Abstract
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.
Highlights
Projective geometry is of fundamental importance in computer vision and object recognition
We present a projective invariant named after the characteristic number of planar algebraic curves, but it relies no more on the existence of an algebraic curve
This paper focuses on the applications of the characteristic number to 2D shape analysis, where the characteristic number can incorporate the geometric information on more points (i.e., n ≥ 1) compared with the cross-ratio
Summary
Projective geometry is of fundamental importance in computer vision and object recognition. In the context of facial analysis, Riccio and Dugelay devise features for recognition based on 2D/3D geometric invariants [15] These invariants are typically defined on a limited number (e.g., five for the classical cross-ratio) of collinear planar points and lack the ability to characterize the curve or surface underlying the given points. The characteristic number reflects the intrinsic properties of the points on an algebraic curve/surface These results are theoretically sound, but the dependency on the existence of the curve or surface curbs the wide application of the invariant to shape recognition and matching in computer vision. Constraints on facial feature points derived from the characteristic number into facial shape matching This incorporation renders accuracy and robustness to pose and view changes.
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