# Point Cloud Denoising Algorithm via Geometric Metrics on the Statistical Manifold
## 3 Point Cloud Denoising Algorithm Based on Geometric Metrics
### Influence Functions of Different Metrics  
In order to analyze the robustness of geometric means when symmetry positive-definite matrices are contaminated by outliers, we adopt the influence function. To show that it is more suitable to assign geometric metrics than the Euclidean metric in Algorithm 1, we will describe the influence functions of the arithmetic mean and the geometric means.  
Let \(\overline{U}\) indicate the mean of m symmetry positive-definite matrices \(U_{1},U_{2},\ldots,U_{m}\) with FM (AIRM, LEM, KLD, or SKLD), and let \(\widehat{U}\) be the mean by supplementing a set of I outliers \(R_{1},R_{2},\ldots,R_{l}\) with a weight \(\tau(\tau\ll 1)\) to \(U_{1},U_{2},\ldots,U_{m}\)[20, 21].  
Next, we can define \(\widehat{U}\) as:  
\[\widehat{U}=\overline{U}+\tau H(\widehat{U},R_{1},R_{2},\ldots,R_{l})+O(\tau^ {2}) \tag{22}\]  
and the norm as  
\[h(\widehat{U},R_{1},R_{2},\ldots,R_{l}):=\|H(\widehat{U},R_{1},R_{2},\ldots,R_ {l})\|_{F}. \tag{23}\]  
The influence functions of the arithmetic mean and the AIRM mean can be found in Refs. [10, 22], as follows.  
**Proposition 1**.: _The influence function of the arithmetic mean for m symmetry positive-definite matrices \(U_{1},U_{2},\ldots,U_{m}\) and \(l\) outliers \(R_{1},R_{2},\ldots,R_{l}\) with a weight \(\tau(\tau\ll 1)\) can be expressed as:_  
\[H(\widehat{U},R_{1},R_{2},\ldots,R_{l})=\frac{1}{l}\sum_{j=1}^{l}R_{j}- \overline{U}. \tag{24}\]  
**Proposition 2**.: _The influence function of the AIRM mean for m symmetry positive-definite matrices \(U_{1},U_{2},\ldots,U_{m}\) and \(l\) outliers \(R_{1},R_{2},\ldots,R_{l}\) is given by_  
\[H(\widehat{U},R_{1},R_{2},\ldots,R_{l})=-\frac{1}{l}\sum_{j=1}^{l}\overline{ U}\log(R_{j}^{-1}\overline{U}). \tag{25}\]  
Before the influence functions of the LEM mean, the KLD mean, as well as the SKLD mean are presented, the following lemmas are given to calculate them [23, 24].  
Figure 1: Flowchart of Algorithm 1.  
**Lemma 1**.: _Let \(V\) and \(Z(\tau),\forall t\in\mathbb{R}\) be an invertible matrix that does not have eigenvalues in the closed negative real line, and let \(\log V\) be its principal logarithm, then we have that_  
1. _Both_ \(V\) _and_ \(\log V\) _commute with_ \([(V-I)t+I]^{-1}\) _for any_ \(t\in\mathbb{R}\)_._
2. _The following identities are valid that_ \[\int_{0}^{1}[(V-I)t+I]^{-2}dt=(I-V)^{-1}[(V-I)t+I]^{-1}|_{t=0}^{1}=V^{-1}\] (26) _and_ \[\frac{d}{d\tau}\log(Z(\tau))=\int_{0}^{1}[(Z(\tau)-I)t+I]^{-1}\frac{dZ(\tau)} {d\tau}[(Z(\tau)-I)t+I]^{-1}dt.\] (27)  
**Lemma 2**.: _Suppose that \(Z(t)\) is a real matrix for \(\forall t\in\mathbb{R}\), then we have_  
\[tr(\int_{a}^{b}Z(t)dt)=\int_{a}^{b}tr(Z(t))dt. \tag{28}\]  
**Lemma 3**.: _For any invertible matrix \(Z(t)\), the following identity holds_  
\[\frac{d}{d\tau}Z(t)=|Z(t)|tr(Z^{-1}\frac{d}{d\tau}Z(t)). \tag{29}\]  
Then, the influence functions of the LEM mean, the KLD mean, and the SKLD mean are presented as:  
**Proposition 3**.: _The influence function of the LEM mean for \(m\) symmetry positive-definite matrices \(U_{1},U_{2},\ldots,U_{m}\) and \(l\) outliers \(R_{1},R_{2},\ldots,R_{l}\) is represented by_  
\[H(\widehat{U},R_{1},R_{2},\ldots,R_{l})=\frac{1}{l}\sum_{j=1}^{l}\overline{U} (\log R_{j}-\log\overline{U}). \tag{30}\]  
Proof.: See Appendix A.  
**Proposition 4**.: _The influence function of the KLD mean for \(m\) symmetry positive-definite matrices \(U_{1},U_{2},\ldots,U_{m}\) and \(l\) outliers \(R_{1},R_{2},\ldots,R_{l}\) is represented by_  
\[H(\widehat{U},R_{1},R_{2},\ldots,R_{l})=\overline{U}-\frac{1}{l}\sum_{j=1}^{ l}\overline{U}R_{j}^{-1}\overline{U}. \tag{31}\]  
Proof.: See Appendix B.  
**Proposition 5**.: _The influence function of the SKLD mean for \(m\) symmetry positive-definite matrices \(U_{1},U_{2},\ldots,U_{m}\) and \(l\) outliers \(R_{1},R_{2},\ldots,R_{l}\) is represented by_  
\[H(\widehat{U},R_{1},R_{2},\ldots,R_{l})=\frac{1}{2l}\sum_{j=1}^{l}(R_{j}- \overline{U}R_{j}^{-1}\overline{U}). \tag{32}\]  
Proof.: See Appendix C.