{
  "id": "1202.1303",
  "version": "v2",
  "published": "2012-02-06T21:43:33.000Z",
  "updated": "2013-03-15T14:00:55.000Z",
  "title": "Object-image correspondence for curves under projections",
  "authors": [
    "Joseph M. Burdis",
    "Irina A. Kogan"
  ],
  "comment": "A significantly improved version of this paper (corrected and completed) has been posted arXiv:1303.3358",
  "categories": [
    "math.AG",
    "cs.CG"
  ],
  "abstract": "We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution. The main advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve. Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. The latter problem is then solved by differential signature construction based on Cartan's moving frame method. A similar approach can be used to decide whether a given finite set of ordered points on a plane is an image of a given finite set of ordered points in R^3. The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-03-15T14:00:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H50",
      "14Q05",
      "68T45",
      "03C10"
    ],
    "keywords": [
      "object-image correspondence",
      "planar curve",
      "spatial curve",
      "finite set",
      "straightforward approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.1303B"
    }
  }
}