# Three-Dimensional Surface Reconstruction from Point Clouds Using [PERSON]'s Elastica Regularization
## 3 Proposed Variational Method  
When reinitializing the symbol distance function, we consider using the projection method instead of the first term in (3), then the (3) is rewritten as follows:  
\[E_{R}(\phi)=\int_{\Omega}d(x)\bigg{(}a+b\bigg{(}\
abla\cdot\frac{\
abla H( \phi)}{|\
abla H(\phi)|}\bigg{)}^{2}\bigg{)}|\
abla H(\phi)|dx,\qquad s.t.\ |\
abla\phi|=1. \tag{9}\]  
The complex form of Euler's elastica affects the computational efficiency. If we use the ADMM algorithm proposed by [PERSON] (1950) to solve Equation (9), we can obtain the following expression:  
\[ \begin{split}\min_{u,v,\vec{u},\vec{u},\vec{m}}E(u,v,\vec{w}, \vec{n},\vec{m},)=&\int_{\Omega}Q(c_{1},\ c_{2})vdx+\gamma\int_{ \Omega}[a+b|\
abla\cdot\vec{n}|^{2}]|\vec{w}|dx\\ &+\beta\int_{\Omega}-d(x)vdx+\frac{\mu}{2}\int_{\Omega}(|\vec{w}| -1)^{2}dx\\ &+\int_{\Omega}\lambda_{1}(|\vec{w}|-\vec{w}\cdot\vec{m})dx+ \gamma_{1}\int_{\Omega}(|\vec{w}|-\vec{w}\cdot\vec{m})dx\\ &+\int_{\Omega}\vec{\lambda}_{2}\cdot(\vec{w}-\
abla u)dx+\frac{ \gamma_{2}}{2}\int_{\Omega}|\vec{w}-\
abla u|^{2}dx\\ &+\int_{\Omega}\lambda_{3}(v-u)dx+\gamma_{3}\int_{\Omega}(v-u)^{ 2}dx\\ &+\int_{\Omega}\vec{\lambda}_{4}\cdot(\vec{n}-\vec{m})dx+\frac{ \gamma_{4}}{2}\int_{\Omega}|\vec{n}-\vec{m}|^{2}dx\end{split}  \tag{10}\]  
Obviously, the ADMM algorithm introduces four splitting operators \(v,\vec{w},\vec{n},\vec{m}\) and four Lagrange multipliers \(\lambda_{1},\vec{\lambda_{2}},\lambda_{3},\vec{\lambda_{4}}\), which undoubtedly impose a significant computational burden on the model. To simplify it, we introduce auxiliary dual variables \(\vec{p}\). This process significantly reduces the computational overhead at each iteration. Furthermore, ADMM requires the tuning of four extra parameters, e.g., \(\gamma_{1}\), \(\gamma_{2}\), \(\gamma_{3}\), and \(\gamma_{4}\), whereas our method has only one extra parameter (e.g., \(\gamma\)) to tune. This introduces a considerable level of complexity and difficulty in utilizing ADMM effectively. With these variables, it is easy to rewrite the energy function in (3) as an Augmented Lagrangian Function, which forms the basis of our proposed dual algorithm for the problem:  
\[ \begin{split}\min_{u}\max_{|\vec{p}|\leq g}E(\phi,\vec{p})& =\gamma\int_{\Omega}d(x)(\
abla\cdot\vec{p})H(\phi)dx+\lambda \int_{\Omega}Q(c_{1},c_{2})H(\phi)dx\\ &+\beta\int_{\Omega}d(x)H(-\phi)dx+\frac{\mu}{2}\int_{\Omega}(| \
abla\phi|-1)^{2}dx\\ s.t.\ g=\bigg{(}a+b\bigg{(}\
abla\cdot\frac{\
abla\phi}{| \
abla\phi|}\bigg{)}^{2}\bigg{)}\end{split}  \tag{11}\]  
The solution of \(c_{1}\), \(c_{2}\) is the same as (7), but the formula of \(\phi\) is simplified to the following gradient reduction equation:\[\frac{\partial\phi}{\partial t}=\!\!d(x)(\
abla\cdot\vec{p})\delta(\phi)+\lambda Q (c_{1},c_{2})\delta(\phi)-\beta d(x)\delta(\phi)+\mu\left(\Delta\phi-\
abla \cdot\frac{\
abla\phi}{|\
abla\phi|}\right), \tag{12}\]  
where \(t\) is the time step. After \(\phi\) is solved, the other problem \(\vec{p}\) can be optimized as follows:  
\[ \begin{cases}\frac{\partial\vec{p}(x,\ t)}{\partial t}=\gamma\
abla(d(x)\delta (\phi))\\ |\vec{p}|\leq g\end{cases} , \tag{13}\]  
where the value of \(g\) is directly determined by the \(\phi\) obtained in the previous step. Therefore, the numerical implementations of the function (13) can be calculated as follows:  
\[ \begin{cases}\vec{p}_{i,j}^{k+1}=\vec{p}_{i,j}^{k}+t\gamma\
abla\left(d\delta \left(\phi_{i,j}^{k+1}\right)\right)\\ \vec{p}_{i,j}^{k+1}=\frac{g_{i,j}^{k+1}\vec{p}_{i,j}^{\frac{T}{2}}}{\max\{g_{ i,j}^{k+1},|\vec{p}_{i,j}^{\frac{T}{2}}|\}}\end{cases} , \tag{14}\]  
In each iteration, the following error tolerances should be checked to determine convergence, i.e.,  
\[\Sigma^{k+1}=\frac{\|E^{k+1}-E^{k}\|}{\|E^{k}\|}\leq\text{Tol.} \tag{15}\]  
where \(\text{Tol}=0.01\). The advantage of this method is very obvious; not only is the calculation process greatly simplified but also the convergence speed of the model will increase. We give detailed experimental explanations in the result section.