
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Cited by 76 (4 self)
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AbstractIn this paper we show that the twosided Lanczos procedure combined with implicit restarts, offers significant advantages over Pad6 approximations used typically for model reduction in circuit simulation.
A Generalized Entropy Criterion for NevanlinnaPick Interpolation with Degree Constraint
 IEEE Trans. Automat. Control
, 2001
"... In this paper, we present a generalized entropy criterion for solving the rational NevanlinnaPick problem for +1 interpolating conditions and the degree of interpolants bounded by . The primal problem of maximizing this entropy gain has a very wellbehaved dual problem. This dual is a convex opti ..."
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Cited by 75 (31 self)
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In this paper, we present a generalized entropy criterion for solving the rational NevanlinnaPick problem for +1 interpolating conditions and the degree of interpolants bounded by . The primal problem of maximizing this entropy gain has a very wellbehaved dual problem. This dual is a convex optimization problem in a finitedimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most . The criterion requires a selection of a monic Schur polynomial of degree . It follows that this class of monic polynomials completely parameterizes all such rational interpolants, and it therefore provides a set of design parameters for specifying such interpolants. The algorithm is implemented in statespace form and applied to several illustrative problems in systems and control, namely sensitivity minimization, maximal power transfer and spectral estimation.
A Convex Optimization Approach to the Rational Covariance Extension Problem
 SIAM J. Control Optim
, 1999
"... 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 ..."
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Cited by 72 (27 self)
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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 [7]. K3 words. rational covariance extension, partial stochastic realization, trigonometric moment problem, spectral estimation, speech processing, stochastic modeling AMS subject classifications.30ERR 60G35, 62M15, 93A30,93E0 1.