# Three-Dimensional Surface Reconstruction from Point Clouds Using [PERSON]'s Elastica Regularization
## 2 Variational Image 3D Reconstruction Theory
This method offers a more straightforward way to calculate the initial image bounded by the point cloud data. The proposed method is a simpler way to compute the initial image enclosed by the point clouds compared to other methods. By thresholding the distance function \(d(x)\) obtained from (2), an annular binary image \(I(x)\) is created. This image is then located using the fast sweeping algorithm. On the right-hand side of (2), \(W(x)\) is substituted with \(I(x)\), which makes \(d(x)\) no longer a distance function. The grid points on the six border faces of the domain \(\Omega\) are set to zero, and other inner grid points are set to very large values, to solve \(d(x)\) using the rapid sweeping procedure with eight directional sweeping Gauss-Seidel iterations. Once \(d(x)\) is calculated, the new function \(f(x)\) is determined by setting \(f(x)=d(x)\). The result of thresholding \(f(x)\) to create a new binary image is demonstrated in the final two images of Figure 2. The border points of the object in the binary picture can be found using a simple algorithm such as Marching Cube, and these new points \(\{\widetilde{x}_{i}\}\) and the original disorganized point clouds \(\{x_{i}\}\) are close to each other. To obtain the signed distance function \(\phi_{0}\), the fast sweeping algorithm is performed again on the new point clouds \(\{\widetilde{x}_{i}\}\). The inside of the object is set to be negative, and the outside is set to be positive. The signed distance function \(\phi_{0}\) shown in Figure 3 is used as the initial level set function in the proposed method. The functional of the new method called TV\({}_{g}\) combined balloon force (TV\({}_{g}\)-B) was introduced by [PERSON] in [21] and is composed of three separate energy terms, as follows:  
\[E(\phi,c_{1},c_{2})=E_{R}(\phi)+\lambda E_{I}(\phi,c_{1},c_{2})+\beta E_{B}( \phi). \tag{3}\]  
The first term, \(E_{R}\), is responsible for maintaining the level set function \(\phi\) as a signed distance function, which serves as a regularization term to ensure that the reconstructed surface is smooth:  
\[E_{R}(\phi)=\int_{\Omega}d(x)|\
abla H(\phi)|dx+\frac{\mu}{2}\int_{\Omega}(| \
abla\phi|-1)^{2}dx, \tag{4}\]  
The second term \(E_{R}\) in the functional of the TV\({}_{g}\)-B method helps keep the level set function \(\phi\) as a signed distance function. The weight \(\mu\) ensures that \(\phi\) remains a signed distance function, with larger values of \(\mu\) making \(\phi\) closer to a signed distance function. This term corresponds to the GAC model and uses a weighted TV-norm. The third term ensures that the level set function \(\phi\) stays as a signed distance function, eliminating the need for repeated initialization. The information acquired from the data set is incorporated into the second data fitting term \(E_{I}\), which is:  
\[|\
abla d(x)|=w(x),\ \forall x\in\Omega\backslash\{x_{i}\},w(x)=1,d(x_{i})=0,\ \forall x\in\{x_{i}\}. \tag{2}\]0where the values \(c_{1}\) and \(c_{2}\) denote the mean inside and outside the zero level set of \(\phi\), respectively. The function \(Q(c_{1},c_{2})=(c_{1}-f(x))^{2}-(c_{2}-f(x))^{2}\) is established to enhance the precision of the reconstruction by accounting for equal intensity levels of the fine details in the object. The penalty factor \(\lambda\) mentioned in Equation (3) has a positive value. The area information included in the zero level set of \(\phi\) is incorporated into the third balloon force term \(E_{B}\) in order to speed up surface evolution and segment concave objects. The term is expressed as:  
\[|\
abla d(x)|=w(x),\ \forall x\in\Omega\backslash\{x_{i}\},w(x)=1,d(x_{i})=0,\ \forall x\in\{x_{i}\}. \tag{2}\]1  
The value of the parameter \(\beta\) in \(E_{B}\) depends on whether the zero level set is reached or not. If the initial zero level set function is outside of the object, then the inside of the level set is less than 0 and the outside is greater than 0. In this case, \(\beta\) takes on a positive value, which ensures that the contour line shrinks during evolution.