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## Optimizing the Viewing Graph for Structure-from-Motion

### Citations

4738 |
Multiple View Geometry in Computer Vision
- Hartley, Zisserman
- 2000
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Citation Context ...asible when considering all three edges [26, 33]. Ideally, the edges in these loops would be consistent with each other. Condition 1. A triplet of fundamental matrices is consistent when they satisfy =-=[15]-=-: e>ikFijejk = e > ijFikekj = e > jiFjkeki = 0 , (1) where eij is the epipole of Fij corresponding to the image of camera center j in view i and eij 6= eik i.e., the noncollinearity condition is satis... |

676 | Photo tourism: Exploring photo collections
- Snavely, Seitz, et al.
- 2006
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Citation Context ...lative geometries from the viewing graph as an input and output a reconstruction consisting of camera poses and 3D points. The traditional method for computing a SfM reconstruction is incremental SfM =-=[28, 32]-=- which progressively grows a reconstruction by adding one new view at a time. Incremental SfM requires repeatedly performing nonlinear optimization (i.e., bundle adjustment) as the reconstruction grow... |

129 |
On benchmarking camera calibration and multi-view stereo for high resolution imagery”,
- Strecha, Hansen, et al.
- 2008
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Citation Context ...been resized or cropped. Using the median focal length is very inaccurate and is not sufficient for use in a SfM pipeline. 6.3. Structure-from-Motion We ran our pipeline on the small-scale dataset of =-=[30]-=- and the large-scale datasets of [31] to measure the performance and feasibility of our method on real data. We compare our SfM pipeline to several alternative global SfM pipelines, and the results ar... |

82 |
Group-Theoretical Methods in Image Understanding.
- Kanatani
- 1990
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Citation Context ...form t×R for a given relative translation t and rotation R if and only if E is rank 2 with its two non-zero singular values equal [15]. This property may be encapsulated by the scalar invariants of E =-=[17]-=-: C = ||EE>||2 − 1 2 ||E||4 . (10) For a valid essential matrix E, the cost function C will be 0. Kanatani and Matsunaga [18] show that Eq. (10) may be used to recover the two focal lengths from a fun... |

75 | Robust rotation and translation estimation in multiview reconstruction
- Martinec, Pajdla
- 2007
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Citation Context ...raph) to simultaneously estimate all camera poses in a single step [3, 11, 12]. These methods operate on calibrated image sets by first estimating the global orientation of all cameras simultaneously =-=[6, 13, 14, 21]-=-, then solving for the camera positions simultaneously [3, 16, 22, 31]. Finally, structure is estimated and a global bundle adjustment is applied. Since bundle adjustment is generally the most expensi... |

63 | Camera network calibration from dynamic silhouettes.
- Sinha, Pollefeys, et al.
- 2004
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Citation Context .... Triplet Projection Error We define here the triplet projection error used in Section 4.1. Given three views, i, j, and k, and the corresponding fundamental matrices Fij , Fik, and Fjk, Sinha et al. =-=[27]-=- compute a consistent triplet of fundamental matrices. We use their technique to define a triplet projection error that measures the consistency of a triplet of fundamental matrices. We will briefly s... |

60 | Discrete-continuous optimization for largescale structure from motion - Crandall, Owens, et al. |

52 | Towards linear-time incremental structure from motion. In:
- Wu
- 2013
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Citation Context ...ative rotation and translation errors of the initial viewing graph G, the subgraph G′ and the viewing graph after optimization GOPT when executed on the uncalibrated images from the Colosseum dataset =-=[32]-=-. The subgraph G′ has lower relative pose errors than the initial viewing graph and the viewing graph optimization greatly improves the quality of relative poses. 5.2. Focal Lengths from the Viewing G... |

51 | Skeletal graphs for efficient structure from motion
- Snavely, Seitz, et al.
- 2008
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Citation Context ...our knowledge, ours is the first global SfM pipeline capable of handling uncalibrated image sets. 1. Introduction The viewing graph is a fundamental tool in the context of Structure-from-Motion (SfM) =-=[20, 26, 29]-=-. This graph encapsulates the cameras that are to be estimated as vertices and the relative geometries between cameras as edges. SfM algorithms take the relative geometries from the viewing graph as a... |

35 | Disambiguating visual relations using loop constraints - Zach, Klopschitz, et al. |

30 | Combining two-view constraints for motion estimation
- Govindu
- 2001
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Citation Context ...ent. Much recent work has focused on so-called “global SfM” techniques that consider all relative poses (i.e., edges in the viewing graph) to simultaneously estimate all camera poses in a single step =-=[3, 11, 12]-=-. These methods operate on calibrated image sets by first estimating the global orientation of all cameras simultaneously [6, 13, 14, 21], then solving for the camera positions simultaneously [3, 16, ... |

28 | Nonlinear estimation of the fundamental matrix with minimal parameters.
- Bartoli, Sturm
- 2004
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Citation Context ...e nonlinear optimization must be done carefully to ensure that the resulting 3×3 matrix remains a valid fundamental matrix. We use the nonlinear fundamental matrix representation of Bartoli and Sturm =-=[4]-=- to update the fundamental matrices and briefly summarize their method here. Note that a fundamental matrix F may be decomposed into matrices U , S, and V by singular value decomposition F = USV >, wh... |

22 |
L1 rotation averaging using the Weiszfeld algorithm. In:
- Hartley, Aftab, et al.
- 2011
(Show Context)
Citation Context ...raph) to simultaneously estimate all camera poses in a single step [3, 11, 12]. These methods operate on calibrated image sets by first estimating the global orientation of all cameras simultaneously =-=[6, 13, 14, 21]-=-, then solving for the camera positions simultaneously [3, 16, 22, 31]. Finally, structure is estimated and a global bundle adjustment is applied. Since bundle adjustment is generally the most expensi... |

18 | Non-sequential structure from motion. In:
- Enqvist, Kahl, et al.
- 2011
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Citation Context ...Bundle Adjustment 9: end procedure graph. Given an input viewing graph G = {V, E}, we aim to create a subgraph G′ that sufficiently covers the viewing graph with a minimum number of edges. Similar to =-=[9]-=-, we first select the maximum spanning tree G′ = GMST where edge weights are the number of inliers from fundamental matrix estimation between two views then find all edges ET ∈ E that, if added to G′ ... |

18 |
V.M.: Robustness in motion averaging
- Govindu
(Show Context)
Citation Context ...ent. Much recent work has focused on so-called “global SfM” techniques that consider all relative poses (i.e., edges in the viewing graph) to simultaneously estimate all camera poses in a single step =-=[3, 11, 12]-=-. These methods operate on calibrated image sets by first estimating the global orientation of all cameras simultaneously [6, 13, 14, 21], then solving for the camera positions simultaneously [3, 16, ... |

16 | Rotation averaging
- Hartley, Trumpf, et al.
(Show Context)
Citation Context ...raph) to simultaneously estimate all camera poses in a single step [3, 11, 12]. These methods operate on calibrated image sets by first estimating the global orientation of all cameras simultaneously =-=[6, 13, 14, 21]-=-, then solving for the camera positions simultaneously [3, 16, 22, 31]. Finally, structure is estimated and a global bundle adjustment is applied. Since bundle adjustment is generally the most expensi... |

15 | Global motion estimation from point matches
- Arie-Nachimson, Kovalsky, et al.
- 2012
(Show Context)
Citation Context ...ent. Much recent work has focused on so-called “global SfM” techniques that consider all relative poses (i.e., edges in the viewing graph) to simultaneously estimate all camera poses in a single step =-=[3, 11, 12]-=-. These methods operate on calibrated image sets by first estimating the global orientation of all cameras simultaneously [6, 13, 14, 21], then solving for the camera positions simultaneously [3, 16, ... |

13 |
The viewing graph
- Levi, Werman
(Show Context)
Citation Context ...our knowledge, ours is the first global SfM pipeline capable of handling uncalibrated image sets. 1. Introduction The viewing graph is a fundamental tool in the context of Structure-from-Motion (SfM) =-=[20, 26, 29]-=-. This graph encapsulates the cameras that are to be estimated as vertices and the relative geometries between cameras as edges. SfM algorithms take the relative geometries from the viewing graph as a... |

12 | Video stabilization using epipolar geometry,”
- Goldstein, Fattal
- 2012
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Citation Context ...we define a cost function based on the epipolar point transfer: C(x)jki = ||xi − x̂ijk||2 . (3) This cost is a geometric error in terms of pixel distance and has previously been shown to be effective =-=[10, 25]-=-; however, care must be taken to avoid numerical instabilities (see Section 3.4). 3.2. Updating Fundamental Matrices We seek to adjust fundamental matrix edges Fij ∈ E in G based on Eq. (3). Fundament... |

12 | Closed-form expression for focal lengths from the fundamental matrix
- Kanatani, Matsunaga
- 2000
(Show Context)
Citation Context ...y decomposing the essential matrix [15]. For uncalibrated cameras, only the fundamental matrix is available between two views. Focal lengths may be obtained from the fundamental matrix in closed form =-=[18]-=- and the resulting essential matrix may be decomposed into relative rotations and translations. The relative rotations and translations obtained through fundamental matrix decomposition, however, are ... |

8 | 3d reconstruction from image collections with a single known focal length
- Bujnak, Kukelova, et al.
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Citation Context ...ix is solely dependent on the quality of the fundamental matrix estimation. Focal lengths are not a lie group and so a simple averaging of focal lengths does not give statistically meaningful results =-=[5]-=- and a more meaningful metric is needed to effectively “average” focal lengths. In this section we propose a new calibration method for simultaneously determining the focal lengths of all cameras in a... |

8 | Global fusion of relative motions for robust, accurate and scalable structure from motion.
- Moulon, Monasse, et al.
- 2013
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Citation Context ... 11, 12]. These methods operate on calibrated image sets by first estimating the global orientation of all cameras simultaneously [6, 13, 14, 21], then solving for the camera positions simultaneously =-=[3, 16, 22, 31]-=-. Finally, structure is estimated and a global bundle adjustment is applied. Since bundle adjustment is generally the most expensive part of SfM, global SfM methods are generally more efficient and sc... |

6 |
Efficient and robust large-scale rotation averaging.
- Chatterjee, Govindu
- 2013
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Citation Context |

6 |
A global linear method for camera pose registration.
- Jiang, Cui, et al.
- 2013
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Citation Context ...h compared to a referFocal Length Error 0 0.1 0.2 0.3 0.4 CD FsofsC am er as 0 0.2 0.4 0.6 0.8 1 Our Method Median Focal EXIF Figure 6. We show the accuracy of calibration methods on the Pisa dataset =-=[16]-=- and show the focal length error |f − fgt|/fgt compared to ground truth focal lengths obtained from a reconstruction from VisualSfM [32]. Our method is at least as accurate as using EXIF, and is signi... |

6 | Stable structure from motion for unordered image collections - Olsson, Enqvist |

5 | Robust global translations with 1dsfm.
- Wilson, Snavely
- 2014
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Citation Context ...imization (i.e., bundle adjustment) as the reconstruction grows in size. As a reFigure 1. Reconstructions computed from global SfM methods on the Pisa dataset [16]. Top: Standard global SfM pipelines =-=[31]-=- struggle to handle image sets with poor calibration or inaccurate relative geometries. Bottom: Our method optimizes the relative geometries in the viewing graph to enforce global consistency, resulti... |

3 |
solver. http: //ceres-solver.org
- Ceres
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Citation Context ...aph optimization to converge to a better minimum. Finally, we use all selected edges and feature tracks to optimize the viewing graph by minimizing Eq. (7) using the Ceres Solver optimization library =-=[2]-=-. We use a Huber loss function to remain robust to outliers from feature matching. 4.2. Estimating Motion The resulting optimized viewing graph provides accurate fundamental matrices that nearly form ... |

2 |
Robust camera location estimation by convex programming.
- Ozyesil, Singer
- 2014
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Citation Context ...nning times for each reconstruction method. TOPT is the time our method takes for the viewing graph optimization. Our method is 2 to 9 times faster than alternative global SfM methods. 1DSfM [31] LUD =-=[24]-=- Cui et al. [8] Our Pipeline Name TBA TΣ TBA TΣ TBA TΣ TOPT TBA TΣ Piccadilly 2425 3483 - - - - 310 702 1246 Union Square 340 452 - - - - 98 102 243 Roman Forum 1245 1457 - - - - 284 847 1232 Vienna C... |

1 |
Certifying the existence of epipolar matrices. arXiv preprint arXiv:1407.5367
- Agarwal, Lee, et al.
- 2014
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Citation Context ...stabilities (see Section 3.4). 3.2. Updating Fundamental Matrices We seek to adjust fundamental matrix edges Fij ∈ E in G based on Eq. (3). Fundamental matrices are a special class of rank-2 matrices =-=[1]-=-. Thus, updating a fundamental matrix during the nonlinear optimization must be done carefully to ensure that the resulting 3×3 matrix remains a valid fundamental matrix. We use the nonlinear fundamen... |

1 |
Linear global translation estimation from feature tracks
- Cui, Jiang, et al.
- 2015
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Citation Context ...t the relative rotations in each triplet in GC are also consistent (i.e., Algorithm 1 Standard Global SfM Pipeline 1: procedure GLOBAL SFM(G = {V, E}, Focal lengths) 2: Filter G from loop constraints =-=[8, 22, 33]-=- 3: Robust orientation estimation [6] 4: Filter relative poses [16, 22, 31] 5: Robust Position Estimation [8, 16, 22, 31] 6: Triangulate 3D points 7: Bundle Adjustment 8: end procedure C1sC2sC3sC5sC4s... |

1 |
Decomposing three fundamental matrices for initializing 3-d reconstruction from three views
- Kanazawa, Sugaya, et al.
(Show Context)
Citation Context ... lower relative pose errors than the initial viewing graph and the viewing graph optimization greatly improves the quality of relative poses. 5.2. Focal Lengths from the Viewing Graph Kanazawa et al. =-=[19]-=- extend Eq. (10) to a triplet of fundamental matrices with a simple cost function: C = C(F12) + C(F13) + C(F23) . (12) When image noise is present, this non-negative cost function is no longer guarant... |

1 |
Consistent averaging of multi-camera epipolar geometries
- Pillai
- 2008
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Citation Context ... the solvability of viewing graphs in the context of creating reconstructions. Both of these works, however, only analyze characteristics of consistent viewing graphs. In contrast, Pillai and Govindu =-=[25]-=- assume they are given a non-consistent viewing graph and present a method that attempts to modify it to form a consistent viewing graph. They iteratively re-estimate pixels locations of observed feat... |

1 | Linear solvability in the viewing graph
- Rudi, Pizzoli, et al.
- 2011
(Show Context)
Citation Context ...our knowledge, ours is the first global SfM pipeline capable of handling uncalibrated image sets. 1. Introduction The viewing graph is a fundamental tool in the context of Structure-from-Motion (SfM) =-=[20, 26, 29]-=-. This graph encapsulates the cameras that are to be estimated as vertices and the relative geometries between cameras as edges. SfM algorithms take the relative geometries from the viewing graph as a... |