## Bundle Adjustment -- A Modern Synthesis (2000)

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Venue: | VISION ALGORITHMS: THEORY AND PRACTICE, LNCS |

Citations: | 420 - 10 self |

### BibTeX

@INPROCEEDINGS{Triggs00bundleadjustment,

author = {Bill Triggs and Philip McLauchlan and Richard Hartley and Andrew Fitzgibbon},

title = {Bundle Adjustment -- A Modern Synthesis},

booktitle = {VISION ALGORITHMS: THEORY AND PRACTICE, LNCS},

year = {2000},

pages = {298--375},

publisher = {Springer Verlag}

}

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### Abstract

This paper is a survey of the theory and methods of photogrammetric bundle adjustment, aimed at potential implementors in the computer vision community. Bundle adjustment is the problem of refining a visual reconstruction to produce jointly optimal structure and viewing parameter estimates. Topics covered include: the choice of cost function and robustness; numerical optimization including sparse Newton methods, linearly convergent approximations, updating and recursive methods; gauge (datum) invariance; and quality control. The theory is developed for general robust cost functions rather than restricting attention to traditional nonlinear least squares.

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Citation Context ...t computations, with an emphasis on surveying. Hartley & Zisserman [62] is an excellent recent textbook covering vision geometry from a computer vision viewpoint. For nonlinear optimization, Fletcher =-=[29]-=- and Gill et al [42] are the traditional texts, and Nocedal & Wright [93] is a good modern introduction. For linear least squares, Björck [11] is superlative, and Lawson & Hanson is a good older text.... |

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Citation Context ...r. nonlinearly over just the current state, assuming all previous ones to be linearized). The effects of variable window size on the Variable State Dimension Filter (VSDF) sequential bundle algorithm =-=[85, 86, 83, 84]-=- are shown in figure 7. 9 Gauge Freedom Coordinates are a very convenient device for reducing geometry to algebra, but they come at the price of some arbitrariness. The coordinate system can be change... |

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Citation Context ...cially lengthening the update steps by a heuristic factor 1 <γ<2. Most if not all of the above alternations have been applied to both the bundle problem and the independent model one many times, e.g. =-=[19, 95, 2, 108, 91, 20]-=-. Brown considered the relatively sophisticated SOR method for aerial cartography problems as early as 1964, before developing his recursive decomposition method [19]. None of these alternations are v... |

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Citation Context ...ence is significantly faster if the eigenvalues of H are tightly clustered away from� zero: if the eigenvalues are covered by intervals [ai,bi]i=1...k, conver�k � � gence occurs in O bi/ai iterations =-=[99, 47, 48]-=- 11 . Preconditioning (see below) i=1 11 For other eigenvalue based based analyses of the bundle adjustment covariance, see [103, 92].s328 B. Triggs et al. 0.05 0.04 0.03 0.02 0.01 0 −0.01 −0.02 −0.03... |

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Citation Context ... within this framework: pure translation, affine, quadratic or homographic patch deformation models, 3D model based intensity predictions, coupled affine or spline patches for surface coverage, etc., =-=[1, 52, 55, 9, 110, 94, 53, 97, 76, 104, 102]-=-. The structure of intensity based bundle problems is very similar to that of feature based ones, so all of the techniques studied below can be applied. We will not go into more detail on intensity ma... |

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Citation Context ...for fixed z and x) are small and for typical f rather simple. So in spite of the extra parameters z, optimizing this model is not significantly more expensive than optimizing an explicit one z = z(x) =-=[14, 13, 105, 106]-=-. For example, when estimating matching constraints between image pairs or triplets [60, 62], instead of using an explicit 3D representation, pairs or triplets of corresponding image points can be use... |

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Citation Context ...[ai,bi]i=1...k, conver�k � � gence occurs in O bi/ai iterations [99, 47, 48] 11 . Preconditioning (see below) i=1 11 For other eigenvalue based based analyses of the bundle adjustment covariance, see =-=[103, 92]-=-.s328 B. Triggs et al. 0.05 0.04 0.03 0.02 0.01 0 −0.01 −0.02 −0.03 −0.04 Gauss−Newton 11 steps 1.2e+06 flops Diag. Precond. Conjugate Gradient 12 steps 7.5e+06 flops Resect−Intersect with line search... |

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Citation Context ...ting sets. Finding a globally minimal partition sequence is NP complete but several effective heuristics exist. This is currently an active research field. One promising family are multilevel schemes =-=[70, 71, 65, 4]-=- which decimate (subsample) the graph, partition using e.g. a spectral method, then refine the result to the original graph. (These algorithms should also be very well-suited to graph based visual seg... |

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Citation Context ...t along epipolar lines in the other images. Initial versions assumed known camera matrices, but a full patch-based bundle method was later investigated [9]. Related methods in computer vision include =-=[94, 98, 67]-=-. Globally enforced least squares matching [53, 97, 76] further stabilizes the solution in low-signal regions by enforcing continuity constraints between adjacent patches. Patches are arranged in a gr... |

42 |
Novelle méthodes pour la détermination des orbites des comètes, Paris. List of Tables 1 Standard coefficients of variation of errors of various prediction models in a) compliance, b) shrinkage, c) relative compliance, and d) relative shrinkage (cubes a
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Citation Context ...ncluding literature references. Least squares: The theory of combining measurements by minimizing the sum of their squared residuals was developed independently by Gauss and Legendre around 1795–1820 =-=[37, 74]-=-, [36, Vol.IV, 1–93], about 40 years after robust L1 estimation [15]. Least squares was motivated by estimation problems in astronomy and geodesy and extensively applied to both fields by Gauss, whose... |

40 |
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Citation Context ...ptimizing this model is not significantly more expensive than optimizing an explicit one z = z(x) [14, 13, 105, 106]. For example, when estimating matching constraints between image pairs or triplets =-=[60, 62]-=-, instead of using an explicit 3D representation, pairs or triplets of corresponding image points can be used as features zi, subject to the epipolar or trifocal geometry contained in x [105, 106]. Ho... |

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Citation Context ...r. nonlinearly over just the current state, assuming all previous ones to be linearized). The effects of variable window size on the Variable State Dimension Filter (VSDF) sequential bundle algorithm =-=[85, 86, 83, 84]-=- are shown in figure 7. 9 Gauge Freedom Coordinates are a very convenient device for reducing geometry to algebra, but they come at the price of some arbitrariness. The coordinate system can be change... |

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Citation Context ...r. nonlinearly over just the current state, assuming all previous ones to be linearized). The effects of variable window size on the Variable State Dimension Filter (VSDF) sequential bundle algorithm =-=[85, 86, 83, 84]-=- are shown in figure 7. 9 Gauge Freedom Coordinates are a very convenient device for reducing geometry to algebra, but they come at the price of some arbitrariness. The coordinate system can be change... |

37 |
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Citation Context ...seful statistical interpretation, the Wi should be chosen to approximate the inverse measurement covariance of zi . Even for non-Gaussian noise with this mean and covariance, the Gauss-Markov theorem =-=[37, 11]-=- states that if the models zi(x) are 3 The latter case corresponds to a hard inlier / outlier decision rule: for any observation in the ‘outlier’ region, the density is constant so the observation has... |

34 |
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Citation Context ... and sparsity. We will consider several such schemes below for bundle adjustment. Classically, bundle adjustment and similar adjustment computations are formulated as nonlinear least squares problems =-=[19, 46, 100, 21, 22, 69, 5, 73, 109]-=-. The cost function is assumed to be quadratic in the feature reprojection errors, and robustness is provided by explicit outlier screening. Although it is already very flexible, this model is not rea... |

33 |
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Citation Context ...is significantly faster if the eigenvalues of H are tightly clustered � away from zero: if the eigenvalues are covered by intervals �k � � [ai,bi]i=1...k, convergence occurs in O i=1 bi/ai iterations =-=[99, 47, 48]-=- 11 . Preconditioning (see below) aims at achieving such clustering. As with alternation methods, there is a range of possible update / re-linearization choices, ranging from a fully nonlinear method ... |

29 |
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Citation Context ... for this, even if application constraints make it difficult to use them. The extraordinary extent to which weak geometry and lack of redundancy can mask gross errors is too seldom appreciated, c.f . =-=[34, 50, 30, 33]-=-. • “Point P is reconstructed accurately”: In reconstruction, just as there are no absolute references for position, there are none for uncertainty. The 3D coordinate frame is itself uncertain, as it ... |

28 |
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Citation Context ... and sparsity. We will consider several such schemes below for bundle adjustment. Classically, bundle adjustment and similar adjustment computations are formulated as nonlinear least squares problems =-=[19, 46, 100, 21, 22, 69, 5, 73, 109]-=-. The cost function is assumed to be quadratic in the feature reprojection errors, and robustness is provided by explicit outlier screening. Although it is already very flexible, this model is not rea... |

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Citation Context ... asymptotic limit is usually assumed to be a valid for well-designed highlyredundant photogrammetric measurement networks, but recent sampling-based empirical studies of posterior likelihood surfaces =-=[35, 80, 68]-=- suggest that the case is much less clear for small vision geometry problems and weaker networks. More work is needed on this. 1 Cost is additive, so as measurements of the same type are added the ent... |

23 | Gauge independence in optimization algorithms for 3D vision
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Citation Context ...ation has occurred’ but with uneven scaling it becomes ‘with the largest scale’. (iii) The choice of gauge (datum, §9) may depend on variable scaling, and this can significantly influence convergence =-=[82, 81]-=-. For all of these reasons, it is important to choose variable scalings that relate meaningfully to the problem structure. This involves a judicious comparison of the relative influence of, e.g., a un... |

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Citation Context ...ation has occurred’ but with uneven scaling it becomes ‘with the largest scale’. (iii) The choice of gauge (datum, §9) may depend on variable scaling, and this can significantly influence convergence =-=[82, 81]-=-. For all of these reasons, it is important to choose variable scalings that relate meaningfully to the problem structure. This involves a judicious comparison of the relative influence of, e.g., a un... |

22 |
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Citation Context ...thin this framework: pure translation, affine, quadratic or homographic 12spatch deformation models, 3D model based intensity predictions, coupled affine or spline patches for surface coverage, etc., =-=[1, 52, 55, 9, 110, 94, 53, 97, 76, 104, 102]-=-. The structure of intensity based bundle problems is very similar to that of feature based ones, so all of the techniques studied below can be applied. We will not go into more detail on intensity ma... |

22 | Efficient Iterative Solution to M-View Projective Reconstruction Problem
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(Show Context)
Citation Context ...cially lengthening the update steps by a heuristic factor 1 <γ<2. Most if not all of the above alternations have been applied to both the bundle problem and the independent model one many times, e.g. =-=[19, 95, 2, 108, 91, 20]-=-. Brown considered the relatively sophisticated SOR method for aerial cartography problems as early as 1964, before developing his recursive decomposition method [19]. None of these alternations are v... |