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NearOptimal Sensor Placement for Linear Inverse Problems
"... A classic problem is the estimation of a set of parameters from measurements collected by few sensors. The number of sensors is often limited by physical or economical constraints and their placement is of fundamental importance to obtain accurate estimates. Unfortunately, the selection of the opti ..."
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Cited by 8 (2 self)
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A classic problem is the estimation of a set of parameters from measurements collected by few sensors. The number of sensors is often limited by physical or economical constraints and their placement is of fundamental importance to obtain accurate estimates. Unfortunately, the selection of the optimal sensor locations is intrinsically combinatorial and the available approximation algorithms are not guaranteed to generate good solutions in all cases of interest. We propose FrameSense, a greedy algorithm for the selection of optimal sensor locations. The core cost function of the algorithm is the frame potential, a scalar property of matrices that measures the orthogonality of its rows. Notably, FrameSense is the first algorithm that is nearoptimal in terms of mean square error, meaning that its solution is always guaranteed to be close to the optimal one. Moreover, we show with an extensive set of numerical experiments that FrameSense achieves the stateoftheart performance while having the lowest computational cost, when compared to other greedy methods.
Optimal placement of bearingonly sensors for target localization
 in Proc. of 2012 American Control Conference, 2012
"... Abstract—We investigate optimal placements of multiple bearingonly sensors for target localization in both 2D and 3D spaces. The target is assumed to be static, and sensortarget ranges are arbitrary but fixed. The Fisher information matrix is used to characterize the localization uncertainty. By e ..."
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Cited by 5 (2 self)
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Abstract—We investigate optimal placements of multiple bearingonly sensors for target localization in both 2D and 3D spaces. The target is assumed to be static, and sensortarget ranges are arbitrary but fixed. The Fisher information matrix is used to characterize the localization uncertainty. By employing frame theory, we show that there are two types of optimal sensor placements, regular and irregular. Necessary and sufficient conditions of optimal placements are presented. It is proved that an irregular optimal placement can be converted to a regular one in a lower dimensional space. We furthermore propose explicit algorithms to construct some important specific regular optimal placements. I.
Necessary and sufficient conditions to perform Spectral Tetris, Linear Algebra Appl
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The geometry of structured . . . Frame Potentials
, 2015
"... In this dissertation, we study the geometric character of structured Parseval frames, which are families of vectors that provide perfect Hilbert space reconstruction. Equiangular Parseval frames (EPFs) satisfy that the magnitudes of the pairwise inner products between frame vectors are constant. ..."
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In this dissertation, we study the geometric character of structured Parseval frames, which are families of vectors that provide perfect Hilbert space reconstruction. Equiangular Parseval frames (EPFs) satisfy that the magnitudes of the pairwise inner products between frame vectors are constant. These types of frames are useful in many applications. However, EPFs do not always exist and constructing them is often difficult. To address this problem, we consider two generalizations of EPFs, equidistributed frames and Grassmannian equalnorm Parseval frames, which include EPFs when they exist. We provide several examples of each type of Parseval frame. To characterize and locate these classes of frames, we develop an optimization program involving families of real analytic frame potentials, which are realvalued functions of frames. With the help of the Łojasiewicz gradient inequality, we prove that the gradient descent of these functions on the manifold of Gram matrices of
Author manuscript, published in "SAMPTA'09, Marseille: France (2009)" Gradient descent of the frame potential
, 2010
"... Unit norm tight frames provide Parsevallike decompositions of vectors in terms of possibly nonorthogonal collections of unit norm vectors. One way to prove the existence of unit norm tight frames is to characterize them as the minimizers of a particular energy functional, dubbed the frame potential ..."
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Unit norm tight frames provide Parsevallike decompositions of vectors in terms of possibly nonorthogonal collections of unit norm vectors. One way to prove the existence of unit norm tight frames is to characterize them as the minimizers of a particular energy functional, dubbed the frame potential. We consider this minimization problem from a numerical perspective. In particular, we discuss how by descending the gradient of the frame potential, one, under certain conditions, is guaranteed to produce a sequence of unit norm frames which converge to a unit norm tight frame at a geometric rate. This makes the gradient descent of the frame potential a viable method for numerically constructing unit norm tight frames. 1.
Acta Applicandae Mathematicae manuscript No. (will be inserted by the editor) Minimizing Fusion Frame Potential
"... Abstract Fusion frames are an emerging topic of frame theory, with applications to encoding and distributed sensing. However, little is known about the existence of tight fusion frames. In traditional frame theory, one method for showing that unit norm tight frames exist is to characterize them as t ..."
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Abstract Fusion frames are an emerging topic of frame theory, with applications to encoding and distributed sensing. However, little is known about the existence of tight fusion frames. In traditional frame theory, one method for showing that unit norm tight frames exist is to characterize them as the minimizers of an energy functional, known as the frame potential. We generalize the frame potential to the fusion frame setting. In particular, we introduce the fusion frame potential, and show how its minimization is equivalent to the minimization of the traditional frame potential over a particular domain. We then study this minimization problem in detail. Specifically, we show that if the fusion frame’s subspaces are large in number but small in dimension compared to the dimension of the underlying space, then fusion frames will always exists, with each being a minimizer of the fusion frame potential. Key words frames, fusion, potential, tight
RESEARCH ARTICLE Optimal Sensor Placement for Target Localization and Tracking in 2D and 3D
, 2013
"... This paper analytically characterizes optimal sensor placements for target localization and tracking in 2D and 3D. Three types of sensors are considered: bearingonly, rangeonly, and receivedsignalstrength. The optimal placement problems of the three sensor types are formulated as an identical pa ..."
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This paper analytically characterizes optimal sensor placements for target localization and tracking in 2D and 3D. Three types of sensors are considered: bearingonly, rangeonly, and receivedsignalstrength. The optimal placement problems of the three sensor types are formulated as an identical parameter optimization problem and consequently analyzed in a unified framework. Recently developed frame theory is applied to the optimality analysis. We prove necessary and sufficient conditions for optimal placements in 2D and 3D. A number of important analytical properties of optimal placements are further explored. In order to verify the analytical analysis, we present a gradient control law that can numerically construct generic optimal placements.
1Optimal Sensor Placement for Target Localization and Tracking in 2D and 3D
"... This paper analytically characterizes optimal sensor placements for target localization and tracking in 2D and 3D. Three types of sensors are considered: bearingonly, rangeonly, and receivedsignalstrength. The optimal placement problems of the three sensor types are formulated as an identical pa ..."
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This paper analytically characterizes optimal sensor placements for target localization and tracking in 2D and 3D. Three types of sensors are considered: bearingonly, rangeonly, and receivedsignalstrength. The optimal placement problems of the three sensor types are formulated as an identical parameter optimization problem and consequently analyzed in a unified framework. Recently developed frame theory is applied to the optimality analysis. We prove necessary and sufficient conditions for optimal placements in 2D and 3D. A number of important analytical properties of optimal placements are further explored. In order to verify the analytical analysis, we present a gradient control law that can numerically construct generic optimal placements. Index Terms Fisher information matrix; gradient control; optimal sensor placement; target tracking; tight frame. I.