Reconstruction from Projections Using Grassmann Tensors
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In this paper a general procedure is given for reconstruction of a set of feature points in an arbitrary dimensional projective space from their projections into lower dimensional spaces. This extends the methods applied in the well-studied problem of reconstruction of scene points in ℘3 given their projections in a set of images. In this case, the bifocal, trifocal and quadrifocal tensors are used to carry out this computation. It is shown that similar methods will apply in a much more general context, and hence may be applied to projections from ℘ n to ℘ m , which have been used in the analysis of dynamic scenes, and in radial distortion correction. For sufficiently many generic projections, reconstruction of the scene is shown to be unique up to projectivity, except in the case of projections onto one-dimensional image spaces (lines), in which case there are two solutions.
Projections from ℘ n to ℘2 have been considered by Wolf and Shashua (in International Journal of Computer Vision 48(1): 53–67, 2002), where they were applied to several different problems in dynamic scene analysis. They analyzed these projections using tensors, but no general way of defining such tensors, and computing the projections was given. This paper settles the general problem, showing that tensor definition and retrieval of the projections is always possible.
KeywordsGrassman tensor Projective reconstruction Multiview tensor Exterior algebra
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- Hartley, R. I. (1998a). Computation of the quadrifocal tensor. In Lecture notes in computer science : Vol. 1406. Proceedings of the 5th European conference on computer vision (pp. 20–35). Freiburg, Germany. Berlin: Springer. Google Scholar
- Hartley, R., & Vidal, R. (2008). Perspective nonrigid shape and motion recovery. In Proceedings of the European conference on computer Vision (pp. 276–289). Google Scholar
- Heyden, A. (1998a). A common framework for multiple-view tensors. In Proceedings of the 5th European conference on computer vision (pp. 3–19). Freiburg, Germany. Google Scholar
- Semple, J. G., & Kneebone, G. T. (1979). Algebraic projective geometry. Oxford: Oxford University Press. Google Scholar
- Thirthala, S., & Pollefeys, M. (2005a). Multi-view geometry of 1d radial cameras and its application to omnidirectional camera calibration. In Proceedings of the 10th international conference on computer vision (pp. 1539–1546). Beijing, China. Google Scholar
- Thirthala, S., & Pollefeys, M. (2005b). The radial trifocal tensor: A tool for calibrating the radial distortion of wide-angle cameras. In Proceedings of the IEEE conference on computer vision and pattern recognition (Vol. 1, pp. 321–328) San Diego. Google Scholar