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Sensor network localization by eigenvector synchronization over the Euclidean group
 In press
"... We present a new approach to localization of sensors from noisy measurements of a subset of their Euclidean distances. Our algorithm starts by finding, embedding and aligning uniquely realizable subsets of neighboring sensors called patches. In the noisefree case, each patch agrees with its global ..."
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Cited by 25 (15 self)
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We present a new approach to localization of sensors from noisy measurements of a subset of their Euclidean distances. Our algorithm starts by finding, embedding and aligning uniquely realizable subsets of neighboring sensors called patches. In the noisefree case, each patch agrees with its global positioning up to an unknown rigid motion of translation, rotation and possibly reflection. The reflections and rotations are estimated using the recently developed eigenvector synchronization algorithm, while the translations are estimated by solving an overdetermined linear system. The algorithm is scalable as the number of nodes increases, and can be implemented in a distributed fashion. Extensive numerical experiments show that it compares favorably to other existing algorithms in terms of robustness to noise, sparse connectivity and running time. While our approach is applicable to higher dimensions, in the current paper we focus on the two dimensional case.
A Cheeger inequality for the graph connection laplacian. available online
, 2012
"... Abstract. The O(d) Synchronization problem consists of estimating a set of n unknown orthogonal d × d matrices O1,..., On from noisy measurements of a subset of the pairwise ratios OiO −1 j. We formulate and prove a Cheegertype inequality that relates a measure of how well it is possible to solve t ..."
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Cited by 23 (13 self)
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Abstract. The O(d) Synchronization problem consists of estimating a set of n unknown orthogonal d × d matrices O1,..., On from noisy measurements of a subset of the pairwise ratios OiO −1 j. We formulate and prove a Cheegertype inequality that relates a measure of how well it is possible to solve the O(d) synchronization problem with the spectra of an operator, the graph Connection Laplacian. We also show how this inequality provides a worst case performance guarantee for a spectral method to solve this problem.
Representation theoretic patterns in three dimensional cryoelectron macroscopy I  The Intrinsic reconstitution algorithm
"... Abstract. In this paper, we continue to develop the representation theoretic setup for 3D cryoelectron microscopy (cryoEM) that was initiated in [7]. In particular, we provide a complete spectral analysis of the local parallel transport operator on the two dimensional sphere. This is then used to ..."
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Cited by 20 (8 self)
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Abstract. In this paper, we continue to develop the representation theoretic setup for 3D cryoelectron microscopy (cryoEM) that was initiated in [7]. In particular, we provide a complete spectral analysis of the local parallel transport operator on the two dimensional sphere. This is then used to prove the admissibility (correctness) and the numerical stability of the intrinsic classification algorithm for identifying raw projection images of similar viewing directions in cryoEM, that was recently introduced in [9]. This preliminary classification is of fundamental importance in determining the three dimensional structure of macromolecules from cryoEM images. The goal in cryoEM is to determine the 3D structure of a molecule from noisy projection images taken at unknown random orientations by an electron microscope, i.e., a random Computational Tomography (CT). Determining 3D structures of
EXACT AND STABLE RECOVERY OF ROTATIONS FOR ROBUST SYNCHRONIZATION
, 1211
"... Abstract. The synchronization problem over the special orthogonal group SO(d) consists of estimating a set of unknown rotations R1, R2,..., Rn from noisy measurements of a subset of their pairwise ratios R −1 i Rj. The problem has found applications in computer vision, computer graphics, and sensor ..."
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Cited by 20 (7 self)
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Abstract. The synchronization problem over the special orthogonal group SO(d) consists of estimating a set of unknown rotations R1, R2,..., Rn from noisy measurements of a subset of their pairwise ratios R −1 i Rj. The problem has found applications in computer vision, computer graphics, and sensor network localization, among others. Its least squares solution can be approximated by either spectral relaxation or semidefinite programming followed by a rounding procedure, analogous to the approximation algorithms of MaxCut. The contribution of this paper is threefold: First, we introduce a robust penalty function involving the sum of unsquared deviations and derive a relaxation that leads to a convex optimization problem; Second, we apply the alternating direction method to minimize the penalty function; Finally, under a specific model of the measurement noise and the measurement graph, we prove that the rotations are exactly and stably recovered, exhibiting a phase transition behavior in terms of the proportion of noisy measurements. Numerical simulations confirm the phase transition behavior for our method as well as its improved accuracy compared to existing methods. Key words. Synchronization of rotations; least unsquared deviation; semidefinite relaxation; and alternating direction method 1. Introduction. The
Global Motion Estimation from Point Matches
"... Abstract—Multiview structure recovery from a collection of images requires the recovery of the positions and orientations of the cameras relative to a global coordinate system. Our approach recovers camera motion as a sequence of two global optimizations. First, pairwise Essential Matrices are used ..."
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Cited by 15 (3 self)
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Abstract—Multiview structure recovery from a collection of images requires the recovery of the positions and orientations of the cameras relative to a global coordinate system. Our approach recovers camera motion as a sequence of two global optimizations. First, pairwise Essential Matrices are used to recover the global rotations by applying robust optimization using either spectral or semidefinite programming relaxations. Then, we directly employ feature correspondences across images to recover the global translation vectors using a linear algorithm based on a novel decomposition of the Essential Matrix. Our method is efficient and, as demonstrated in our experiments, achieves highly accurate results on collections of real images for which ground truth measurements are available. Keywordsstructure from motion; 3D reconstruction; camera motion estimation; convex relaxation; linear estimation I.
Cramérrao bounds for synchronization of rotations
 CoRR
"... Synchronization of rotations is the problem of estimating a set of rotations Ri ∈ SO(n), i = 1... N based on noisy measurements of relative rotations RiR ⊤ j. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchron ..."
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Cited by 4 (4 self)
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Synchronization of rotations is the problem of estimating a set of rotations Ri ∈ SO(n), i = 1... N based on noisy measurements of relative rotations RiR ⊤ j. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchronization as estimation on Riemannian manifolds for arbitrary n under a large family of noise models. The noise models we address encompass zeromean isotropic noise, and we develop tools for Gaussianlike as well as heavytail types of noise in particular. As a main contribution, we derive the CramérRao bounds of synchronization, that is, lowerbounds on the variance of unbiased estimators. We find that these bounds are structured by the pseudoinverse of the measurement graph Laplacian, where edge weights are proportional to measurement quality. We leverage this to provide interpretation in terms of random walks and visualization tools for these bounds in both the anchored and anchorfree scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussianlike noise. Synchronization of rotations, estimation on manifolds, estimation on graphs, graph Laplacian, Fisher information, CramérRao bounds, distributions on the rotation group, Langevin. 2000 Math Subject Classification: 62F99, 94C15, 22C05, 05C12, 1
RESEARCH SUMMARY
"... My research concerns the study of the symmetries and the geometry of the basic Hilbert spaces that appear in harmonic analysis and applied mathematics, i.e., the Hilbert spaces of functions (or more generally sections of certain vector bundels) on geometric spaces. For example when I study the Hilbe ..."
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My research concerns the study of the symmetries and the geometry of the basic Hilbert spaces that appear in harmonic analysis and applied mathematics, i.e., the Hilbert spaces of functions (or more generally sections of certain vector bundels) on geometric spaces. For example when I study the Hilbert spaces of functions on vector spaces of the form FN where F is a
eld which can be either a
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eld or an in
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eld such as the
elds of real or complex numbers I use two intertwining points of view: group representation theory and algebraic geometry. The group representation theory is the HeisenbergWeil representation of the Heisenberg group H(2N;F) and the symplectic group Sp(2N;F): The algebraic geometry is the Grothendieck theory of `adic sheaves in the
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eld setting and the theory of algebraic Dmodules in the real and complex cases. Other examples which arise naturally in my research, are the Hilbert spaces of global section of certain vector bundels on the (N 1)dimensional sphere SN1 RN: Now, the harmonic analysis tools that I use in the study of these Hilbert spaces are the representation theory of the orthogonal group O(N) and the related di¤erential geometry. My motivation to study these objects comes from the fact that these Hilbert spaces appear naturally in concrete problems of applied mathematics and mathetheory to discrete harmonic analysis, digital signal processing, arithmetic quantum chaos, and threedimensional structuring of molecules. This summary shall describe some of my works in pure and applied mathematics. Pure Mathematics 1. The Weil representation over finite fields The discrete Fourier transform (DFT for short) is in a natural way a member of a group of unitary operators that act on the Hilbert space H = C(F) of complex valued functions on the
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eld F = Fp, where p is an odd prime number: The DFT operator satis
es the following system w of p2 linear equation: DFT (t; w) = (w (t; w)) DFT; for every t; w 2 F; where w =