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).
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