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**1 - 5**of**5**### 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 Erdős-Rényi 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 poly-logarithmic 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.

### 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 obser-vations 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 obser-vations 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 Erdös-Rényi 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 Structure-from-Motion

"... The viewing graph represents a set of views that are re-lated by pairwise relative geometries. In the context of Structure-from-Motion (SfM), the viewing graph is the in-put 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 re-lated by pairwise relative geometries. In the context of Structure-from-Motion (SfM), the viewing graph is the in-put 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 fun-damentally different approach to SfM and instead focus on improving the quality of the viewing graph before apply-ing SfM. Our main contribution is a novel optimization that improves the quality of the relative geometries in the view-ing 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 view-ing graph and removes the need for filtering steps or robust algorithms typically used in global SfM methods. In addi-tion, 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.

### Scalable Structure from Motion for Densely Sampled Videos

"... Videos consisting of thousands of high resolution frames are challenging for existing structure from motion (SfM) and simultaneous-localization and mapping (SLAM) tech-niques. We present a new approach for simultaneously com-puting extrinsic camera poses and 3D scene structure that is capable of han ..."

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Videos consisting of thousands of high resolution frames are challenging for existing structure from motion (SfM) and simultaneous-localization and mapping (SLAM) tech-niques. We present a new approach for simultaneously com-puting extrinsic camera poses and 3D scene structure that is capable of handling such large volumes of image data. The key insight behind this paper is to effectively exploit co-herence in densely sampled video input. Our technical con-tributions include robust tracking and selection of confident video frames, a novel window bundle adjustment, frame-to-structure verification for globally consistent reconstructions with multi-loop closing, and utilizing efficient global linear camera pose estimation in order to link both consecutive and distant bundle adjustment windows. To our knowledge we describe the first system that is capable of handling high resolution, high frame-rate video data with close to real-time performance. In addition, our approach can robustly integrate data from different video sequences, allowing mul-tiple video streams to be simultaneously calibrated in an efficient and globally optimal way. We demonstrate high quality alignment on large scale challenging datasets, e.g., 2-20 megapixel resolution at frame rates of 25-120 Hz with thousands of frames. 1.

### DEEPFOCAL: A METHOD FOR DIRECT FOCAL LENGTH ESTIMATION

"... Estimating the focal length of an image is an important preprocessing step for many applications. Despite this, exist-ing methods for single-view focal length estimation are lim-ited in that they require particular geometric calibration ob-jects, such as orthogonal vanishing points, co-planar circle ..."

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Estimating the focal length of an image is an important preprocessing step for many applications. Despite this, exist-ing methods for single-view focal length estimation are lim-ited in that they require particular geometric calibration ob-jects, such as orthogonal vanishing points, co-planar circles, or a calibration grid, to occur in the field of view. In this work, we explore the application of a deep convolutional neu-ral network, trained on natural images obtained from Internet photo collections, to directly estimate the focal length using only raw pixel intensities as input features. We present quan-titative results that demonstrate the ability of our technique to estimate the focal length with comparisons against several baseline methods, including an automatic method which uses orthogonal vanishing points. Index Terms — focal length estimation, camera calibra-tion, convolutional neural network 1.