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Recognition of projectively transformed planar figures. XVI The octet of projectively stable vertices of the oval and new methods for its reference description using the octet

© 2022 P. P. Nikolaeva,b

aInstitute for Information Transmission Problems “Kharkevich Institute” RAS 127994 Moscow, Bolshoy Karetny per., 19, Russia
bSmart Engines Service LLC 117312 Moscow, Prospect 60-Letiya Oktyabrya, 9, Russia

Received 25 Aug 2021

I propose the idea of the existence of an octet of projectively invariant vertices of an oval (o), which are obtained under the condition of the numerical localization of an external straight line, HL (“horizon line”), which is fixed at the stage of optical registration of the figure o or is calculated in the presence of central symmetry properties of the curve by fast algorithms searching for the center that determines the position of HL. The proposed idea is then illustrated by model experiments. According to the theorem on an arbitrary external line, L, composed with o, the line L always contains at least two pairs of stable points (called dual points – DP), and each DP defines an invariant quartet of vertices on o, as a result of successful positional estimation of which the contour appears to have a set of eight ordered vertices, and it is expedient to use this set for a projectively-invariant description of o. Two hypotheses for HL are expressed and investigated in simulations: about the projective relationship between the positions of two DP pairs and about the possibility, for an axially symmetric o, to estimate the position of its inherent HL on the basis of some projective features revealed for DP. Two novel methods for finding the center of o with hidden radial symmetry are also described. Finally, I have proposed and tested in numerical experiments a pair of new methods for using the octet of stable contour points, found for o (either symmetric or not), for a projectively-invariant description of o.

Key words: oval, center and axis of symmetry, Plucker pole and polar curve, dual pairs, harmonic wurf, wurf function, descriptor, descriptor template, Lamé curve

DOI: 10.31857/S023500922201005X

Cite: Nikolaev P. P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. xvi. oktet proektivno stabilnykh vershin ovala i novye metody etalonnogo ego opisaniya, ispolzuyushchie oktet [Recognition of projectively transformed planar figures. xvi the octet of projectively stable vertices of the oval and new methods for its reference description using the octet]. Sensornye sistemy [Sensory systems]. 2022. V. 36(1). P. 61–89 (in Russian). doi: 10.31857/S023500922201005X

References:

  • Akimova G.P., Bogdanov D.S., Kuratov P.A. Zadacha proektivno invariantnogo opisanija ovalov s nejavno vyrazhennoj central’noj i osevoj simmetriej i princip dvojstvennosti Pljukkera [Task projectively the invariant description of ovals with implicitly expressed central and axial symmetry and the principle of a duality of Plucker]. Trudy ISA RAN [Proceedings of the ISA RAS]. 2014. V. 64 (1). P. 75–83 (in Russian).
  • Balitsky A.M., Savchik A.V., Gafarov R.F., Konovalenko I.A. O proektivno invariantnyh tochkah ovala s vydelennoj vneshnej prjamoj [On projective invariant points of oval coupled with external line]. Problemy peredachi informacii [Problems of Information Transmission]. 2017. V. 53 (3). P. 84–89 (in Russian).
  • Glagolev N.A. Proektivnaja geometrija [Projective geometry]. Moscow, Vysshaja shkola [High school]. 1963. 344 p. (in Russian).
  • Deputatov V.N. K voprosu o prirode ploskostnyh vurfov [On the nature of the plane wurfs]. Matematicheskij sbornik [Mathematical collection]. 1926. V. 33 (1). P. 109–118. (in Russian).
  • Kartan Je. Metod podvizhnogo repera, teoriya nepreryvnykh grupp i obobshchennye prostranstva. Sb. Sovremennaya matematika. Kniga 2-ya [The method of a moving ranging mark, the theory of continuous groups and generalized spaces]. Moscow, Leningrad, Gosudarstvennoe tehniko-teoreticheskoe izdatel’stvo [State technical and theoretical publishing]. 1933. 72 p. (in Russian).
  • Modenov P.S. Analiticheskaja geometrija [Analytic geometry]. Moscow, Izdatel’stvo moskovskogo universiteta [Moscow University Press]. 1969. 699 p. (in Russian).
  • Nikolaev P.P. Metod proektivno invariantnogo opisaniya ovalov s osevoi libo tsentral’noi simmetriei [A method for projectively-invariant description of ovals having axial or central symmetry]. Informatsionnye tekhnologii i vychislitel’nye sistemy. 2014. No. 2. P. 46–59. (in Russian).
  • Nikolaev P.P. O zadache proektivno invariantnogo opisaniya ovalov s simmetriyami trekh rodov [A projective invariant description of ovals with three possible symmetry genera]. Vestnik RFFI [RFBR Information Bulletin]. 2016. V. 92 (4). P. 38–54. (in Russian). https://doi.org/10.22204/2410-4639-2016-092-04-38-54
  • Nikolayev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. II. Oval v kompozitsii s dual’nym ehlementom ploskosti [Recognition of projectively transformed planar figures. II. An oval in a composition with a dual element of a plane]. Sensornye sistemy [Sensory systems]. 2011. V. 25 (3). P. 245–266. (in Russian).
  • Nikolayev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. VIII. O vychislenii ansamblja rotacionnoj korrespondencii ovalov s simmetriej vrashhenija [Recognition of projectively transformed planar figures. VIII. On computation of the ensemble of correspondence for “rotationally symmetric” ovals]. Sensornye sistemy [Sensory systems]. 2015. V. 29 (1). P. 28–55 (in Russian).
  • Nikolayev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. IX. Metody opisanija ovalov s fiksirovannoj tochkoj na konture [Recognition of projectively transformed planar figures. IX. Methods for description of ovals with a fixed point on the contour]. 2015. V. 29 (3). P. 213–244 (in Russian).
  • Nikolayev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. X. Metody poiska okteta invariantnykh tochek kontura ovala – itog vklyucheniya razvitoi teorii v skhemy ego opisaniya [Recognition of projectively transformed planar figures. X. Methods for finding an octet of invariant points of an oval contour – the result of introducing a developed theory into the schemes of oval description]. Sensornye sistemy [Sensory systems]. 2017. V. 31 (3). P. 202–226. (in Russian).
  • Nikolaev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. XII. O novykh metodakh proektivno invariantnogo opisaniya ovalov v kompozitsii s lineinym elementom ploskosti [Recognition of projectively transformed planar figures. XII. On new methods for projectively-invariant description of ovals in composition with a linear element of a plane]. Sensornye sistemy [Sensory systems]. 2019. V. 33(1). P. 15–29. https://doi.org/10.1134/S0235009219010104 (in Russian).
  • Nikolaev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. XV. Metody poiska osej i centrov ovalov s simmetrijami, ispol’zujushhie set dual’nyh par libo triady chevian [Recognition of projectively transformed planar figures. XV. Methods for searching for axes and centers of ovals with symmetries, using a set of dual pairs of cevian triads]. Sensornye sistemy [Sensory systems]. 2021. V. 35 (1). P. 55–78. https://doi.org/10.31857/S0235009221010054 (in Russian).
  • Ovsienko I.Ju., Tabachnikov S.L. Proektivnaya differentsial’naya geometriya. Staroe i novoe: ot proizvodnoi Shvartsa do kogomologii grupp diffeomorfizmov [Projective differential geometry. Old and new from the schwarzian derivative to the cohomology of diffeomorphism groups]. Moscow, MCNMO. 2008. 280 p. (in Russian).
  • Savelov A.A. Ploskie krivye. Sistematika, svojstva, primenenija [Flat curves. Systematics, properties, applications]. M. Gos. izd-vo fiziko-matematicheskoj literatury [Moscow. State publishing house of physical and mathematical literature], 1960. 293 p. (in Russian).
  • Savchik A.V., Nikolaev P.P. Teorema o peresechenii T- i Hpoljar [The Theorem of T- and H- Polars Intersections Count]. Informacion-nye process [Information processes]. 2016. V. 16 (4). P. 430–443 (in Russian).
  • Savchik A.V. Sablina V.A. Finding the correspondence between closed curves under projective distortions [Ustanovlenie sootvetstvija mezhdu zamknutymi konturami ob#ektov pri proektivnyh iskazhenijah]. Sensornye sistemy [Sensory systems]. 2018. V. 32 (1). P. 60–66 (in Russian).
  • Brugalle E. Symmetric plane curves of degree 7: Pseudoholomorphic and algebraic classifications. Journal fur Die Reine und Angewandte Mathematic (Crelles Journal). 2007. V. 612. P. 1–38. https://doi.org/10.1515/CRELLE.2007.086
  • Carlsson S. Projectively invariant decomposition and recognition of planar shapes. International Journal of Computer Vision. 1996. V. 17 (2). P. 193–209.
  • Faugeras O. Cartan’s moving frame method and its application to the geometry and evolution of curves in the euclidean, affine and projective planes. Joint EuropeanUS Workshop on Applications of Invariance in Computer Vision. Berlin, Heidelberg. Springer, 1993. P. 9–46.
  • Gardner M. Piet Hein’s Superellipse, Mathematical Carnival. A New Round-Up of Tantalizers and Puzzles from Scientific American. New York. Vintage Press, 1977. 240–254 p.
  • Hann C.E., Hickman M.S. Projective curvature and integral invariants. Acta Appl. Math. 2002. V. 74 (2). P. 177–193. https://doi.org/10.1023/A:1020617228313
  • Hauer M., Jüttler B. Projective and affine symmetries and equivalences of rational curves in arbitrary dimension. Journal of Symbolic Computation. 2018. V. 87. P. 68–86. https://doi.org/10.1016/j.jsc.2017.05.009
  • Hoff D., Olver P.J. Extensions of invariant signatures for object recognition. J. Math. Imaging Vision. 2013. V. 45. P. 176–185. https://doi.org/10.1007/s10851-012-0358-7
  • Itenberg I.V., Itenberg V.S. Symmetric sextics in the real projective plane and auxiliary conics. Journal of Math. Sciences. 2004. V. 119 (1). P. 78–85. https://doi.org/10.1023/B:JOTH.0000008743.36321.72
  • Lebmeir P., Jurgen R.-G. Rotations, translations and symmetry detection for complexified curves. J. Computer Aided Geometric Design. 2008. V. 25. P. 707–719. https://doi.org/10.1016/j.cagd.2008.09.004
  • Musso E., Nicolodi L. Invariant signature of closed planar curves. J. Math. Imaging and Vision. 2009. V. 35 (1). P. 68–85. https://doi.org/10.1007/s10851-009-0155-0
  • Olver P.J. Geometric foundations of numerical algorithms and symmetry. Appl. Alg. Engin. Comp. Commun. 2001. V. 11. P. 417–436. https://doi.org/10.1007/s002000000053
  • Sanchez-Reyes J. Detecting symmetries in polynomial Bezier curves. Journal of Computational and Applied Mathematics. 2015. V. 288. P. 274–283. https://doi.org/10.1016/j.cam.2015.04.025
  • Savchik A.V., Sablina V.A., Nikolaev D.P. Establishing the correspondence between closed contours of objects in images with projective distortions. Proc. SPIE 10696, Tenth International Conference on Machine Vision (ICMV 2017). Verikas A., Bellingham. SPIE, 2018. 1069629. P. 1–9.