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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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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 Fusion of Relative Motions for Robust, Accurate and Scalable Structure from Motion
 ICCV, SYDNEY: AUSTRALIA
, 2013
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Robust Global Translations with 1DSfM
"... Abstract. We present a simple, effective method for solving structure from motion problems by averaging epipolar geometries. Based on recent successes in solving for global camera rotations using averaging schemes, we focus on the problem of solving for 3D camera translations given a network of noi ..."
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Abstract. We present a simple, effective method for solving structure from motion problems by averaging epipolar geometries. Based on recent successes in solving for global camera rotations using averaging schemes, we focus on the problem of solving for 3D camera translations given a network of noisy pairwise camera translation directions (or 3D point observations). To do this well, we have two main insights. First, we propose a method for removing outliers from problem instances by solving simpler lowdimensional subproblems, which we refer to as 1DSfM problems. Second, we present a simple, principled averaging scheme. We demonstrate this new method in the wild on Internet photo collections.
Analysis of an Exact
 Fractional Step Method, J. Comp. Physics
"... and stable recovery of rotations for robust synchronization R ..."
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and stable recovery of rotations for robust synchronization R
ShapeFit: Exact location recovery from corrupted pairwise directions
"... Abstract Let t 1 , . . . , t n ∈ R d and consider the location recovery problem: given a subset of pairwise direction observations {( , where a constant fraction of these observations are arbitrarily corrupted, find {t i } n i=1 up to a global translation and scale. We propose a novel algorithm for ..."
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Abstract Let t 1 , . . . , t n ∈ R d and consider the location recovery problem: given a subset of pairwise direction observations {( , where a constant fraction of these observations are arbitrarily corrupted, find {t i } n i=1 up to a global translation and scale. We propose a novel algorithm for the location recovery problem, which consists of a simple convex program over dn real variables. We prove that this program recovers a set of n i.i.d. Gaussian locations exactly and with high probability if the observations are given by an ErdősRényi graph, d is large enough, and provided that at most a constant fraction of observations involving any particular location are adversarially corrupted. We also prove that the program exactly recovers Gaussian locations for d = 3 if the fraction of corrupted observations at each location is, up to polylogarithmic factors, at most a constant. Both of these recovery theorems are based on a set of deterministic conditions that we prove are sufficient for exact recovery.
Temporal ordering and registration of images in studies of developmental dynamics
"... Abstract Dynamics of developmental progress is commonly reconstructed from imaging snapshots of chemical or mechanical processes in fixed embryos. As a first step in these reconstructions, snapshots must be spatially registered and ordered in time. Currently, image registration and ordering is ofte ..."
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Abstract Dynamics of developmental progress is commonly reconstructed from imaging snapshots of chemical or mechanical processes in fixed embryos. As a first step in these reconstructions, snapshots must be spatially registered and ordered in time. Currently, image registration and ordering is often done manually, requiring a significant amount of expertise with a specific system. However, as the sizes of imaging data sets grow, these tasks become increasingly difficult, especially when the images are noisy and the examined developmental changes are subtle. To address these challenges, we present an automated approach to simultaneously register and temporally order imaging data sets. The approach is based on vector diffusion maps, a manifold learning technique that does not require a priori knowledge of image features or a parametric model of the developmental dynamics. We illustrate this approach by registering and ordering data from imaging studies of pattern formation and morphogenesis in three different model systems. We also provide software to aid in the application of our methodology to other experimental data sets.
ShapeFit: Exact location recovery from corrupted pairwise directions
, 2015
"... Let t1,..., tn ∈ Rd and consider the location recovery problem: given a subset of pairwise direction observations {(ti − tj)/‖ti − tj‖2}i<j∈[n]×[n], where a constant fraction of these observations are arbitrarily corrupted, find {ti}ni=1 up to a global translation and scale. We propose a novel a ..."
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Let t1,..., tn ∈ Rd and consider the location recovery problem: given a subset of pairwise direction observations {(ti − tj)/‖ti − tj‖2}i<j∈[n]×[n], where a constant fraction of these observations are arbitrarily corrupted, find {ti}ni=1 up to a global translation and scale. We propose a novel algorithm for the location recovery problem, which consists of a simple convex program over dn real variables. We prove that this program recovers a set of n i.i.d. Gaussian locations exactly and with high probability if the observations are given by an ErdösRényi graph, d is large enough, and provided that at most a constant fraction of observations involving any particular location are adversarially corrupted. 1
Optimizing the Viewing Graph for StructurefromMotion
"... The viewing graph represents a set of views that are related by pairwise relative geometries. In the context of StructurefromMotion (SfM), the viewing graph is the input to the incremental or global estimation pipeline. Much effort has been put towards developing robust algorithms to overcome po ..."
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The viewing graph represents a set of views that are related by pairwise relative geometries. In the context of StructurefromMotion (SfM), the viewing graph is the input to the incremental or global estimation pipeline. Much effort has been put towards developing robust algorithms to overcome potentially inaccurate relative geometries in the viewing graph during SfM. In this paper, we take a fundamentally different approach to SfM and instead focus on improving the quality of the viewing graph before applying SfM. Our main contribution is a novel optimization that improves the quality of the relative geometries in the viewing graph by enforcing loop consistency constraints with the epipolar point transfer. We show that this optimization greatly improves the accuracy of relative poses in the viewing graph and removes the need for filtering steps or robust algorithms typically used in global SfM methods. In addition, the optimized viewing graph can be used to efficiently calibrate cameras at scale. We combine our viewing graph optimization and focal length calibration into a global SfM pipeline that is more efficient than existing approaches. To our knowledge, ours is the first global SfM pipeline capable of handling uncalibrated image sets. 1.