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Recognition of projectively transformed planar figures. XIII. New methods for projectively-invariant description of ovals, using a T-polar curve

© 2019 P. P. Nikolaev

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

Received 18 Jan 2019

For scenes where the input object is a composition of an arbitrary oval with point P fixed in its field (called a test pole), and this P determines the position (outside the oval) of a projectively invariant curve, the so-called T-polar curve, methods of engaging the triad of wurf functions, w1(n), w2(n), and w3(n) (n being the number of vertices used to approximate the oval), are described. These functions allow detecting on the T-polar curve a set of its projectively- invariant points, previously called elliptic points (EP) or dual points (DP), which can be used to compute descriptors of the “Oval + P” composition, while obtaining solely a triad of new wurf functions is enough to quickly calculate several independent forms of wurf mapping of such a composition. The developed fast processing algorithms have been successfully tested on a representative series of numerical models. The proposed approaches of discrete analysis of the curve use the previously obtained theoretical statements. For an oval with a specified internal point, intP, these statements guarantee the presence of the EP triad, while for an oval combined with extL (externally positioned line), they guarantee the presence of two pairs of DP. On the basis of the new features obtained for the elements of implicit symmetry (IS) of the oval (the axis or the center), fast iterative schemes of IS detection are described, which implement the projectively-invariant description of the “oval + IS” curve. Thus, such an “old tool” as the T-polar curve has introduced new possibilities of invariant recognition of figures of the oval family.

Key words: test pole, elliptic and dual points of a T-polar curve, wurf function, projectively-invariant mapping, axial or radial symmetry of an oval

DOI: 10.1134/S0235009219030077

Cite: Nikolaev P. P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. xiii. novye metody proektivno invariantnogo opisaniya ovalov s ispolzovaniem t-polyary [Recognition of projectively transformed planar figures. xiii. new methods for projectively-invariant description of ovals, using a t-polar curve]. Sensornye sistemy [Sensory systems]. 2019. V. 33(3). P. 238-266 (in Russian). doi: 10.1134/S0235009219030077

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