## 3 Methodology
### Accurate Transformation Estimation
#### 3.3.1 Feature Matching  
The goal of feature matching is to find feature correspondences between two sets of multi-scale covariance matrix descriptors and then find keypoint correspondences between the source point cloud and the target point cloud. To establish feature correspondences, we must determine the similarity of two multi-scale covariance matrix descriptors followed by the feature correspondences between two sets of multi-scale covariance matrix descriptors. * Similarity between two multi-scale covariance matrix descriptors. The distance between two covariance matrix descriptors can be used to represent their similarity. The distance is closer, and the similarity is greater. Because the covariance matrix is a symmetric positive definite matrix, the geodesic distance between two covariance matrices can be used to measure their similarity. In recent years, various geodesic distance formulas have been proposed. In this paper, Jensen-Bregman LogDet Divergence [33] is employed, and it is defined as follows: \[d_{JBLD}(X,\;Y)=log\Big{|}\frac{X+Y}{2}\Big{|}-\frac{1}{2}log|XY|\] (23) where \(|\cdot|\) is the matrix determinant and \(\;log(\cdot)\) is the natural logarithm. This metric has been proven to be more efficient in terms of speed without sacrificing application performance [28], which is why we use it to compare the similarity of two covariance matrix descriptors in this paper. Additionally, the similarity between two multi-scaled covariance matrix descriptors can be defined using formula (24) as follows:  
\[d\left(C_{M}^{1},\ C_{M}^{2}\right)=\frac{1}{s}\sum_{i=r_{1},..,r_{s}}d_{BLD} \left(C_{i}^{1},\ C_{i}^{2}\right) \tag{24}\]  
where \(C_{M}^{1}\) and \(C_{M}^{2}\) correspond to two covariance matrix descriptors relating to different radius scales ranging from \(r_{1}\) to \(r_{s}\) and \(d_{BLD}\left(C_{i}^{1},\ C_{i}^{2}\right)\) represents the similarity of two covariance matrix descriptors with a fixed scale \(r_{i}\). Formula (24) considers similarity under multi-scales and utilizes the mean of similarity under multi-scales as the similarity of multi-scaled covariance matrix descriptors. A covariance matrix descriptor stated in the rest of this paper refers to a multi-scale covariance matrix descriptor unless otherwise specified. * Correspondence between two sets of covariance matrix descriptors. We present a bidirectional nearest neighbor distance ratio (BNNDR) to detect correspondences from two sets of covariance matrix descriptors, which differs from existing techniques, such as the nearest neighbor (NN) and the nearest neighbor distance ratio (NNDR). The detailed matching procedure via the BNNDR is as follows. First, assume that \(D_{S}=\left\{d_{S}^{i}\right\}\) and \(D_{T}=\left\{d_{T}^{i}\right\}\) are the sets of covariance matrix descriptors from a source point \(\mathbf{\mathcal{P}_{S}}\) and a target point cloud \(\mathbf{\mathcal{P}_{T}}\), respectively. Next, each covariance matrix descriptor \(d_{T}^{i}\) in \(D_{T}\) is matched against all the covariance matrix descriptors in \(D_{S}\) to yield the first and second nearest covariance matrix descriptors to \(d_{T}^{i}\cdot d_{S}^{j^{\prime}}\) and \(d_{S}^{j^{\prime\prime}}\), respectively.