\sqrt{n_i}$ with $\alpha$ being a given parameter and $n_i$ being the average (of the nine cell-pairs) amount of correspondences.

**Locality Preserving Matching [@Ma2017Locality].** This algorithm removes mismatches by digging out the local geometric structure consensus. With the hypothesis that the local structure around a correspondence may not change freely, a cost function is defined as $$\label{eq:LRF24}
\begin{aligned}
L({\mathcal{K}_{inlier}},\lambda_{lpm} )=\sum\limits_{i\in {\mathcal{K}_{inlier}}}{\left( \sum\limits_{j|{\mathbf{x}_{j}}\in {\mathcal{K}_{{\mathbf{x}_{i}}}}}{{{\left( d\left( {\mathbf{x}_{i}},{\mathbf{x}_{j}} \right)-d\left( \mathbf{x}_{i}^{'},\mathbf{x}_{j}^{'} \right) \right)}^{2}}} \right.}\\
+\left. \sum\limits_{j|{\mathbf{x}_{j}}^{'}\in {\mathcal{K}^{'}_{\mathbf{x}_{i}^{'}}}}{{{\left( d\left( {\mathbf{x}_{i}},{\mathbf{x}_{j}} \right)-d\left( \mathbf{x}_{i}^{'},\mathbf{x}_{j}^{'} \right) \right)}^{2}}} \right)+\lambda_{lpm} \left(N-\left| {\mathcal{K}_{inlier}} \right| \right),
\end{aligned}$$ where $\lambda_{lpm}$ is a regularization parameter, $d$ is the Euclidean distance between two keypoints, $\mathcal{K}_{\mathbf{x}_i}$ and $\mathcal{K}_{\mathbf{x}_{i}^{'}}^{'}$ respectively are sets of the $k$ nearest neighbors of $\mathbf{x}_i$ and $\mathbf{x}_{i}^{'}$, $N$ is the size of $\mathcal{K}$, and $\mathcal{K}_{inlier}$ is an inlier subset of $\mathcal{K}$. Under non-rigid transformations such as deformation, the absolute distance in Eq. \[eq:LRF24\] may not be preserved well. To address this issue, LPM converts the cost function to $$\label{eq:LRF25}
\begin{aligned}
L(\mathcal{W},\lambda_{lpm})=\sum\limits_{i=1}^{N}{{{w}_{i}}\left(\sum\limits_{j|{\mathbf{x}_{j}}\in {\mathcal{K}_{{\mathbf{x}_{i}}}}}{d\left( \mathbf{x}_{i}^{'},\mathbf{x}_{j}^{'} \right)} \right.}\\
+\left.\sum\limits_{j|\mathbf{x}_{j}^{'}\in {\mathcal{K}^{'}_{\mathbf{x}_{i}^{'}}}}{d\left( {\mathbf{x}_{i}},{\mathbf{x}_{j}} \right)} \right)+\lambda_{lpm} \left( N-\sum\limits_{i=1}^{N}{{{w}_{i}}} \right),
\end{aligned}$$ where $\mathcal{W}$ is a set of indicators where $w_i=1$ indicates the inlier and $w_i=0$ otherwise. This equation can be further reorganized by merging the related items of $w_i$ as $$\label{eq:LRF26}
