Analytical Tensor Voting in ND Space and its Properties

This article aims to propose a novel Analytical Tensor Voting (ATV) mechanism, which enables robust perceptual grouping and salient information extraction for noisy
$N-$
dimensional (ND) data. Firstly, the approximation of the decaying function is investigated and adopted based on the idea of penalizing the
$1-$
tensor votes by distance and curvature, respectively, followed by the derivation of analytical solution to the
$1-$
tensor voting in ND space from the geometric view. Secondly, a novel spherical representation mechanism is proposed to facilitate the representation of the elementary tensors in various dimensional spaces, where the high dimensional spherical coordinate system is utilized to construct the controllable unit vectors and corresponding
$1-$
tensors. Accordingly, any elementary
$K-$
tensor is represented by the surface integration of the constructed
$1-$
tensors over the unit
$K-$
sphere. Thirdly, the ATV mechanism is constructed using the adopted decaying function and proposed spherical representation mechanism, where the analytical solution to tensor voting in ND space is derived, which enables the robust and accurate salient information extraction from noisy ND data. Finally, several interesting properties of the proposed ATV mechanism are investigated. Experimental results on synthetic and real data validate the effectiveness, efficiency and robustness of the proposed method in perceptual grouping tasks in 3D,10D or higher dimensional spaces.