# Point Cloud Denoising Algorithm via Geometric Metrics on the Statistical Manifold
## 2 Preliminaries
### Metrics on \(\mathrm{P}(n)\)  
On \(\mathrm{P}(n)\), some metrics can be used as the measurements. Here, we introduce five specific measures including the Frobenius metric, the affine-invariant Riemannian metric, the log-Euclidean metric, the Kullback-Leibler divergence, and the symmetric Kullback-Leibler divergence.  
(1) Euclidean Distance  
The Frobenius metric (FM) on \(\mathrm{P}(n)\) is indicated by  
\[\langle U,W\rangle_{F}=\mathrm{tr}(U^{T}W), \tag{3}\]  
where \(\mathrm{tr}(\cdot)\) refers to the trace of \"\(\cdot\)\" and \(U,W\in\mathrm{P}(n)\). Then, \(\mathrm{P}(n)\) with the metric (3) becomes a manifold. The tangent space at the point \(U\in\mathrm{P}(n)\) can be represented by \(T_{U}\mathrm{P}(n)\). Also, the tangent space \(T_{U}\mathrm{P}(n)\) at \(U\) can be recognized as \(\mathrm{S}(n)\), as \(\mathrm{P}(n)\) is an open subset of \(\mathrm{S}(n)\). The Euclidean distance with the metric (3) corresponds to  
\[d_{F}(U,W)=\|U-W\|_{F}. \tag{4}\]  
(2) Riemannian Distance  
The manifold \(\mathrm{P}(n)\) is transformed into a Riemannian manifold, which is equipped with the affine-invariant Riemannian metric (AIRM) at point \(U\)  
\[gu(A,B)=g_{I}(U^{-1}A,U^{-1}B)=\mathrm{tr}(U^{-1}AU^{-1}B), \tag{5}\]  
where \(I\) represents the identity matrix, and \(A,B\in T_{U}\mathrm{P}(n)\). With the metric (5), the curvature of \(\mathrm{P}(n)\) is non-positive [11; 12]. The distance of two points \(U,W\in\mathrm{P}(n)\) induced by the AIRM is provided by the length of the local geodesic:  
\[d_{\mathrm{AIRM}}(U,W)=\|\log(U^{-\frac{1}{2}}WU^{-\frac{1}{2}})\|_{F}. \tag{6}\]  
However, the calculation cost of the AIRM distance is usually time-consuming in practical applications. An alternative option is the log-Euclidean metric (LEM), which can be indicated as:  
\[gu(A,B)=g_{I}(D_{U}\log A,D_{U}\log B), \tag{7}\]  
where \(A,B\in T_{U}\mathrm{P}(n)\) and \(D_{U}\) indicate the differential map at the point \(U\).  
The LEM distance between \(U\) and \(W\) could be written as  
\[d_{\mathrm{LEM}}(U,W)=\|\log U-\log W\|_{F}, \tag{8}\]  
which is obviously more concise than the AIRM distance.  
(3) Divergences  
Other geometric measures can be equipped on the manifold \(\mathrm{P}(n)\)[13]. One widely used measure on Riemannian manifolds is the Kullback-Leibler divergence (KLD), also known as Stein loss or log-determinant divergence [14]. The KLD between two points \(U\) and \(W\) can be indicated as:  
\[d_{\mathrm{KLD}}(U,W)=\mathrm{tr}(W^{-1}U-I)-\log\det(W^{-1}U). \tag{9}\]  
The symmetric Kullback-Leibler divergence (SKLD) is the Jeffreys divergence [15], and the divergence between \(U\) and \(W\) is represented by  
\[d_{\mathrm{SKLD}}(U,W)=\frac{1}{2}(d_{\mathrm{KLD}}(U,W)+d_{\mathrm{KLD}}(U,W)) =\frac{1}{2}\mathrm{tr}(W^{-1}U+U^{-1}W-2I). \tag{10}\]When dealing with optimization problems on manifolds, it is often necessary to calculate the gradient of the objective function \(F(U)\), which can be achieved using the covariant/directional derivative correlated with a given metric, such as a Riemannian metric etc:  
\[\langle\
abla F(U),A\rangle=\frac{\mathrm{d}}{\mathrm{d}\tau}\bigg{|}_{\tau=0}F (\gamma(\tau)),\forall A\in T_{U}P(n), \tag{11}\]  
with the curve \(\gamma:[0,1]\rightarrow\mathrm{P}(n)\) and \(\gamma(0)=U,\dot{\gamma}(0)=A\). The curve \(\gamma\) is linearized and can be re-expressed by  
\[\langle\
abla F(U),A\rangle=\frac{\mathrm{d}}{\mathrm{d}\tau}\bigg{|}_{\tau=0} F(U+\tau A)),\forall A\in T_{U}P(n). \tag{12}\]