# 3D Ear Normalization and Recognition Based on Local Surface Variation
## 3 Technical Approach
### Ear Recognition  
The ICPIF algorithm combines the positional and feature distances to get a better performance in iteration. Each vertex is represented by its three positional coordinates and \(k\) feature coordinates.  
Figure 5: Ear normalization: (**a**) The coarsely segmented ear; (**b**) The normalized ear; (**c**) The time-error curve of five iterations.  
Points are matched using the \(L_{2}\) norm in the \(k+3\) dimensional space [13]. The positional components are denoted as \(P_{e}\), and its \(k\) feature components are denoted as \(P_{f}\). That is  
\[\begin{array}{l}P_{e}=(p_{x},p_{y},p_{z})\in\mathrm{R}^{3}\\ P_{f}=(p_{f1},p_{f2},\cdots,p_{fk})\in\mathrm{R}^{k}\\ p^{\prime}=(p_{e},p_{f})\in\mathrm{R}^{3+k}.\end{array} \tag{5}\]  
The weighted feature distance between \(p\) and \(q\) is denoted as  
\[d^{\prime}(p,q)=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2 }+\left(z_{1}-z_{2}\right)^{2}+\alpha^{2}\left(f_{11}-f_{12}\right)^{2}+ \alpha^{2}\left(f_{21}-f_{22}\right)^{2}+\cdots\alpha^{2}\left(f_{11}-f_{12} \right)^{2}} \tag{6}\]  
where \(\alpha\) controls the relative contribution of the positions and features. Curvature representation is perhaps the most familiar of all invariant features that researchers used. However, it will bring in two invariant features (maximum and minimum values of the curvatures) and a rise in time consumption in iteration. Therefore, a modified ICPIF algorithm is proposed in this paper. The LSV value of each vertex is utilized to be an invariant feature in this paper. The \(d^{\prime}(p,q)\) is denoted as  
\[d^{\prime}(p,q)=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{ 2}+\left(z_{1}-z_{2}\right)^{2}+\alpha^{2}\left(\sigma_{n1}-\sigma_{n2}\right) ^{2}} \tag{7}\]  
where \(\alpha\) is the weighting coefficient and \(\sigma_{n}\) represents the LSV value. It considers only one invariant feature in the algorithm to get a faster alignment. Additionally, the LSV value of each vertex has been extracted and saved in the procedure of ear normalization, so the extra time of invariant feature extraction is not necessary for this modified algorithm in this system.  
Some rules are defined during the process of iteration to enhance its performance [17]. Firstly, only the correspondent closest point pairs whose distance is less than a threshold \(d_{\mathrm{t}}\) are chosen to calculate the transformation. \(d_{\mathrm{t}}\) is given by  
\[d_{\mathrm{t}}=\overline{d}+2R \tag{8}\]  
where \(\overline{d}\) is the mean distance between two point clouds, and \(R\) is the resolution of the probe set. Second, to avoid the influence of the noise data, the distances between point pairs are sorted and only the lower 60% are employed to calculate the average distance. Finally, the ear pits of all the ears which are detected from the normalized ear data are aligned before ICP iteration to avoid mismatching.  
Since the ICP-LSV algorithm applies the local feature in the iteration process, utilizing one third the size of original data, the proposed recognition procedure obtains a better performance than the traditional coarse-to-fine registration algorithm utilizing one-step ICP matching.