On the basis of numerical simulations performed, we state and discuss approaches and procedures for projectively-
invariant description of ovals, which use our previously developed theory and technique of ht-contours in the problem of
nding eight invariant points on a gure contour, given that the curve shows either axial or radial implicit symmetry. It
is demonstrated that a similar octet of stable contour points can be obtained by introducing subsidiary structures of
ht-polar lines and notes of the developed theory, including those for curves showing no symmetry features and being de
ned along with an a priori selected point on the plane of the gure (either inside or outside of its contour). The
detected octet, {Ci}, allows computing a compact descriptor of the oval shape on the wurf plane. This descriptor can be
used for low-costidenti cation of a curve with respect to the projective equivalence class. Using the examples of ovals
of two types of symmetry, we consider some new iterative schemes for detecting their basic elements (the positions of
the centers or the axes), the adequate localization of which ensures computation of {Ci}. The key stages of curve
recognition procedures (successively engaged schemes of analysis, search, and representation) do not exceed the o(n)
threshold of the algorithm complexity, which shows high potentials of their use in systems of automatic analysis of
object geometry. We outline the aspects of the suggested theory, further development of which might turn the developed
heuristic methods into universal schemes for description of smooth planar curves.
Key words:
projective invariant, wurf mapping, symmetry elements, ht-contour, descriptor octet {Ci}, theorems on the number and the
properties of invariant points of an oval
Cite:
Nikolayev P. P.
Raspoznavanie proektivno preobrazovannykh ploskikh figur. xi. novye metody poiska proektivno invariantnykh tochek ovala
[Recognition of projectively transformed planar gures. xi. a new methods for detecting projectively-invariant points of an oval].
Sensornye sistemy [Sensory systems].
2017.
V. 31(4).
P. 343-362 (in Russian).
References:
- Balitsky A.M., Savchik A.V., Gafarov R.F., Konovalenko I.A. On projective invariant points of oval coupled with external line // Problemy peredachi informacii. 2017. V. 53(3). P. 84–89. [in Russian]
- Gill P., Murray W., Wright M. Practical optimization. New York. Academic Press, 1981. 401 p. [in English]
- Deputatov V.N. On the nature of planar wurfs // Matematichesk sbornik. 1926. V. 33 (1). P. 109– 118. [in Russian]
- Kartan Je. The method of a moving ranging mark, the theory of continuous groups and generalized spaces. Moscow, Leningrad. Gosudarstvennoe tehniko-teoreticheskoe izdatel’stvo, 1933. 72 p. [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 figures. 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. VI. Invariant representation and methods for detecting the center image of an oval having implicit central symmetry // Sensornye sistemy. 2014a. V. 28(1). P. 43–71 [in Russian]
- Nikolayev P.P. A method for projectively invariant description of ovals having axial or central symmetry // Informacionnye tekhnologii i vychislitel’nye sistemy. 2014б. I.2. P. 46–59 [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. 2015. V. 29(3). P. 213–244 [in Russian]
- Nikolayev P.P. Recognition of projectively transformed planar gures. X. Methods for nding an octet of invariant points of an oval contour – the result of introducing a developed theory into the schemes of oval description // Sensornye sistemy. 2017. V. 31(3). P. 202– 226 [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]
- Savchik A.V., Nikolaev P.P. The theorem on the intersection of the T-and H-polars // Informacionnye processy. 2016. V. 16 (4). P. 430–443 [in Russian]
- Alt H., Godau M. Computing the Frechet distance between two polygonal curves // International Journal of Computational Geometry and Applications. 1995. V. 5 (1–2). P. 75–91.
- Boutin M. Polygon recognition and symmetry detection // Found. Comput. Math. 2003. V. 3. Р. 227–271.
- Faugeras O. Cartan’s moving frame method and its application to the geometry and evolution of curves in the euclidean, a ne and projective planes // Applicat. Invar. Comput. Vision / Springer Verlag, Lecture Notes in Computer Science. 1994. V. 825. P. 11–46.
- Fels M., Olver P.J. Moving coframes. I. A practical algorithm // Acta Appl. Math. 1998. V. 51. P. 161–213.
- Hann C.E., Hickman M.S. Projective curvature and integral invariants // Acta Appl. Math. 2002. V. 74. No 2. P. 177–193.
- Musso E., Nicolodi L. Invariant signature of closed planar curves // J. Math. Imaging and Vision. 2009. V. 35. No 1. P. 68–85.
- Olver P.J. Equivalence, Invariants and Symmetry // Cambridge. Cambridge Univ. Press. 1995. 525 р.
- Olver P.J. Geometric foundations of numerical algorithms and symmetry // Appl. Alg. Engin. Comp. Commun. 2001. V. 11. P. 417–436.
