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

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

Citations: | 555 - 12 self |

### Citations

4704 |
Multiple View Geometry in Computer Vision
- Hartley, Zisserman
- 2000
(Show Context)
Citation Context ...w quite dated, although its presentation of bundle adjustment is still relevant. Wolf & Ghiliani [109] is a text devoted to adjustment 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 & Wrigh... |

3249 | Numerical optimization
- Nocedal, SJ
- 1999
(Show Context)
Citation Context ...s 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. For more general numerical linear algebra, Golub & Van Loan [44] is the ... |

1950 |
Practical Optimization
- Gill, Murray, et al.
- 1981
(Show Context)
Citation Context ... 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. For more general nu... |

1573 |
Practical Methods of Optimization
- Fletcher
- 1987
(Show Context)
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.... |

866 |
Numerical Methods for Least Squares Problems
- Björck
- 1996
(Show Context)
Citation Context ...er 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. For more general numerical linear algebra, Golub & Van Loan [44] is the standard. Duff et al [26] and George & Liu [40] are the standard text... |

732 | Theoretical Statistics - Cox, Hinkley - 1974 |

665 | Direct methods for sparse matrices - Duff, Erisman, et al. - 1987 |

634 | Parallel multilevel k-way partitioning scheme for irregular graphs
- Karypis, Kumar
- 1999
(Show Context)
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... |

631 |
der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition
- Barrett, Berry, et al.
- 1994
(Show Context)
Citation Context ...an decomposition in C with several ordering methods. For iterative linear system solvers implementation is seldom difficult, but there are again many methods and implementations. The ‘Templates’ book =-=[10]-=- contains potted code. For nonlinear optimization there are various older codes such as MINPACK, and more recent codes designed mainly for very large problems such as MINPACK-2 (ftp://info.mcs.anl.gov... |

602 |
Computer Solution of Large Sparse Positive Definite Systems
- George, Liu
- 1981
(Show Context)
Citation Context ...least squares, Björck [11] is superlative, and Lawson & Hanson is a good older text. For more general numerical linear algebra, Golub & Van Loan [44] is the standard. Duff et al [26] and George & Liu =-=[40]-=- are the standard texts on sparse matrix techniques. We will not discuss initialization methods for bundle adjustment in detail, but appropriate reconstruction methods are plentiful and well-known in ... |

544 |
What can be seen in three dimensions with an uncalibrated stereo rig
- Faugeras
- 1992
(Show Context)
Citation Context ...uge invariance. Examples of trivial gauges include: (i) using several visible 3D points as a ‘projective basis’ for reconstruction (i.e. fixing their projective 3D coordinates to simple values, as in =-=[27]-=-); and (ii) fixing the components of one projective 3 × 4 camera matrix as (I 0), as in [61] (this only partially fixes the 3D projective gauge — 3 projective 3D degrees of freedom remain unfixed). Li... |

447 |
Iterative Methods for Solving Linear Systems
- Greenbaum
- 1997
(Show Context)
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 ... |

308 |
Kanade T. A multiple-baseline stereo
- Okutomi
- 1993
(Show Context)
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... |

269 |
T.: Stereo from uncalibrated cameras
- Hartley, Gupta, et al.
- 1992
(Show Context)
Citation Context ... ‘projective basis’ for reconstruction (i.e. fixing their projective 3D coordinates to simple values, as in [27]); and (ii) fixing the components of one projective 3 × 4 camera matrix as (I 0), as in =-=[61]-=- (this only partially fixes the 3D projective gauge — 3 projective 3D degrees of freedom remain unfixed). Linearized gauge: Let the local linearizations of the gauge functions be: d(x + δx) ≈ d(x)+D δ... |

265 |
Nested dissection of a regular finite element mesh
- George
- 1973
(Show Context)
Citation Context ... into smaller sub-problems which are solved quasi-independently and later merged. 6.3.2 Top-down Ordering Methods The most common top-down method is called nested dissection or recursive partitioning =-=[64, 57, 19, 38, 40, 11]-=-. The basic idea is to recursively split the factorization problem into smaller sub24sHessian Natural Cholesky Minimum Degree Reverse Cuthill-McKee Figure 4: A bundle Hessian for an irregular coverage... |

262 | A maximum-flow formulation of the n-camera stereo correspondence problem
- Roy, Cox
- 1998
(Show Context)
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... |

252 | Close-range camera calibration
- Brown
- 1971
(Show Context)
Citation Context ...ion), and simple internal parameters such as focal length and principal point, real cameras also require various types of additional parameters to model internal aberrations such as radial distortion =-=[17, 18, 19, 100, 69, 5]-=-. 4sFor simplicity, suppose that the scene is modelled by individual static 3D features Xp, p =1...n, imaged in m shots with camera pose and internal calibration parameters Pi, i =1...m.Theremay also ... |

245 | Automatic camera recovery for closed or open image sequences
- Fitzgibbon, Zisserman
- 1998
(Show Context)
Citation Context ...view geometry and numerics (the latter being mainly C++ wrappers to well established routines from Netlib — see below). A bundle adjustment code exists for it but is not currently planned for release =-=[28, 62]-=-. C.2 Software Resources A great deal of useful numerical linear algebra and optimization software is available on the Internet, although more commonly in FORTRAN than in C/C++. The main repository is... |

172 | Linear pushbroom cameras
- Gupta, Hartley
- 1997
(Show Context)
Citation Context ...rd, but the affine and orthographic projections are sometimes useful for distant cameras, and more exotic models such as push-broom and rational polynomial cameras are needed for certain applications =-=[56, 63]-=-. In addition to pose (position and orientation), and simple internal parameters such as focal length and principal point, real cameras also require various types of additional parameters to model int... |

161 |
Methods for Modifying Matrix Factorizations
- Golub, Murray, et al.
- 1974
(Show Context)
Citation Context ...akes O � n 2� operations. The same recursion rule (and several equivalent forms) can be derived by reducing (L b) ⊤ to an upper triangular matrix using Givens rotations or Householder transformations =-=[43, 11]-=-. C Software C.1 Software Organization For a general purpose bundle adjustment code, an extensible object-based organization is natural. The measurement network can be modelled as a network of objects... |

111 |
Multiphoto Geometrically Constrained Matching
- Baltsavias
- 1991
(Show Context)
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... |

103 |
A fast and highly quality multilevel scheme for partitioning irregular graphs
- Karypis, Kumar
- 1999
(Show Context)
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... |

96 |
A testing procedure for use in geodetic networks
- Baarda
- 1968
(Show Context)
Citation Context ...known in vision, especially among researchers working on statistical issues. Förstner [33, 34] and Grün [49, 50] give introductions with some sobering examples of the effects of poor design. See also =-=[7, 8, 21, 22]-=-. All of these papers use least squares cost functions and scalar measurements. Our treatment generalizes this to allow robust cost functions and vector measurements, and is also slightly more self-co... |

96 | On determining the Fundamental matrix: analysis of different methods and experimental results
- Luong, Deriche, et al.
- 1993
(Show Context)
Citation Context ...points in ‘two point’ line parametrizations, with their two 1D motion-along-line groups. (iii) The underspecified 3×3 homographies used for ‘homography + epipole’ parametrizations of matching tensors =-=[77, 62, 106]-=-. For example, the fundamental matrix can be parametrized as F =[e ] × H where e is its left epipole and H is the inter-image homography induced by any 3D plane. The choice of plane gives a freedom H ... |

91 |
LAPACK User's Guide, Third Edition
- Anderson, Bai, et al.
- 1999
(Show Context)
Citation Context ...e/, which is based in part on Moré & Wright’s guide book [90]; and the Object Oriented Numerics page http://oonumerics.org. For large-scale dense linear algebra, LAPACK (http://www.netlib.org/lapack, =-=[3]-=-) is the best package available. However it is optimized for relatively large problems (matrices of size 100 or more), so if you are solving many small ones (size less than 20 or so) it may be faster ... |

91 | Lines and points in three views and the trifocal tensor
- Hartley
- 1997
(Show Context)
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... |

80 | A unifying framework for structure and motion recovery from image sequences
- McLauchlan, Murray
- 1995
(Show Context)
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... |

76 |
On the rates of convergence of the Lanczos and the block-Lanczos methods
- Saad
- 1980
(Show Context)
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... |

75 | A unified factorization algorithm for points, line segments and planes with uncertainty models, ICCV
- Morris, Kanade
- 1998
(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... |

66 |
A stable and efficient algorithm for nonlinear orthogonal distance regression
- Boggs, Byrd, et al.
- 1987
(Show Context)
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... |

66 |
1805) Nouvelle Methodes pour la Determination des Orbites des Cometes
- Legendre
(Show Context)
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... |

62 |
Optimization Software Guide
- Moré, Wright
- 1993
(Show Context)
Citation Context ...ul sites include: the ‘Guide to Available Mathematical Software’ GAMS at http://gams.nist.gov; the NEOS guide http://wwwfp.mcs.anl.gov/otc/Guide/, which is based in part on Moré & Wright’s guide book =-=[90]-=-; and the Object Oriented Numerics page http://oonumerics.org. For large-scale dense linear algebra, LAPACK (http://www.netlib.org/lapack, [3]) is the best package available. However it is optimized f... |

58 | S.B.: Shape ambiguities in structure from motion
- Szeliski, Kang
- 1997
(Show Context)
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... |

55 |
18 21). "Theoria Combinationis Observatorium Pars Posterior
- Gauss
(Show Context)
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... |

50 | Robust ordering of sparse matrices using multisection
- Ashcraft, Liu
- 1998
(Show Context)
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... |

43 | Direct recovery of planar-parallax from multiple frames
- Irani, Anandan, et al.
- 2002
(Show Context)
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... |

41 |
and points in three views and the trifocal tensor
- Lines
- 1997
(Show Context)
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... |

40 | Active camera calibration for a head-eye platform using the variable state-dimension filter
- McLauchlan, Murray
- 1996
(Show Context)
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... |

39 |
The bundle adjustment û progress and prospects
- Brown
- 1976
(Show Context)
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... |

39 |
Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences
- Greenbaum
- 1989
(Show Context)
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 ... |

39 | A batch/recursive algorithm for 3d scene reconstruction
- McLauchlan
(Show Context)
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... |

38 |
Bundle adjustment methods in engineering photogrammetry. The Photgrammetric Record
- Granshaw
- 1980
(Show Context)
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... |

37 | Improving the run time and quality of nested dissection ordering
- Hendrickson, Rothberg
- 1998
(Show Context)
Citation Context |

35 |
Reliability analysis of parameter estimation in linear models with application to mensuration problems in computer vision
- Forstner
- 1987
(Show Context)
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 ... |

30 |
A solution to the general problem of multiple station analytical stereotriangulation
- Brown
- 1958
(Show Context)
Citation Context ... the camera one. HSS is block diagonal, so the reduction can be calculated rapidly by a sum of contributions from the individual 3D features ‘p’inS. Brown’s original 1958 method for bundle adjustment =-=[16, 19, 100]-=- was based on finding the reduced camera system as above, and solving it using Gaussian elimination. Profile Cholesky decomposition (§B.3) offers a more streamlined method of achieving this. Occasiona... |

28 |
Digital image correlation: Performance and potential application in photogrammetry
- Ackermann
- 1984
(Show Context)
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... |

25 | Ecient iterative solution to m-view projective reconstruction problem
- Chen, Medioni
- 1999
(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... |

25 | Bayesian structure from motion
- Forsyth, Ioffe, et al.
- 1999
(Show Context)
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
- McLauchlan
- 2000
(Show Context)
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... |

23 | Gauge invariance in projective 3d reconstruction
- McLauchlan
- 1999
(Show Context)
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... |

20 |
A feature-based correspondence algorithm for image matching
- Förstner
- 1984
(Show Context)
Citation Context ... extracted image points. Automated image processing was clearly desirable, but it only gradually became feasible owing to the sheer size and detail of photogrammetric images. Both feature based, e.g. =-=[31,32]-=-, and direct (region based) [1, 52, 55, 110] methods were studied, the latter especially for matching low-contrast natural terrain in cartographic applications. Both rely on some form of least squares... |

18 | The cubic rational polynomial camera model - Hartley, Saxena |

17 | A parallel feature tracker for extended image sequences
- Szeliski, Kang, et al.
- 1995
(Show Context)
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... |

16 |
On the geometric precision of digital correlation
- Förstner
- 1982
(Show Context)
Citation Context ... extracted image points. Automated image processing was clearly desirable, but it only gradually became feasible owing to the sheer size and detail of photogrammetric images. Both feature based, e.g. =-=[31,32]-=-, and direct (region based) [1, 52, 55, 110] methods were studied, the latter especially for matching low-contrast natural terrain in cartographic applications. Both rely on some form of least squares... |

16 |
Large-scale geodetic least-squares adjustment by dissection and orthogonal decomposition
- Golub, Plemmons
- 1980
(Show Context)
Citation Context ...follows (and similarly but transposed for U): � � � j−1 (L−1 )ii =(Lii) −1 , (L−1 )ji = −L −1 jj Ljk (L k=i −1 )ki = − (L k=i+1 −1 )jk Lki i=1...n , j=i+1...n i=n...1 , j=n...i+1 � � j� Alternatively =-=[45, 11]-=-, the diagonal and the (zero) upper triangle of the linear system UA −1 = D −1 L −1 can be combined with the (zero) lower triangle of A −1 L = U −1 D −1 to give the direct recursion (i = n...1 and j =... |

15 | Statistical Concepts in Geodesy - Baarda - 1967 |

14 |
Reducing the profile of sparse symmetric matrices
- Snay
- 1976
(Show Context)
Citation Context ...ransposed for U): (L−1 )ii =(Lii) −1 , (L−1 )ji = −L −1 � �j−1 jj Ljk (L k=i −1 � � j� )ki = − (L k=i+1 −1 )jk Lki i=1...n , j=i+1...n i=n...1 , j=n...i+1 � L −1 ii 30 Snay’s Banker’s strategy (§6.3, =-=[101, 24]-=-) seems to be one of the most effective ordering strategies. (52)s360 B. Triggs et al. Alternatively [45, 11], the diagonal and the (zero) upper triangle of the linear system UA −1 = D −1 L −1 can be ... |

13 |
An automatic reordering scheme for simultaneous equations derived from network systems
- King
- 1970
(Show Context)
Citation Context ...sophisticated are the so-called banker’s strategies, which maintain an active set of all the variables coupled to the already-eliminated ones, and choose the next variable — from the active set (King =-=[72]-=-), it and its neighbours (Snay [101]) or all uneliminated variables (Levy [75]) — to minimize the new size of the active set at each step. In particular, Snay’s banker’s algorithm is reported to perfo... |

12 |
Die mathematischen und physikalischen Theorien der höheren Geodäsie. 1
- Helmert
- 1880
(Show Context)
Citation Context ... into smaller sub-problems which are solved quasi-independently and later merged. 6.3.2 Top-down Ordering Methods The most common top-down method is called nested dissection or recursive partitioning =-=[64, 57, 19, 38, 40, 11]-=-. The basic idea is to recursively split the factorization problem into smaller sub24sHessian Natural Cholesky Minimum Degree Reverse Cuthill-McKee Figure 4: A bundle Hessian for an irregular coverage... |

11 |
A comparison of some methods for solving sparse linear least-squares problems
- George, Heath, et al.
- 1983
(Show Context)
Citation Context ...ast squares Jacobian. Sparse QR methods can be implemented either with Givens rotations or (more efficiently) with sparse Householder transformations. Row ordering is important for the Givens methods =-=[39]-=-. For Householder ones (and some Givens ones too) the multifrontal organization is now usual [41, 11], as it captures the natural parallelism of the problem. 7 Implementation Strategy 2: First Order A... |

10 |
3 Königlichen Gesellschaft der Wissenschaften. Göttingen (1866
- Gauss, Band
(Show Context)
Citation Context ... 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 remarkable 1823 monograph =-=[37, 36]-=- already contains almost the complete modern theory of least squares including elements of the theory of probability distributions, the definition and properties of the Gaussian distribution, and a di... |

10 |
The variable state dimension filter
- McLauchlan
- 1999
(Show Context)
Citation Context |

8 |
Calibration of Close Range Camera
- Brown, C
(Show Context)
Citation Context ...ion), and simple internal parameters such as focal length and principal point, real cameras also require various types of additional parameters to model internal aberrations such as radial distortion =-=[17, 18, 19, 100, 69, 5]-=-. 4sFor simplicity, suppose that the scene is modelled by individual static 3D features Xp, p =1...n, imaged in m shots with camera pose and internal calibration parameters Pi, i =1...m.Theremay also ... |

8 |
Non-Topographic Photogrammetry. Americal Society for Photogrammetry and Remote
- Karara
- 1989
(Show Context)
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... |

7 |
De Litteraria Expeditione per Pontificiam Ditionem
- Marie, Boscovich
- 1755
(Show Context)
Citation Context ...easurements 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 remarkable 1823 monograph [37, 36] already contains almost the comp... |

7 |
5, Königliche Gesellschaft der Wissenschaften
- Gauss, Werke
- 1867
(Show Context)
Citation Context ... 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 remarkable 1823 monograph =-=[37, 36]-=- already contains almost the complete modern theory of least squares including elements of the theory of probability distributions, the definition and properties of the Gaussian distribution, and a di... |

7 | Householder reflections versus Givens rotations in sparse orthogonal decomposition - George, Liu - 1987 |

7 |
Euclidean reconstruction from multiple views
- Hartley
- 1993
(Show Context)
Citation Context ...pose to calibration parameters can be added in the same way. Brown’s method is probably what most vision researchers think of as ‘bundle adjustment’, following descriptions by Slama [100] and Hartley =-=[58, 59]-=-. It is still a reasonable choice for small dense networks 28 , but it rapidly becomes inefficient for the large sparse ones that arise in aerial cartography and large-scale site modelling. For larger... |

7 |
Resequencing of the structural stiffness matrix to improve computational efficiency
- Levy
- 1971
(Show Context)
Citation Context ...set of all the variables coupled to the already-eliminated ones, and choose the next variable — from the active set (King [72]), it and its neighbours (Snay [101]) or all uneliminated variables (Levy =-=[75]-=-) — to minimize the new size of the active set at each step. In particular, Snay’s banker’s algorithm is reported to perform well on geodesy and aerial cartography problems [101, 24]. For all of these... |

6 |
Statistical concepts and their application in photogrammetry and surveying (continued
- Cooper, Cross
- 1991
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5 |
The Photogrammetric Inner Constraints
- Dermanis
- 1994
(Show Context)
Citation Context ... along gauge orbits, i.e. by applying local coordinate transformations that vary depending on the particular structure involved. Such transformations are called S-transforms (‘similarity’ transforms) =-=[6, 107, 22, 25]-=-. Different gauges through the same central state represent coordinatization rules that agree for the central geometry but differ for perturbed ones — the S-transform is the identity at the centre but... |

5 |
The reliability of block triangulation
- Förstner
- 1985
(Show Context)
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 ... |

5 |
An optimum algorithm for on-line triangulation
- Grün
- 1982
(Show Context)
Citation Context ...becomes inefficient for problems with many features because the sparse structure is not exploited: the full covariance matrix is dense and we would normally avoid calculating it in its entirety. Grün =-=[51, 54]-=- avoids this problem by maintaining a running copy of the reduced camera system (20), using an incremental Schur complement / forward substitution (16) to fold each new observation into this, and then... |

5 |
Adaptive least squares correlation with geometrical constraints
- Grün, Baltsavias
- 1985
(Show Context)
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... |

5 | An object-oriented approach to scene reconstruction
- Hartley
- 1996
(Show Context)
Citation Context ...pose to calibration parameters can be added in the same way. Brown’s method is probably what most vision researchers think of as ‘bundle adjustment’, following descriptions by Slama [100] and Hartley =-=[58, 59]-=-. It is still a reasonable choice for small dense networks 28 , but it rapidly becomes inefficient for the large sparse ones that arise in aerial cartography and large-scale site modelling. For larger... |

4 |
Linear least squares computations using Givens transformations
- Blais
- 1983
(Show Context)
Citation Context ... exploited, they handle any number of images and features and arbitrary problem structure efficiently, and they are numerically more accurate than Woodbury formula methods. The Givens rotation method =-=[12, 54]-=-, which is equivalent to the rank 1 Cholesky update (56), is probably the most common such method. The other updating methods are confusingly named in the literature. Mikhail & Helmering’s method [88]... |

4 |
Hierarchical Multi-point Matching with Simultaneous Detection and Location of Breaklines
- Li
- 1989
(Show Context)
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... |

4 | Recursive methods in photogrammetric data reduction - Mikhail - 1973 |

3 |
S-transformations and criterion matrices. Netherlands Geodetic Commission
- Baarda
- 1967
(Show Context)
Citation Context ... along gauge orbits, i.e. by applying local coordinate transformations that vary depending on the particular structure involved. Such transformations are called S-transforms (‘similarity’ transforms) =-=[6, 107, 22, 25]-=-. Different gauges through the same central state represent coordinatization rules that agree for the central geometry but differ for perturbed ones — the S-transform is the identity at the centre but... |

3 |
reliability and statistics in close range photogrammetry
- Accuracy
- 1978
(Show Context)
Citation Context ...ust a few observations. The photogrammetric literature on quality control deserves to be better known in vision, especially among researchers working on statistical issues. Förstner [33, 34] and Grün =-=[49, 50]-=- give introductions with some sobering examples of the effects of poor design. See also [7, 8, 21, 22]. All of these papers use least squares cost functions and scalar measurements. Our treatment gene... |

3 |
Adaptive least squares correlation — concept and first results. Intermediate Research Report to Helava Associates
- Grün
- 1984
(Show Context)
Citation Context |

3 |
Test of algorithms for sequential adjustment in on-line triangulation
- Holm
- 1989
(Show Context)
Citation Context ...ting ones. Consider first the case where no parameters need be added nor deleted, e.g. adding or deleting an observation of an existing point in an existing image. Several methods have been suggested =-=[54, 66]-=-. Mikhail & Helmering [88] use the Woodbury formula (18) to update the covariance H −1 . This simple approach becomes inefficient for problems with many features because the sparse structure is not ex... |

3 |
Optimal robot self-localization and reliability evaluation
- Kanatani, Ohta
- 1998
(Show Context)
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... |

3 |
Expert system- Based Design of Close-Range Photogrammetric Networks
- Mason
- 1995
(Show Context)
Citation Context ...e estimates of everything that needs to be measured are found. We will not say much about design, merely outlining the basic considerations and giving a few useful rules of thumb. See [5, chapter 6], =-=[79, 78]-=-, [73, Vol.2 §4] for more information. Factors to be considered in network design include: scene coverage, occlusion / visibility and feature viewing angle; field of view, depth of field, resolution a... |

3 |
Expert system based design of photogrammetric networks
- Mason
- 1994
(Show Context)
Citation Context ...e estimates of everything that needs to be measured are found. We will not say much about design, merely outlining the basic considerations and giving a few useful rules of thumb. See [5, chapter 6], =-=[79, 78]-=-, [73, Vol.2 §4] for more information. Factors to be considered in network design include: scene coverage, occlusion / visibility and feature viewing angle; field of view, depth of field, resolution a... |

2 |
Users reference guide for ODRPACK 2.01: Software for weighted orthogonal distance regression
- Boggs, Byrd, et al.
- 1992
(Show Context)
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... |

2 |
Precision and reliability aspects in close range photogrammetry
- Grün
- 1980
(Show Context)
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 ... |

2 |
Algorithmic aspects of on-line triangulation
- Grün
- 1985
(Show Context)
Citation Context ...ting ones. Consider first the case where no parameters need be added nor deleted, e.g. adding or deleting an observation of an existing point in an existing image. Several methods have been suggested =-=[54, 66]-=-. Mikhail & Helmering [88] use the Woodbury formula (18) to update the covariance H −1 . This simple approach becomes inefficient for problems with many features because the sparse structure is not ex... |

2 |
P.: Bootstrapping a heteroscedastic regression model with application to 3D rigid motion evaluation
- Matei, Meer
- 1999
(Show Context)
Citation Context ...asymptotic limit is usually assumed to be a valid for well-designed highly-redundant 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. The effect of incorrect error models: It is clear that incorrect modellin... |

2 |
The adjustment of aerial triangulation by electronic digital computers
- Proctor
- 1962
(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... |

1 |
Digital block adjustment
- Amer
- 1962
(Show Context)
Citation Context |

1 |
de Jonge. A comparative study of algorithms for reducing the fill-in during Cholesky factorization
- J
- 1992
(Show Context)
Citation Context ...inated variables (Levy [75]) — to minimize the new size of the active set at each step. In particular, Snay’s banker’s algorithm is reported to perform well on geodesy and aerial cartography problems =-=[101, 24]-=-. For all of these automatic ordering methods, it often pays to do some of the initial work by hand, e.g. it might be appropriate to enforce the structure / camera division beforehand and only order t... |

1 |
Evaluation of block adjustment results
- Förstner
- 1980
(Show Context)
Citation Context ...d be detected from their residuals (inner reliability), and that any remaining undetected outliers had only a limited effect on the final results (outer reliability). A. Grün [49, 50] and W. Förstner =-=[30, 33, 34]-=- adapted this theory to photogrammetry around 1980, and also gave some early correlation and covariance based model selection heuristics designed to control over-fitting problems caused by over-elabor... |

1 |
Adaptive kleinste Quadrate Korrelation and geometrische Zusatzinformationen
- Grün
- 1985
(Show Context)
Citation Context |

1 |
The inversion of the normal equations of analytical aerotriangulation by the method of recursive partitioning
- Gyer
- 1967
(Show Context)
Citation Context ... into smaller sub-problems which are solved quasi-independently and later merged. 6.3.2 Top-down Ordering Methods The most common top-down method is called nested dissection or recursive partitioning =-=[64, 57, 19, 38, 40, 11]-=-. The basic idea is to recursively split the factorization problem into smaller sub24sHessian Natural Cholesky Minimum Degree Reverse Cuthill-McKee Figure 4: A bundle Hessian for an irregular coverage... |

1 |
Die innere Genauigkeit eines Punkthaufens. Österreichische Zeitschrift für Vermessungswesen, 50(5): 159–165 and 50(6
- Meissl
- 1962
(Show Context)
Citation Context ...that minimizes the weighted size of the state update δx ⊤ G W δxG, the weighted covariance sum Trace(WVG) =Trace(L ⊤ VG L), and the L2 or Frobenius norm of L ⊤ VG L, is given by the inner constraints =-=[87, 89, 6, 22, 25]-=- 16 : D δx = 0 where D ≡ G ⊤ W (38) The corresponding covariance VG is given by (30) with D = G ⊤ W, and the state update is δxG = −VG g as usual. Also, if W is nonsingular, VG is given by the weighte... |

1 |
Zur Ausgleichung freier Netze. Zeitschrift für Vermessungswesen
- Mittermayer
- 1962
(Show Context)
Citation Context ...that minimizes the weighted size of the state update δx ⊤ G W δxG, the weighted covariance sum Trace(WVG) =Trace(L ⊤ VG L), and the L2 or Frobenius norm of L ⊤ VG L, is given by the inner constraints =-=[87, 89, 6, 22, 25]-=- 16 : D δx = 0 where D ≡ G ⊤ W (38) The corresponding covariance VG is given by (30) with D = G ⊤ W, and the state update is δxG = −VG g as usual. Also, if W is nonsingular, VG is given by the weighte... |

1 |
Uncertainty modelling for optimal structure and motion
- Morris, Kanatani, et al.
- 2000
(Show Context)
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... |

1 | Pattern Recongition and Neural Networks - Ripley - 1996 |

1 |
Accuracy improvement of digital matching for elevation of digital terrain models
- Rosenholm
- 1986
(Show Context)
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... |