Affine Equivalences of Trigonometric Curves
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Affine Equivalences of Trigonometric Curves Juan Gerardo Alcázar1
· Emily Quintero1
Received: 16 February 2020 / Accepted: 18 August 2020 © Springer Nature B.V. 2020
Abstract We provide an efficient algorithm to detect whether two given trigonometric curves, i.e. two parametrized curves whose components are truncated Fourier series, in any dimension, are affinely equivalent, i.e. whether there exists an affine mapping transforming one of the curves onto the other. If the coefficients of the parametrizations are known exactly (the exact case), the algorithm boils down to univariate gcd computation, so it is efficient and fast. If the coefficients of the parametrizations are known with finite precision, e.g. floating point numbers (the approximate case), the univariate gcd computation is replaced by the computation of approximate gcds. Our experiments show that the method works well, even for high degrees. Keywords Affine equivalence · Algebraic curves · Elliptic Fourier descriptor (EFD) representations curves · Pattern recognition
1 Introduction Two objects are affinely equivalent if there exists a nonsingular affine transformation mapping one of the objects onto the other one. Detecting whether two objects are related by an affine mapping is a classical problem in applied fields like Pattern Recognition, Image Processing and Computer Vision, and has been addressed in many papers using different strategies: see for instance [6, 9, 17, 18, 21] and the references in these papers. Essentially, the underlying problem is to be able to recognize a same image when it undergoes a smooth J.G. Alcázar is supported by the Spanish Ministerio de Economía y Competitividad and by the European Regional Development Fund (ERDF), under the project MTM2017-88796-P, and is a member of the Research Group ASYNACS (Ref. CCEE 2011/ R 34 and Ref. CT-CE2019/683). E. Quintero is supported by a grant from the Carolina Foundation.
B J.G. Alcázar
[email protected] E. Quintero [email protected]
1
Departamento de Física y Matemáticas, Universidad de Alcalá, 28871 Madrid, Spain
J.G. Alcázar, E. Quintero
deformation, which is modelled as an affine mapping. Furthermore, an important particular instance of affine equivalence is the situation when the two objects are related by a similarity, in which case both objects are the same except for position and scaling. In this paper, we consider the affine equivalence problem for parametrized curves in any dimension whose components are truncated Fourier series. In some references [11, 12], these curves receive the name of trigonometric curves, or generalized trigonometric curves. In other, more applied, references (see for instance [20]), these curves are called elliptic Fourier descriptor (EFD) representations, and are often used to describe closed planar and space curves (see for instance the references in [20]). In particular, for these curves one can compute shape descriptors (see [7, 8, 13], among many others), which are numbers that can be computed from the parametrization, and t
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