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Open AccessArticle

Characteristic Number: Theory and Its Application to Shape Analysis

School of Software, Dalian University of Technology, Tuqiang St. 321, Dalian 116620, China
School of Mathematical Sciences, Dalian University of Technology, Linggong Rd. 2, Dalian 116023, China
Author to whom correspondence should be addressed.
Axioms 2014, 3(2), 202-221;
Received: 27 March 2014 / Revised: 28 April 2014 / Accepted: 28 April 2014 / Published: 15 May 2014
PDF [1880 KB, uploaded 15 May 2014]


Geometric invariants are important for shape recognition and matching. Existing invariants in projective geometry are typically defined on the limited number (e.g., five for the classical cross-ratio) of collinear planar points and also lack the ability to characterize the curve or surface underlying the given points. In this paper, we present a projective invariant named after the characteristic number of planar algebraic curves. The characteristic number in this work reveals an intrinsic property of an algebraic hypersurface or curve, which relies no more on the existence of the surface or curve as its planar version. The new definition also generalizes the cross-ratio by relaxing the collinearity and number of points for the cross-ratio. We employ the characteristic number to construct more informative shape descriptors that improve the performance of shape recognition, especially when severe affine and perspective deformations occur. In addition to the application to shape recognition, we incorporate the geometric constraints on facial feature points derived from the characteristic number into facial feature matching. The experiments show the improvements on accuracy and robustness to pose and view changes over the method with the collinearity and cross-ratio constraints. View Full-Text
Keywords: projective invariant; characteristic number; shape descriptors; facial feature matching projective invariant; characteristic number; shape descriptors; facial feature matching

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Fan, X.; Luo, Z.; Zhang, J.; Zhou, X.; Jia, Q.; Luo, D. Characteristic Number: Theory and Its Application to Shape Analysis. Axioms 2014, 3, 202-221.

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