• 1990 (Vol.4)
  • 1989 (Vol.3)
  • 1988 (Vol.2)
  • 1987 (Vol.1)

Projectively invariant description of non-planar smooth gures. 1. Preliminary analysis of the problem

© 2016 P. P. Nikolayev

Institute for Information Transmission Problems “Kharkevich Institute” RAS, 127994 Moscow, Bolshoy Karetny per., 19

Received 26 Apr 2016

We consider definition of the problem of projectively-invariant recognition of axisymmetric oval curves, whose base surface belongs to the unfolding class (namely – generally cylindrical), while the curves themselves, being non-planar ones, are represented by their central planar projections. It is required to use a single such projection for estimating the parameters that characterize both the bending pro le of the base surface and the 3D shape of the curve, the 2D projection of which is an oval having a symmetry axis. The 3D-transformed shape (which is a rigid “object of observation”) of this curvilinear gure meets an additional condition: in the space of the scene, it shows a bilateral (planar) symmetry, whereas the angle of the sensory registration of the curve is unknown. The 2D projection of the curve is the object of numerical processing, the aim being the development of the “projectively invariant representation of the 3D object”. We discuss some analytical and numerical approaches to solving the problem of estimation of 3D shape parameters for two types of input objects: 1) an oval whose plane of symmetry (PS) is orthogonal to the axis of the base cylinder having no longitudinal PS; 2) a gure where the oval axis belongs to the longitudinal PS. For the list of 3D characteristics of the object, “reachable and computationally unrealizable estimations” are predicted as a result of differences in speci cations of its description: either abstract-geometrical or technical, the latter taking into account circumstances related to the projective parameters of the sensor (camera optics, etc.).

Key words: plane of symmetry, vanishing point, harmonic wurf, projective transformation, double tangent line, wurf-mapping, planar h- and t-curves.

Cite: Nikolayev P. P. Proektivno invariantnoe opisanie neploskikh gladkikh figur. 1. predvaritelnyi analiz zadachi [Projectively invariant description of non-planar smooth gures. 1. preliminary analysis of the problem]. Sensornye sistemy [Sensory systems]. 2016. V. 30(4). P. 290-311 (in Russian).

References:

  • Karpenko S. M., Gladilin S. A., Nikolaev D. P. Method for correction of images with radial distortion. Proc. of Int. Conf. on Informational Technologies and Systems (ITaS’08). 2008. P. 502–505 [in Russian].
  • Nikolayev P. P. Recognition of projectively transformed planar gures. II. An oval in a composition with a dual element of a plane // Sensornye sistemy. 2011a. V. 25(3). P. 245–266 [in Russian].
  • Nikolayev P. P. Recognition of projectively transformed planar gures. III. Processing of axisymmetric ovals by means of polar line analysis methods // Sensornye sistemy. 2011b. V. 25(4). P. 275–296 [in Russian].
  • Nikolayev P. P. Recognition of projectively transformed planar gures. IV. Methods for forming a projective-invariant description of axisymmetric ovals // Sensornye sistemy. 2012. V. 26(4). P. 280–303 [in Russian].
  • Nikolayev P. P. Recognition of projectively transformed planar gures. V. Methods for detecting the center image of an oval having implicit central symmetry // Sensornye sistemy. 2013. V. 27(1). C. 10–34 [in Russian].
  • Nikolayev P. P. A method for projectively-invariant description of ovals having axial or central symmetry // Informacionnye tekhnologii i vychislitel’nye sistemy. 2014. I. 2. P. 46–59 [in Russian].
  • Nikolayev P. P. Recognition of projectively transformed planar gures. VIII. On computation of the ensemble of correspondence for “rotationally symmetric” ovals // Sensornye sistemy. 2015а. V. 29(1). P. 28–55 [in Russian].
  • Nikolayev P. P. Recognition of projectively transformed planar gures. IX. Methods for description of ovals with a xed point on the contour // Sensornye sistemy. 2015b. V. 29(3). P. 213–244 [in Russian].
  • Ovsienko I. U., Tabachnikov S. L. Projective Differential Geometry Old and New From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups. Moscow. MCCME, 2008. 280 p. [in Russian].
  • Finikov S. P. Projective differential geometry. Moscow. URSS, 2010. 264 p. [in Russian].
  • Alt H., Godau M. Computing the Frechet distance between two polygonal curves // Intern. J. Computat. Geom. Applicat. 1995. V. 5 (1–2). P. 75–91.
  • Boutin M. Polygon recognition and symmetry detection // Found. Comput. Math. 2003. V. 3. P. 227–271.
  • Cox D., Katz Sh. Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs // J.Amer. Math. Soc. 1999. V. 68. 469 p.
  • Gross M. Mirror symmetry for P2 and tropical geometry // Adv. Math. 2010. V. 224(1). P. 169–245.
  • Gross M., Siebert B.Mirror symmetry via logarithmic degeneration // I. J. Differential Geom. 72. 2006. P. 169–338.
  • Hann C. E., Hickman M. S. Projective curvature and integral invariants // Acta Appl. Math. 2002. V. 74(2). P. 177–193.
  • Hori K., Katz Sh., Klemm A., Pandharipande R., Thomas R., Vafa C., Vakil R., Zaslow E. Mirror Symmetry // J. Amer. Math. Soc. 2003. V. 1. 952 p.
  • Kontsevich M. Homological algebra of mirror symmetry // Proceedings of the International Congress of Mathematicians. 1994. P. 120–139.
  • Morrison D. Mirror symmetry and Rational Curves on Quintic Threefolds: A Guide for Mathematicians // J. Amer. Math. Soc. 1993. V. 6. P. 223–247.
  • Musso E., Nicolodi L. Invariant signature of closed planar curves // J. Math. Imaging and Vision. 2009. V. 35(1). P. 68–85.
  • Olver P. J. Equivalence, Invariants and Symmetry. Cambridge. Cambridge Univ. Press. 1995, 525 p.
  • Olver P. J. Geometric foundations of numerical algorithms and symmetry // Appl. Alg. Engin. Comp. Commun. 2001. V. 11. P. 417–436.
  • Pritula N., Nikolaev D., Sheshkus A., Pritula M., Nikolaev P. Comparison of two algorithms of projective-invariant recognition of the plane boundaries with the one concavity. Seventh Internat. Conf. Machine Vision (ICMV 2014), Proc. SPIE9445, Milan, Italy, Nov. 2014. P. 19–21.