# ALReg: Registration of 3D Point Clouds Using Active Learning
## 3 Method
### Baseline Methods  
As the baseline models, DCP, DeepBBS and FMR are utilized, whose details are summarized below. The architectural details of these methods are depicted in Figure 1.  
**DCP:** Deep Closest Point framework uses a dynamic graph-based encoder to obtain point cloud features \(F(X)\) and \(F(Y)\) from source \(X\) and target \(Y\). Then, using a transformer [78] with conditional attention, integrated features \(\phi(F(X),F(Y))\) and \(\phi(F(Y),F(X))\) are obtained. To benefit from both local and global information, new point embeddings are created as  
\[\phi(X) =F(X)+\phi(F(X),F(Y))\] \[\phi(Y) =F(Y)+\phi(F(Y),F(X)). \tag{2}\]  
Figure 1: Registration frameworks for (**a**) DCP [33] and DeepBBS [31], (**b**) FMR [32]. To ensure simplicity, some details given in the method section are not shown.  
Then, a probabilistic point-to-point mapping matrix is formed as  
\[m(x_{i},Y) = softmax(\phi(Y)\phi(x_{i})^{T})\] \[softmax(z_{i}) = \frac{e^{z_{i}}}{\sum_{j=1}^{K}e^{z_{j}}}\quad for\ i=1,2,\ldots,K \tag{3}\]  
where \(m(x_{i},Y)\) represents a row of the matrix indicating the probabilistic distribution of \(x_{i}\)'s corresponding point among the points of Y. For every point \(x_{i}\), the predicted corresponding point \(\hat{y}_{i}\) is found, and the cross-covariance matrix \(H(X,Y)\) is filled as  
\[\hat{y}_{i}=Y^{T}m(x_{i},Y),\quad H(X,y)=\sum_{i}^{N}(x_{i}-\overline{x})(\hat {y}_{i}-\overline{y}). \tag{4}\]  
Using singular value decomposition (SVD) on \(H(X,y)=USV^{T}\), R and t are found by \(R=VUT^{T}\) and \(t=-R\overline{x}+\overline{y}\).  
**DeepBBS:** Deep Best Buddies could be considered as an extension to DCP where \(x_{i}\) and \(y_{j}\) are called best buddies if they are the closest points to each other according to their features \(\phi(x_{i})\) and \(\phi(y_{j})\). Given the feature distance matrix \(D_{i,j}\), the BBS score between two points is calculated as  
\[B(x_{i},y_{j})=\frac{e^{\frac{-D_{ij}}{\pi}}}{\sum_{j^{\prime}}e^{\frac{-D_{ ij}}{\pi}}}\cdot\frac{e^{\frac{-D_{ij}}{\pi}}}{\sum_{j^{\prime}}e^{\frac{-D_{ ij}}{\pi}}}. \tag{5}\]  
Similarly to the approach in (6), predicted correspondences and cross-covariance matrix are calculated as  
\[\hat{y}_{i}=Y^{T}\frac{B_{i}}{\sum_{j}B_{ij}},\quad H(X,y)=\sum_{i}^{N}\gamma_ {i}(x_{i}-\overline{x})(\hat{y}_{i}-\overline{y}) \tag{6}\]  
where \(\gamma_{i}\) indicates correspondence weights. For the remaining part of the method, steps in DCP are applied.  
**FMR:** Contrary to the other two studies used as baselines for this work, the feature metric registration framework does not find correspondences between \(X\) and \(Y\). Instead, it focuses on minimizing the feature distance between the source and the target after the alignment is done. To obtain the point features, a PointNet-like encoder is used without a transformation. Due to the nature of the PointNet, the obtained features are rotation-attentive. A decoder is also trained to obtain meaningful features \(F(X)\) and \(F(Y)\) in an unsupervised way. Here, the projection error was defined as  
\[e=||F(X)-F(RY+t)||, \tag{7}\]  
where \(e\) is fed into the inverse compositional algorithm [79] to stepwise increment the transformation as  
\[[\Delta R,\Delta t]=(J^{T}J)^{-1}(J^{T}e), \tag{8}\]  
where \(J\) represents Jacobian of \(e\) with respect to \(R\) and \(t\).