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34
A general framework for motion segmentation: Independent, articulated, rigid, nonrigid, degenerate and nondegenerate
 In ECCV
, 2006
"... Abstract. We cast the problem of motion segmentation of feature trajectories as linear manifold finding problems and propose a general framework for motion segmentation under affine projections which utilizes two properties of trajectory data: geometric constraint and locality. The geometric constra ..."
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Cited by 69 (0 self)
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Abstract. We cast the problem of motion segmentation of feature trajectories as linear manifold finding problems and propose a general framework for motion segmentation under affine projections which utilizes two properties of trajectory data: geometric constraint and locality. The geometric constraint states that the trajectories of the same motion lie in a low dimensional linear manifold and different motions result in different linear manifolds; locality, by which we mean in a transformed space a data and its neighbors tend to lie in the same linear manifold, provides a cue for efficient estimation of these manifolds. Our algorithm estimates a number of linear manifolds, whose dimensions are unknown beforehand, and segment the trajectories accordingly. It first transforms and normalizes the trajectories; secondly, for each trajectory it estimates a local linear manifold through local sampling; then it derives the affinity matrix based on principal subspace angles between these estimated linear manifolds; at last, spectral clustering is applied to the matrix and gives the segmentation result. Our algorithm is general without restriction on the number of linear manifolds and without prior knowledge of the dimensions of the linear manifolds. We demonstrate in our experiments that it can segment a wide range of motions including independent, articulated, rigid, nonrigid, degenerate, nondegenerate or any combination of them. In some highly challenging cases where other stateoftheart motion segmentation algorithms may fail, our algorithm gives expected results. 2 1
An Algebraic Geometric Approach to the Identification of a Class of Linear Hybrid Systems
 In Proc. of IEEE Conference on Decision and Control
, 2003
"... We propose an algebraic geometric solution to the identification of a class of linear hybrid systems. We show that the identification of the model parameters can be decoupled from the inference of the hybrid state and the switching mechanism generating the transitions, hence we do not constraint the ..."
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Cited by 37 (11 self)
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We propose an algebraic geometric solution to the identification of a class of linear hybrid systems. We show that the identification of the model parameters can be decoupled from the inference of the hybrid state and the switching mechanism generating the transitions, hence we do not constraint the switches to be separated by a minimum dwell time. The decoupling is obtained from the socalled hybrid decoupling constraint, which establishes a connection between linear hybrid system identification, polynomial factorization and hyperplane clustering. In essence, we represent the number of discrete states n as the degree of a homogeneous polynomial p and the model parameters as factors of p. We then show that one can estimate n from a rank constraint on the data, the coe#cients of p from a linear system, and the model parameters from the derivatives of p. The solution is closed form if and only if n 4. Once the model parameters have been identified, the estimation of the hybrid state becomes a simpler problem. Although our algorithm is designed for noiseless data, we also present simulation results with noisy data. 1
Identification of piecewise affine systems via mixedinteger programming
 Automatica
, 2004
"... This paper addresses the problem of identification of hybrid dynamical systems, by focusing the attention on hinging hyperplanes (HHARX) and Wiener piecewise affine (WPWARX) autoregressive exogenous models. In particular, we provide algorithms based on mixedinteger linear or quadratic programming ..."
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Cited by 22 (4 self)
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This paper addresses the problem of identification of hybrid dynamical systems, by focusing the attention on hinging hyperplanes (HHARX) and Wiener piecewise affine (WPWARX) autoregressive exogenous models. In particular, we provide algorithms based on mixedinteger linear or quadratic programming which are guaranteed to converge to a global optimum. For the special case where switches occur only seldom in the estimation data, we also suggest a way of trading off between optimality and complexity by using a change detection approach. 1
Hybrid modeling and simulation of genetic regulatory networks: a qualitative approach
 ERCIM News
, 2003
"... The functioning and development of living organisms is controlled by large and complex networks of genes, proteins, small molecules, and their interactions, socalled genetic regulatory networks. The concerted efforts of genetics, molecular biology, biochemistry, and physiology have led to the accum ..."
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Cited by 19 (1 self)
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The functioning and development of living organisms is controlled by large and complex networks of genes, proteins, small molecules, and their interactions, socalled genetic regulatory networks. The concerted efforts of genetics, molecular biology, biochemistry, and physiology have led to the accumulation of enormous amounts of data on the molecular components of genetic regulatory networks and their interactions. Notwithstanding the advances in the mapping of the network structure, surprisingly little is understood about how the dynamic behavior of the system emerges from the interactions between the network components. This has incited an increasingly large group of researchers to turn from the structure to the behavior of genetic regulatory networks, against the background of a broader movement nowadays often referred to as systems biology
A.: Reconstruction of switching thresholds in piecewiseaffine models of genetic regulatory networks
, 2005
"... Abstract. Recent advances of experimental techniques in biology have led to the production of enormous amounts of data on the dynamics of genetic regulatory networks. In this paper, we present an approach for the identification of PieceWiseAffine (PWA) models of genetic regulatory networks from exp ..."
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Cited by 17 (2 self)
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Abstract. Recent advances of experimental techniques in biology have led to the production of enormous amounts of data on the dynamics of genetic regulatory networks. In this paper, we present an approach for the identification of PieceWiseAffine (PWA) models of genetic regulatory networks from experimental data, focusing on the reconstruction of switching thresholds associated with regulatory interactions. In particular, our method takes into account geometric constraints specific to models of genetic regulatory networks. We show the feasibility of our approach by the reconstruction of switching thresholds in a PWA model of the carbon starvation response in the bacterium Escherichia coli. 1
Identification of hybrid systems: a tutorial
"... This tutorial paper is concerned with the identification of hybrid models, i.e. dynamical models whose behavior is determined by interacting continuous and discrete dynamics. Methods specifically aimed at the identification of models with a hybrid structure are of very recent date. After discussing ..."
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Cited by 10 (0 self)
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This tutorial paper is concerned with the identification of hybrid models, i.e. dynamical models whose behavior is determined by interacting continuous and discrete dynamics. Methods specifically aimed at the identification of models with a hybrid structure are of very recent date. After discussing the main issues and difficulties connected with hybrid system identification, and giving an overview of the related literature, this paper focuses on four different approaches for the identification of switched affine and piecewise affine models, namely an algebraic procedure, a Bayesian procedure, a clusteringbased procedure, and a boundederror procedure. The main features of the selected procedures are presented, and possible interactions to still enhance their effectiveness are suggested.
Identification of PWARX Hybrid Models with Unknown and Possibly Different Orders
 In Proceedings of IEEE American Control Conference
, 2004
"... We consider the problem of identifying the orders and the model parameters of PWARX hybrid models from noiseless input/output data. We cast the identification problem in an algebraic geometric framework in which the number of discrete states corresponds to the degree of a multivariate polynomial p a ..."
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Cited by 8 (4 self)
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We consider the problem of identifying the orders and the model parameters of PWARX hybrid models from noiseless input/output data. We cast the identification problem in an algebraic geometric framework in which the number of discrete states corresponds to the degree of a multivariate polynomial p and the orders and the model parameters are encoded on the factors of p. We derive a rank constraint on the input/output data from which one can estimate the coefficients of p. Given p, we show that one can estimate the orders and the parameters of each ARX model from the derivatives of p at a collection of regressors that minimize a certain objective function. Our solution does not require previous knowledge about the orders of the ARX models (only an upper bound is needed), nor does it constraint the orders to be equal. Also the switching mechanism can be arbitrary, hence the switches need not be separated by a minimum dwell time. We illustrate our approach with an algebraic example of a switching circuit and with simulation results in the presence of noisy data.
A new learning method for piecewise linear regression. ICANN
, 2002
"... Abstract. A new connectionist model for the solution of piecewise linear regression problems is introduced; it is able to reconstruct both continuous and non continuous real valued mappings starting from a finite set of possibly noisy samples. The approximating function can assume a different linear ..."
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Cited by 7 (0 self)
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Abstract. A new connectionist model for the solution of piecewise linear regression problems is introduced; it is able to reconstruct both continuous and non continuous real valued mappings starting from a finite set of possibly noisy samples. The approximating function can assume a different linear behavior in each region of an unknown polyhedral partition of the input domain. The proposed learning technique combines local estimation, clustering in weight space, multicategory classification and linear regression in order to achieve the desired result. Through this approach piecewise affine solutions for general nonlinear regression problems can also be found. 1
Recursive Identification of Switched ARX Models with Unknown Number of Models and Unknown Orders
"... Abstract — We consider the problem of recursively identifying the parameters of a switched ARX (SARX) model from input/output data under the assumption that the number of models, the model orders and the switching sequence are unknown. Our approach exploits the fact that applying a polynomial embedd ..."
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Cited by 4 (0 self)
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Abstract — We consider the problem of recursively identifying the parameters of a switched ARX (SARX) model from input/output data under the assumption that the number of models, the model orders and the switching sequence are unknown. Our approach exploits the fact that applying a polynomial embedding to the input/output data leads to a lifted ARX model whose dynamics are linear on the socalled hybrid model parameters and independent of the switching sequence. In principle, one can use a standard recursive algorithm to identify such hybrid parameters. However, when the number of models and the model orders are unknown the embedded regressors may not be persistently exciting, hence the estimates of the hybrid parameters may not converge exponentially to a constant vector. Nevertheless, we show that these estimates still converge to a vector that depends continuously on the initial condition. By identifying the hybrid model parameters starting from two different initial conditions, we show that one can build two homogeneous polynomials whose derivatives at a regressor give an estimate of the parameters of the ARX model generating that regressor. After properly enforcing some of the entries of the hybrid model parameters to be zero, such estimates are shown to converge exponentially to the true ARX model parameters under suitable persistence of excitation conditions on the input/output data. Although our algorithm is designed for the case of perfect input/output data, our experiments also show its performance with noisy data. I.
A learning algorithm for piecewise linear regression
 In Marinaro,M. and Tagliaferri,R. (eds.) Neural Nets: WIRN Vietri01
, 2001
"... A new learning algorithm for solving piecewise linear regression problems is proposed. It is able to train a proper multilayer feedforward neural network so as to reconstruct a target function assuming a different linear behavior on each set of a polyhedral partition of the input domain. The propose ..."
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Cited by 3 (0 self)
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A new learning algorithm for solving piecewise linear regression problems is proposed. It is able to train a proper multilayer feedforward neural network so as to reconstruct a target function assuming a different linear behavior on each set of a polyhedral partition of the input domain. The proposed method combine local estimation, clustering in weight space, classification and regression in order to achieve the desired result. A simulation on a benchmark problem shows the good properties of this new learning algorithm. 1