Projective Bundle Adjustment from Arbitrary Initialization Using the Variable Projection Method
Bundle adjustment is used in structure-from-motion pipelines as final refinement stage requiring a sufficiently good initialization to reach a useful local mininum. Starting from an arbitrary initialization almost always gets trapped in a poor minimum. In
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University of Cambridge, Cambridge, UK {jhh37,rc10001}@cam.ac.uk Toshiba Research Europe, Cambridge, UK [email protected] 3 Microsoft, Cambridge, UK [email protected]
Abstract. Bundle adjustment is used in structure-from-motion pipelines as final refinement stage requiring a sufficiently good initialization to reach a useful local mininum. Starting from an arbitrary initialization almost always gets trapped in a poor minimum. In this work we aim to obtain an initialization-free approach which returns global minima from a large proportion of purely random starting points. Our key inspiration lies in the success of the Variable Projection (VarPro) method for affine factorization problems, which have close to 100 % chance of reaching a global minimum from random initialization. We find empirically that this desirable behaviour does not directly carry over to the projective case, and we consequently design and evaluate strategies to overcome this limitation. Also, by unifying the affine and the projective camera settings, we obtain numerically better conditioned reformulations of original bundle adjustment algorithms.
Keywords: Projective bundle adjustment Nonlinear least squares
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Variable Projection
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Introduction
Standard structure from motion (SfM) approaches are typically multi-stage pipelines comprising of feature matching or tracking, initial structure and camera estimation, and final nonlinear refinement stages. While feature matching and tracking and the nonlinear refinement stage have well-established gold standard implementations (most notably matching using SIFT features, tracking via Lucas-Kanade and nonlinear refinement via Levenberg-Marquardt), no elegant J.H. Hong and R. Cipolla—Much of the work was done while the first author was an intern at Toshiba Research Europe. c Springer International Publishing AG 2016 B. Leibe et al. (Eds.): ECCV 2016, Part I, LNCS 9905, pp. 477–493, 2016. DOI: 10.1007/978-3-319-46448-0 29
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(a) Observed tracks
(b) Initial (377.29)
(c) Best affine (9.38)
(d) Best proj. (0.84)
Fig. 1. Visualization of the Di2 (see Table 7) tracks recovered using our two-stage meta-algorithms. In each run, each meta-algorithm is initialized from random camera poses and points (Fig. 1b). In the first stage, it performs affine bundle adjustment using either a Linear or Nonlinear VarPro-based algorithms, reaching the best affine optimum (Fig. 1c) in 91 % of all runs. The outputs are then used to initialize projective bundle adjustment. Although Di2 has strong perspective effects, our recommended metaalgorithms (TSMA1 and TSMA2) both reach the best projective optimum (Fig. 1d) in 90–98 % of all runs.
and generally accepted framework for estimating the initial poses and 3D structure from feature tracks is known. Even when one has a sensible starting point (initial 3D reconstruction) available, accumulated drift, undetected loop closures, etc., require a large basin of convergence for bundle adjustment to succeed. The essence of this work is widening the convergence basi
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