Johannes Schneider and Falko Schindler and Thomas Läbe and Wolfgang Förstner, "Bundle Adjustment for Multi-camera Systems with Points at Infinity", In ISPRS Annals of Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. I-3, pp. 75-80. 2012.

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We present a novel approach for a rigorous bundle adjustment for omnidirectional and multi-view cameras, which enables an efficient maximum-likelihood estimation with image and scene points at infinity. Multi-camera systems are used to increase the resolution, to combine cameras with different spectral sensitivities (Z/I DMC, Vexcel Ultracam) or - like omnidirectional cameras - to augment the effective aperture angle (Blom Pictometry, Rollei Panoscan Mark III). Additionally multi-camera systems gain in importance for the acquisition of complex 3D structures. For stabilizing camera orientations - especially rotations - one should generally use points at the horizon over long periods of time within the bundle adjustment that classical bundle adjustment programs are not capable of. We use a minimal representation of homogeneous coordinates for image and scene points. Instead of eliminating the scale factor of the homogeneous vectors by Euclidean normalization, we normalize the homogeneous coordinates spherically. This way we can use images of omnidirectional cameras with single-view point like fisheye cameras and scene points, which are far away or at infinity. We demonstrate the feasibility and the potential of our approach on real data taken with a single camera, the stereo camera FinePix Real 3D W3 from Fujifilm and the multi-camera system Ladybug3 from Point Grey.

@inproceedings{Schneider2012Bundle,
  author = {Schneider, Johannes and Schindler, Falko and L\"abe, Thomas and F\"orstner, Wolfgang},
  title = {Bundle Adjustment for Multi-camera Systems with Points at Infinity},
  booktitle = {ISPRS Annals of Photogrammetry, Remote Sensing and Spatial Information Sciences},
  year = {2012},
  volume = {I-3},
  pages = {75--80},
  doi = {10.5194/isprsannals-I-3-75-2012}
}

Wolfgang Förstner, "Minimal Representations for Uncertainty and Estimation in Projective Spaces", In Proc. of Asian Conference on Computer Vision., pp. 619-633, Part II. 2010.

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Estimation using homogeneous entities has to cope with obstacles such as singularities of covariance matrices and redundant parametrizations which do not allow an immediate definition of maximum likelihood estimation and lead to estimation problems with more parameters than necessary. The paper proposes a representation of the uncertainty of all types of geometric entities and estimation procedures for geometric entities and transformations which (1) only require the minimum number of parameters, (2) are free of singularities, (3) allow for a consistent update within an iterative procedure, (4) enable to exploit the simplicity of homogeneous coordinates to represent geometric constraints and (5) allow to handle geometric entities which are at innity or at least very far, avoiding the usage of concepts like the inverse depth. Such representations are already available for transformations such as rotations, motions (Rosenhahn 2002), homographies (Begelfor 2005), or the projective correlation with fundamental matrix (Bartoli 2004) all being elements of some Lie group. The uncertainty is represented in the tangent space of the manifold, namely the corresponding Lie algebra. However, to our knowledge no such representations are developed for the basic geometric entities such as points, lines and planes, as in addition to use the tangent space of the manifolds we need transformation of the entities such that they stay on their specific manifold during the estimation process. We develop the concept, discuss its usefulness for bundle adjustment and demonstrate (a) its superiority compared to more simple methods for vanishing point estimation, (b) its rigour when estimating 3D lines from 3D points and (c) its applicability for determining 3D lines from observed image line segments in a multi view setup.

@inproceedings{Forstner2010Minimal,
  author = {F\"orstner, Wolfgang},
  title = {Minimal Representations for Uncertainty and Estimation in Projective Spaces},
  booktitle = {Proc. of Asian Conference on Computer Vision},
  year = {2010},
  pages = {619--633, Part II},
  note = {Queenstown, New Zealand},
  doi = {10.1007/978-3-642-19309-5_48}
}