We introduce a new method for location recovery from pair-wise directions that leverages an efficient convex program that comes with exact recovery guarantees, even in the presence of adversarial outliers. When pairwise directions represent scaled relative positions between pairs of views (estimated for instance with epipolar geometry) our method can be used for location recovery, that is the determination of relative pose up to a single unknown scale. For this task, our method yields performance comparable to the state-of-the-art with an order of magnitude speed-up. Our proposed numerical framework is flexible in that it accommodates other approaches to location recovery and can be used to speed up other methods. These properties are demonstrated by extensively testing against state-of-the-art methods for location recovery on 13 large, irregular collections of images of real scenes in addition to simulated data with ground truth.
We concentrate on the location recovery problem from relative directions based on known camera rotations. There have been many different approaches to this problem, such as least squares @cite_19 @cite_23 @cite_11 @cite_0 , second-order cone programs, @math methods @cite_2 @cite_20 @cite_0 @cite_21 @cite_13 , spectral methods @cite_23 , similarity transformations for pair alignment @cite_15 , Lie-algebraic averaging @cite_24 , Markov random fields @cite_1 , and several others @cite_14 @cite_7 @cite_15 @cite_17 . Unfortunately, many location recovery algorithms either lack robustness to mismatches, at times produce collapsed solutions @cite_7 , or suffer from convergence to local minima, in sum causing large errors in (or even complete degradation of) the recovered locations.
@cite_14 @cite_7 @cite_21 @cite_1 @cite_17 @cite_0 @cite_19 @cite_24 @cite_23 @cite_2 @cite_15 @cite_13 @cite_20 @cite_11
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3D structure recovery from a collection of 2D images requires the estimation of the camera locations and orientations, i.e. the camera motion. For large, irregular collections of images, existing methods for the location estimation part, which can be formulated as the inverse problem of estimating n locations t 1 , t 2 , …, tn in ℝ 3 from noisy measurements of a subset of the pairwise directions t i −t j ∥t i −t j ∥, are sensitive to outliers in direction measurements. In this paper, we firstly provide a complete characterization of well-posed instances of the location estimation problem, by presenting its relation to the existing theory of parallel rigidity. For robust estimation of camera locations, we introduce a two-step approach, comprised of a pairwise direction estimation method robust to outliers in point correspondences between image pairs, and a convex program to maintain robustness to outlier directions. In the presence of partially corrupted measurements, we empirically demonstrate that our convex formulation can even recover the locations exactly. Lastly, we demonstrate the utility of our formulations through experiments on Internet photo collections. It is known that the problem of multiview reconstruction can be solved in two steps: first estimate camera rotations and then translations using them. This paper presents new robust techniques for both of these steps, (i) Given pair-wise relative rotations, global camera rotations are estimated linearly in least squares, (ii) Camera translations are estimated using a standard technique based on Second Order Cone Programming. Robustness is achieved by using only a subset of points according to a new criterion that diminishes the risk of choosing a mismatch. It is shown that only four points chosen in a special way are sufficient to represent a pairwise reconstruction almost equally as all points. This leads to a significant speedup. In image sets with repetitive or similar structures, non-existent epipolar geometries may be found. Due to them, some rotations and consequently translations may be estimated incorrectly. It is shown that iterative removal of pairwise reconstructions with the largest residual and reregistration removes most non-existent epipolar geometries. The performance of the proposed method is demonstrated on difficult wide base-line image sets. One of the potentially effective means for large-scale 3D scene reconstruction is to reconstruct the scene in a global manner, rather than incrementally, by fully exploiting available auxiliary information on the imaging condition, such as camera location by Global Positioning System (GPS), orientation by inertial measurement unit (or compass), focal length from EXIF, and so on. However, such auxiliary information, though informative and valuable, is usually too noisy to be directly usable. In this paper, we present an approach by taking advantage of such noisy auxiliary information to improve structure from motion solving. More specifically, we introduce two effective iterative global optimization algorithms initiated with such noisy auxiliary information. One is a robust rotation averaging algorithm to deal with contaminated epipolar graph, the other is a robust scene reconstruction algorithm to deal with noisy GPS data for camera centers initialization. We found that by exclusively focusing on the estimated inliers at the current iteration, the optimization process initialized by such noisy auxiliary information could converge well and efficiently. Our proposed method is evaluated on real images captured by unmanned aerial vehicle, StreetView car, and conventional digital cameras. Extensive experimental results show that our method performs similarly or better than many of the state-of-art reconstruction approaches, in terms of reconstruction accuracy and completeness, but is more efficient and scalable for large-scale image data sets. Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. This paper applies a smoothing technique and an accelerated first-order algorithm, both from Nesterov [Math. Program. Ser. A, 103 (2005), pp. 127-152], and demonstrates that this approach is ideally suited for solving large-scale compressed sensing reconstruction problems as (1) it is computationally efficient, (2) it is accurate and returns solutions with several correct digits, (3) it is flexible and amenable to many kinds of reconstruction problems, and (4) it is robust in the sense that its excellent performance across a wide range of problems does not depend on the fine tuning of several parameters. Comprehensive numerical experiments on realistic signals exhibiting a large dynamic range show that this algorithm compares favorably with recently proposed state-of-the-art methods. We also apply the algorithm to solve other problems for which there are fewer alternatives, such as total-variation minimization and convex programs seeking to minimize the @math norm of @math under constraints, in which @math is not diagonal. The code is available online as a free package in the MATLAB language.