Untwisting a Projective Reconstruction

A method for upgrading a projective reconstruction to metric is presented. The method compares favourably to state of the art algorithms and has been found extremely reliable for both large and small reconstructions in a large number of experiments on real data. The notion of a twisted pair is generalized to the uncalibrated case. The reconstruction is first transformed by considering cheirality so that it does not contain any twisted pairs. It is argued that this is essential, since the method then proceeds by local perturbation. As is preferrable, we utilize a cost function that reflects the prior likelihood of calibration matrices well. It is shown that with any such cost function, there is little hope of untwisting a twisted pair by local perturbation. The results show that in practice it is most often sufficient to start with a reconstruction that is free from twisted pairs, provided that the minimized objective function is a geometrically meaningful quantity. When subjected to the common degeneracy of little or no rotation between the views, the proposed method still yields a very reasonable member of the family of possible solutions. Furthermore, the method is very fast and therefore suitable for the purpose of viewing reconstructions.

[1]  J. G. Semple,et al.  Algebraic Projective Geometry , 1953 .

[2]  H. C. Longuet-Higgins,et al.  A computer algorithm for reconstructing a scene from two projections , 1981, Nature.

[3]  Jorge Stolfi,et al.  Oriented projective geometry , 1987, SCG '87.

[4]  William H. Press,et al.  Numerical Recipes in FORTRAN - The Art of Scientific Computing, 2nd Edition , 1987 .

[5]  Richard I. Hartley,et al.  Euclidean Reconstruction from Uncalibrated Views , 1993, Applications of Invariance in Computer Vision.

[6]  Olivier D. Faugeras,et al.  Relative 3D positioning and 3D convex hull computation from a weakly calibrated stereo pair , 1995, Image Vis. Comput..

[7]  S. P. Mudur,et al.  Three-dimensional computer vision: a geometric viewpoint , 1993 .

[8]  Olivier D. Faugeras,et al.  Oriented Projective Geometry for Computer Vision , 1996, ECCV.

[9]  Bill Triggs,et al.  Autocalibration and the absolute quadric , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[10]  Reinhard Koch,et al.  Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[11]  Andrew Zisserman,et al.  Resolving ambiguities in auto–calibration , 1998, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  S. Bougnoux,et al.  From projective to Euclidean space under any practical situation, a criticism of self-calibration , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[13]  Andrew W. Fitzgibbon,et al.  Bundle Adjustment - A Modern Synthesis , 1999, Workshop on Vision Algorithms.

[14]  Anders Heyden,et al.  Flexible calibration: minimal cases for auto-calibration , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[15]  Peter F. Sturm,et al.  Critical motion sequences for the self-calibration of cameras and stereo systems with variable focal length , 1999, Image Vis. Comput..

[16]  M. Pollefeys.  Self-calibration and metric 3d reconstruction from uncalibrated image sequences , 1999 .

[17]  Ian D. Reid,et al.  Camera calibration and the search for infinity , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[18]  David Nistér,et al.  Frame Decimation for Structure and Motion , 2000, SMILE.

[19]  David Nistér,et al.  Reconstruction from Uncalibrated Sequences with a Hierarchy of Trifocal Tensors , 2000, ECCV.

[20]  David Nistér.  Calibration with robust use of cheirality by quasi-affine reconstruction of the set of camera projection centres , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[21]  Bernhard P. Wrobel,et al.  Multiple View Geometry in Computer Vision , 2001 .

[22]  Andrew Zisserman,et al.  Multiple view geometry in computer visiond , 2001 .

[23]  David Nister,et al.  Automatic Dense Reconstruction from Uncalibrated Video Sequences , 2001 .

[24]  William H. Press,et al.  Numerical recipes in C , 2002 .

[25]  Reinhard Koch,et al.  Self-Calibration and Metric Reconstruction Inspite of Varying and Unknown Intrinsic Camera Parameters , 1999, International Journal of Computer Vision.

[26]  David Nistér,et al.  An efficient solution to the five-point relative pose problem , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[27]  Paul A. Beardsley,et al.  Sequential Updating of Projective and Affine Structure from Motion , 1997, International Journal of Computer Vision.

[28]  Olivier D. Faugeras,et al.  A theory of self-calibration of a moving camera , 1992, International Journal of Computer Vision.