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**11 - 18**of**18**### RIESZ OUTER PRODUCT HILBERT SPACE FRAMES: QUANTITATIVE BOUNDS, TOPOLOGICAL PROPERTIES, AND FULL GEOMETRIC CHARACTERIZATION

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### 1Near-Optimal Sensor Placement for Linear Inverse Problems

"... Abstract—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 ..."

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Abstract—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 near-optimal 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 experi-ments that FrameSense achieves the state-of-the-art performance while having the lowest computational cost, when compared to other greedy methods. Index Terms—Sensor placement, inverse problem, frame po-tential, greedy algorithm. I.

### Optimal Deployment of Mobile Sensors for Target Tracking in 2D and 3D Spaces

"... Abstract: This paper proposes a control strategy to autonomously deploy optimal placements of range-only mobile sensors in 2D and 3D spaces. Based on artificial potential approaches, the control strategy is designed to minimize the inter-sensor and external potentials. The inter-sensor potential is ..."

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Abstract: This paper proposes a control strategy to autonomously deploy optimal placements of range-only mobile sensors in 2D and 3D spaces. Based on artificial potential approaches, the control strategy is designed to minimize the inter-sensor and external potentials. The inter-sensor potential is the objective function for optimal sensor placements. A placement is optimal when the inter-sensor potential is minimized. The external potential is introduced to fulfill constraints on sensor trajectories. Since artificial potential approaches can handle various issues such as obstacle avoidance and collision avoidance among sensors, the proposed control strategy provides a flexible solution to practical autonomous optimal sensor deployment. The control strategy is applied to several optimal sensor deployment problems in 2D and 3D spaces. Simulation results illustrate how the proposed control strategy can improve target tracking performance.

### Operators associated to frames:

, 2004

"... The recently introduced notion of a frame potential has led to useful characterizations of finite-dimensional tight frames consisting of vectors with prescribed lengths. It is natural to ask whether the frame potential leads to similar characterizations for systems with additional im-posed structure ..."

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The recently introduced notion of a frame potential has led to useful characterizations of finite-dimensional tight frames consisting of vectors with prescribed lengths. It is natural to ask whether the frame potential leads to similar characterizations for systems with additional im-posed structure. We will describe how such a generaliza-tion can be obtained for the class of shift-invariant sys-tems. The fast algorithms associated with convolution make shift-invariant systems advantageous in applications. (joint work with M. Fickus, K. Kornelson, & K. Okoudjou) 0 What is a frame? • A collection X: = {xj}j∈J ⊂ H is a frame for H if and only if there exist constants 0 < B1 ≤ B2 < ∞ such that for each x ∈ H B1‖x‖2 ≤ j∈J ∣∣〈x, xj〉∣∣2 ≤ B2‖x‖2; (1) • X is called a tight frame if it is possible that B1 = B2; • Frames generalize the notion of an orthonormal basis.? Frame coefficients uniquely determine elements: 〈x, xj 〉 = 〈y, xj〉, ∀j ∈ J ⇒ x = y;? However, representation of an element in terms of frame vec-tors is not, in general, unique: x = j

### Date

, 2012

"... This material is declared a work of the U.S. Government and is not subject to copyright ..."

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This material is declared a work of the U.S. Government and is not subject to copyright

### EXPLOITING DATA-DEPENDENT STRUCTURE FOR IMPROVING SENSOR ACQUISITION AND INTEGRATION

, 2014

"... This thesis deals with two approaches to building efficient representations of data. The first is a study of compressive sensing and improved data acquisition. We outline the development of the theory, and proceed into its uses in matrix com-pletion problems via convex optimization. The aim of this ..."

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This thesis deals with two approaches to building efficient representations of data. The first is a study of compressive sensing and improved data acquisition. We outline the development of the theory, and proceed into its uses in matrix com-pletion problems via convex optimization. The aim of this research is to prove that a general class of measurement operators, bounded norm Parseval frames, sat-isfy the necessary conditions for random subsampling and reconstruction. We then demonstrate an example of this theory in solving 2-dimensional Fredholm integrals with partial measurements. This has large ramifications in improved acquisition of nuclear magnetic resonance spectra, for which we give several examples. The second part of this thesis studies the Laplacian Eigenmaps (LE) algorithm and its uses in data fusion. In particular, we build a natural approximate inversion algorithm for LE embeddings using L1 regularization and MDS embedding tech-niques. We show how this inversion, combined with feature space rotation, leads to a novel form of data reconstruction and inpainting using a priori information. We demonstrate this method on hyperspectral imagery and LIDAR. We also aim to understand and characterize the embeddings the LE algorithm gives. To this end, we characterize the order in which eigenvectors of a disjoint graph emerge and the support of those eigenvectors. We then extend this characterization to weakly connected graphs with clusters of differing sizes, utilizing the theory of invariant subspace perturbations and proving some novel results.