Recognition of projectively transformed planar figures. XVII. Using plucker’s reciprocity theorem to describe ovals with an external fixed point

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Resumo

An approach to a projectively invariant description of a family of ovals (o) in scenes where the figure o is given in a composition with an external point, P, fixed in its plane is considered, and in cases where o has hidden symmetries (central or axial), the position of P is not specified in the form of an additional condition defining the scene, but can be calculated through the symmetry parameters. The invariant description, as a general universal method for numerical processing of compositions like “o + ext-P”, is proposed to be implemented in the form of Wurf mappings.The method uses the apparatus of dual pairs (DP) and wurf functions,previously developed and described by us, which are a product of decomposition of statements of the reciprocity theorem proposed by J. Plьcker to describe the properties of quadratic curves (conics).Illustrated examples of special cases of the “o + ext-P”composition are considered and discussed, actually completing the topic of studying the scenes like “an oval and a linear element of the plane”, which are classified according to the types of symmetry of o.

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Sobre autores

P. Nikolaev

A. A. Kharkevich Institute of Information Transmission Problems of the Russian Academy of Sciences; Smart Engines Service LLC

Autor responsável pela correspondência
Email: nikol@iitp.ru
Rússia, Moscow; Moscow

Bibliografia

  1. 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).
  2. Balitsky A.M., Savchik A.V., Gafarov R.F., Konovalenko I.A. O proektivno invariantnyh tochkah ovala s vydelennoj vneshnej pryamoj. [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). https://doi.org/10.1134/S0032946017030097
  3. Glagolev N.A. Proektivnaya geometriya [Projective geometry]. Moscow, Vysshaya shkola [High school]. 1963. 344 p. (in Russian).
  4. 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).
  5. 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).
  6. Modenov P.S. Analiticheskaya geometriya [Analytic geometry]. Moscow, Izdatel’stvo moskovskogo universiteta [Moscow University Press]. 1969. 699 p. (in Russian).
  7. 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).
  8. 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. doi: 10.22204/2410-4639-2016-092-04-38-54 (in Russian).
  9. 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).
  10. Nikolayev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. VIII. O vychislenii ansamblya rotacionnoj korrespondencii ovalov s simmetriej vrashheniya [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).
  11. 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).
  12. 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. (in Russian). https://doi.org/10.1134/S0235009219030077
  13. Nikolaev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. XV. Metody poiska osej i centrov ovalov s simmetriyami, 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. (in Russian). https://doi.org/10.31857/S0235009221010054
  14. Nikolayev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. XVI. Oktet proektivno stabil’nykh vershin ovala i novye metody etalonnogo ego opisaniya, ispol’zuyushchie oktet. [Recognition of projectively transformed planar figures. XIV. The octet of projectively stable vertices of the oval and new methods for its reference description using the octet]. Sensornye sistemy [Sensory sistems]. 2022. V. 36(1). P. 61–89. (in Russian). https://doi.org/10.31857/S023500922201005X
  15. Savelov A.A. Ploskie krivye. Sistematika, svojstva, primeneniya [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).
  16. Savchik A.V., Nikolaev P.P. Teorema o peresechenii T- i H- polyar [The Theorem of T- and H-Polars Intersections Count]. Informacionnye process [Information processes]. 2016. V. 16(4). P. 430–443 (in Russian).
  17. 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
  18. Carlsson S. Projectively invariant decomposition and recognition of planar shapes. International Journal of Computer Vision. 1996. V. 17(2). P. 193–209. https://doi.org/10.1007/BF00058751
  19. 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 European-US Workshop on Applications of Invariance in Computer Vision. Berlin, Heidelberg. Springer, 1993. P. 9–46. https://doi.org/10.1007/3-540-58240-1_2
  20. 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.
  21. Hann C.E., Hickman M.S. Projective curvature and integral invariants. Acta Applicandae Mathematica. 2002. V. 74(2). P. 177–193. https://doi.org/10.1023/A:1020617228313
  22. Hoff D., Olver P.J. Extensions of invariant signatures for object recognition. Journal of mathematical imaging and vision. 2013. V. 45. P. 176–185. https://doi.org/10.1007/s10851-012-0358-7
  23. Itenberg I.V., Itenberg V.S. Symmetric sextics in the real projective plane and auxiliary conics. Journal of Mathematical Sciences. 2004. V. 119(1). P. 78–85. https://doi.org/10.1023/B:JOTH.0000008743.36321.72
  24. Lebmeir P., Jurgen R.-G. Rotations, translations and symmetry detection for complexified curves. Computer Aided Geometric Design. 2008. V. 25. P. 707–719. https://doi.org/10.1016/j.cagd.2008.09.004
  25. Musso E., Nicolodi L. Invariant signature of closed planar curves. Journal of mathematical imaging and vision. 2009. V. 35(1). P. 68–85. https://doi.org/10.1007/s10851-009-0155-0
  26. Olver P.J. Geometric foundations of numerical algorithms and symmetry. Applicable Algebra in Engineering, Communication and Computing. 2001. V. 11. P. 417–436. https://doi.org/10.1007/s002000000053
  27. 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

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2. Fig. 1. Pole-pole relations for a family of quadratic curves (conics) on the example of an ellipse, the formula for calculating the Wurf (through the lengths of three segments) and the main points of J. Plücker's reciprocity theorem. Other explanations are in the text

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3. Fig. 2. Method of calculation of the radial Wurf function and its typical (two-phase) form on the model numerical example about the general form. Other explanations are in the text

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4. Fig. 3. Method for optimizing a pair of tangent W-functions by means of a square root extraction operation for the ordinates of the W3(n) function with a significantly stronger extremum - compared to the maximum at W2(n). Other explanations are in the text

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5. Fig. 4. View of the Wurf images W2(W1) and w3(W1) in the functional dependence of the “tangential - on radial” kind (with the same numbering of the approximation vertices o for all three functions). On the inset: the method of obtaining W-functions by scanning the contour o with a ray from the pole P (the position of tangential functions coincidence for vertex i is shown). Other explanations are in the text

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6. Fig. 5. Problem formulation for scenes of the form “o + ext-P”, illustrating the possibility of obtaining triads of DPs (in agreement with the reciprocity theorem), a view of histograms of DP search for poles P and D (in the inset on the right) and the use of triads P..D, P..E to calculate additional DPs M..S (blue color) and N..G on lines PD and PE. Explanations are in the text

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7. Fig. 6. View of the combined reference projection of the two found octets ({B, K, I, T, A, J, R, Q} and {B, F, Z, H, A, U, L, W}) on the contour o of the general view, giving a projectively stable description of o given the positions of the 14 vertices of its contour in the scene with pole P and pluperfect polar AB (two quartets are projected onto a square, two onto a rhombus, the diagonal of the polar is common to the black and brown projections). Other explanations are in the text

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8. Fig. 7. Histogram view of the search for DP positions on HL based on the results of the program, leading in a sequential search of the approximation vertices on the projection of the superellipse to select those satisfying the criterion of “Plücker's duality”. Explanations are in the text

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9. Fig. 8. Summary picture of the properties of the two-axis hidden symmetry of the Lame curve projection used to obtain its invariant description in the form of a W-image for two cases of ext-P selection: as a random position Z and as a variant of the intersection point V of the tangents defining the map of two DPs (D..P and G..N) on HL. In the insets are views of the W-functions of the W-tangents computed for the poles V and Z, and a descriptor image (bottom left) for the cyclic octet chain {A, T, S, R, B, M, J, I}. Other explanations are in the text

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10. Fig. 9. Summary of the methods and numerical results for the hidden axial symmetry oval (oo). The two W-functions W1(n) and W2(n) computed for image AB (as Plücker polars) are nonlinearly dependent, so it is required to construct a function W3(n) that attracts an additional point N into the estimation scheme. The box on the bottom right gives the formulas for computing the W-functions for the Plücker pole S of axial symmetry (right) and for the random position P (left). The remaining boxes show the structural formulas and the Wurf-curves for poles S and P. Other explanations are in the text

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11. Fig. 10. Histogram view searching (on HL) for DPs and clustering them into quartets (blue and brown sets labeled “square”) for o implicit radial symmetry. Numerical model for 1,200 approximation vertices. Comments are in the text

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12. Fig. 11. Summary picture of dislocation of two DPs on HL in the neighborhood of implicit radial symmetry, determining the positions of poles H and V, relative to which all required (to obtain invariant mappings) W-functions are calculated. The Wurf-products for H and V are shown in the insets (above for H, below for V). All comments are in the text

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13. Fig. 12. View of W-functions and W-objections for a random position p, whose localization with respect to o and HL is shown in Fig. 11. Other explanations are in the text

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14. Fig. 13. Histogram view of the detection of 12 DPs on HL and their clustering into quartets of vertices (array of 1,800 discrete contour positions) for a projection about type R3 possessing three axes of hidden symmetry (48 vertices are marked and merged). Comments are in the text

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15. Fig. 14. A picture of the preliminary analysis of the scene with HL in the neighborhood of the projection about type R3, showing the DPs M..N and T..P, determining the position of the pole V, and by it - and S, relative to which the required Wurf products are calculated; their view is shown in the insets on the left. The insets on the right show the structural formulas (top) and a view of the orthoform with dislocation of one of the symmetry axes and the RL polar for the N pole (bottom). Explanations are in the text

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16. Fig. 15. A picture of the symmetric projective organization of the 12 nodes of the complete DP map for the type R3, for its orthoform manifested by the composition of three parallel bundles combining five lines each, where the lines are incident to the regular structure of the pairs of nodes (the inset on the upper right shows a bundle with projective center in D1, for which d1 on HL forms a DP). The inset on the upper left shows the appearance of the three W-functions and a pair of W-reflections computed for the pole d2 and the polar CD. Other explanations are in the text

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17. Fig. 16. Histogram view of the search histogram of DPs on HL at o type R3, which does not have three hidden symmetry axes, and their clustering into vertex quartets for six DPs. Explanations are in the text

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18. Fig. 17. A picture of the pairwise regular organization of DP nodes in an orthoform of type R3 without three hidden axes of symmetry, for which the chord AB, incident to nodes O1 and O2, is chosen as the plukker polar, with respect to the pole P of which the necessary W-functions and a pair of W-objections are calculated (their appearance is shown in the insets on the left), and the uniqueness of the choice of P is based on the identity of the Wurf-products obtained for polars with pairs of nodes O3-O4 and O5-O6. Other explanations are in the text

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19. Fig. 18. A more detailed map of the connections of six DPs on HL with a regular composition of nodes in the projection field of type R3 (which has no hidden three-axis symmetry), showing the dislocation of 24 stable nodes on its contour and showing (in the upper right inset) a view of the triads of the W-function and W-objections computed for a polara with a pair of nodes O3-O6. The bottom inset (right) gives a view of the 6-node graph with a choice of pairs for two examples of computing Wurf-products over polaras with O1-O2 and O3-O6 pairs. Comments are in the text

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20. Fig. 19. A summary map of the choice of the ext-P position for the computation of the necessary invariant mappings for a non-axial o of type R3, which does not use the DP apparatus on HL, but sets the unique ext-P cordinates by evaluating the dislocation of elliptic points (Theorem 1) with the choice of one of five (namely E2) according to some design criterion. The insets show the view of the W-products according to the unified scheme. All explanations are in the text

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21. Fig. 20. Results of the computation of the required invariant descriptions for a non-axial o of type R3 in the case when in the “o + ext-P” scene the position P is set randomly (without reliance on the analysis of hidden symmetries). “Starting” position P (as in the considered case of general-type o, see Fig. 5) provides a triad of DPs: P..D and P..d, creating the possibility of computing W-objections not only for the pole P, but also with respect to an additional dual D, as shown in the insets (on the right for P, on the left for D). Other explanations are in the text

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