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155
Spatiallydistributed coverage optimization and control with limitedrange interactions
 ESAIM Control, Optimisation Calculus Variations
, 2005
"... Abstract. This paper presents coordination algorithms for groups of mobile agents performing deployment and coverage tasks. As an important modeling constraint, we assume that each mobile agent has a limited sensing/communication radius. Based on the geometry of Voronoi partitions and proximity grap ..."
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Cited by 78 (34 self)
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Abstract. This paper presents coordination algorithms for groups of mobile agents performing deployment and coverage tasks. As an important modeling constraint, we assume that each mobile agent has a limited sensing/communication radius. Based on the geometry of Voronoi partitions and proximity graphs, we analyze a class of aggregate objective functions and propose coverage algorithms in continuous and discrete time. These algorithms have convergence guarantees and are spatially distributed with respect to appropriate proximity graphs. Numerical simulations illustrate the results.
Means and Averaging in the Group of Rotations
, 2002
"... In this paper we give precise definitions of different, properly invariant notions of mean or average rotation. Each mean is associated with a metric in SO(3). The metric induced from the Frobenius inner product gives rise to a mean rotation that is given by the closest special orthogonal matrix to ..."
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Cited by 74 (1 self)
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In this paper we give precise definitions of different, properly invariant notions of mean or average rotation. Each mean is associated with a metric in SO(3). The metric induced from the Frobenius inner product gives rise to a mean rotation that is given by the closest special orthogonal matrix to the usual arithmetic mean of the given rotation matrices. The mean rotation associated with the intrinsic metric on SO(3) is the Riemannian center of mass of the given rotation matrices. We show that the Riemannian mean rotation shares many common features with the geometric mean of positive numbers and the geometric mean of positive Hermitian operators. We give some examples with closedform solutions of both notions of mean.
Riemannian geometry of Grassmann manifolds with a view on algorithmic computation
 Acta Appl. Math
"... Abstract. We give simple formulas for the canonical metric, gradient, Lie ..."
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Cited by 58 (14 self)
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Abstract. We give simple formulas for the canonical metric, gradient, Lie
Inverse eigenvalue problems
 SIAM Rev
, 1998
"... Abstract. A collection of inverse eigenvalue problems are identi ed and classi ed according to their characteristics. Current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper. This exposition also reveals many open questions that deserves furth ..."
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Cited by 48 (7 self)
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Abstract. A collection of inverse eigenvalue problems are identi ed and classi ed according to their characteristics. Current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper. This exposition also reveals many open questions that deserves further study. An extensive bibliography of pertinent literature is attached.
O.: Regularizing flows for constrained matrixvalued images
 J. Math. Imaging Vision
, 2004
"... Abstract. Nonlinear diffusion equations are now widely used to restore and enhance images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a differentialgeometric framework to define PDEs acting on some manifol ..."
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Cited by 39 (11 self)
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Abstract. Nonlinear diffusion equations are now widely used to restore and enhance images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a differentialgeometric framework to define PDEs acting on some manifold constrained datasets. We consider the case of images taking value into matrix manifolds defined by orthogonal and spectral constraints. We directly incorporate the geometry and natural metric of the underlying configuration space (viewed as a Lie group or a homogeneous space) in the design of the corresponding flows. Our numerical implementation relies on structurepreserving integrators that respect intrinsically the constraints geometry. The efficiency and versatility of this approach are illustrated through the anisotropic smoothing of diffusion tensor volumes in medical imaging. Note: This is the draft
Constrained flows of matrixvalued functions: Application to diffusion tensor regularization
 In European Conference on Computer Vision
, 2002
"... Abstract. Nonlinear partial differential equations (PDE) are now widely used to regularize images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a geometric framework to design PDE flows actingon constrained d ..."
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Cited by 33 (9 self)
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Abstract. Nonlinear partial differential equations (PDE) are now widely used to regularize images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a geometric framework to design PDE flows actingon constrained datasets. We focus our interest on flows of matrixvalued functions undergoing orthogonal and spectral constraints. The correspondingevolution PDE’s are found by minimization of cost functionals, and depend on the natural metrics of the underlyingconstrained manifolds (viewed as Lie groups or homogeneous spaces). Suitable numerical schemes that fit the constraints are also presented. We illustrate this theoretical framework through a recent and challenging problem in medical imaging: the regularization of diffusion tensor volumes (DTMRI).
Parallel Distributed Solvers For Large Stable Generalized Lyapunov Equations
, 1998
"... In this paper we study the solution of stable generalized Lyapunov matrix equations with largescale, dense coefficient matrices. Our iterative algorithms, based on the matrix sign function, only require scalable matrix algebra kernels which are highly efficient on parallel distributed architectures ..."
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Cited by 24 (14 self)
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In this paper we study the solution of stable generalized Lyapunov matrix equations with largescale, dense coefficient matrices. Our iterative algorithms, based on the matrix sign function, only require scalable matrix algebra kernels which are highly efficient on parallel distributed architectures. This approach avoids therefore the difficult parallelization of direct methods based on the QZ algorithm. The experimental analysis reports a remarkable performance of our solvers on an ibm sp2 platform. Keywords: Generalized Lyapunov matrix equations, mathematical software, matrix sign function, parallel distributed multiprocessors. 1. Introduction Consider the generalized Lyapunov equation A T XE +E T XA+Q = 0; (1) where A; E; X;Q 2 IR n\Thetan , Q = Q T , and X = X T is the unknown matrix. Lyapunov equations are of fundamental importance in many analysis and synthesis algorithms in control theory. They arise naturally in linear control problems driven by linear autonomous ...
Analog Computation with Dynamical Systems
 Physica D
, 1997
"... This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete th ..."
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Cited by 22 (0 self)
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This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems, discrete or continuous in time, on the basis of an intrinsic time scale of the system. Dissipative dynamical systems are classified into the computational complexity classes P d , CoRP d , NP d
CONSENSUS OPTIMIZATION ON MANIFOLDS
 VOL. 48, NO. 1, PP. 56–76 C ○ 2009 SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS
, 2009
"... The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of th ..."
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Cited by 22 (6 self)
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The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to efficient gradient algorithms to synchronize (i.e., maximizing the consensus) or balance (i.e., minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and timevarying. The cost function is linked to a specific centroid definition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group SO(n) andthe Grassmann manifold Grass(p, n) are treated as original examples. A link is also drawn with the many existing results on the circle.