Efficient 3D Volume Reconstruction from a Point Cloud Using a Phase-Field Method

We propose an explicit hybrid numerical method for the efficient 3D volume reconstruction from unorganized point clouds using a phase-field method. The proposed three-dimensional volume reconstruction algorithm is based on the 3D binary image segmentation method. First, we define a narrow band domain embedding the unorganized point cloud and an edge indicating function. Second, we define a good initial phase-field function which speeds up the computation significantly. Third, we use a recently developed explicit hybrid numerical method for solving the three-dimensional image segmentation model to obtain efficient volume reconstruction from point cloud data. In order to demonstrate the practical applicability of the proposed method, we perform various numerical experiments.


Introduction
In this paper, we propose an efficient and robust algorithm for volume reconstruction from a point cloud.Reconstructing the three-dimensional model from a point cloud is important in medical applications.Surface reconstruction from a point cloud is a process of finding a surface model that approximates an unknown surface for a given set of sample points lying on or near the unknown surface [1].
Hoppe et al. developed an algorithm to reconstruct a surface in the three-dimensional space from unorganized points scattered on or near the unknown surface.The algorithm is based on the idea of determining the zero level set of a signed distance function [2].Kazhdan proposed a surface reconstruction method which takes an oriented point set and returns a solid model.The method uses Stokes' theorem to calculate the characteristic function (one inside the model and zero outside of it) of the solid model [3].To reconstruct implicit surfaces from scattered unorganized data set, Li et al. presented a novel numerical method for surface embedding narrow volume reconstruction from unorganized points [4,5].Yang  Beneš et al. used the Allen-Cahn equation with a forcing term to achieve image segmentation [9].Caselles et al. proposed a model for active contours which could extract smooth shapes and could be adapted to find several contours simultaneously [10].Methods using geometric active contour were introduced in [11][12][13][14].Zhang et al. developed a weighted sparse penalty and a weighted grouping effect penalty in modeling the subspace structure [15].Chen used an ICKFCM method (ICA analysis and KFCM algorithm) in medical image segmentation and made a good result in extracting the complicated images [16].Zhang et al. proposed a novel fuzzy level set method based on finding the minimum of energy function to locate the true object boundaries effectively [17].Other numerical studies based on level set method were also introduced in [18,19].In this article, we propose an explicit hybrid algorithm for volume reconstruction from a point cloud.Therefore, it does not need implicit solvers such as multigrid methods.The computation is fast and efficient because the proposed algorithm uses a narrow band domain and a good initial condition.
This paper is organized as follows.In Section 2, we describe a mathematical model and a numerical solution algorithm for volume reconstruction from a point cloud.We present the numerical results for several examples in Section 3. In Section 4, we conclude.

Mathematical Model and Numerical Solution Algorithm
Now, we propose an explicit hybrid numerical method for volume reconstruction from a point cloud using a phase-field method.For X  = (  ,   ) in the two-dimensional space or X  = (  ,   ,   ) in the three-dimensional space,  = {X  | 1 ≤  ≤ } denote the point cloud in the two-or the threedimensional space, respectively.The geometric active contour model based on the mean curvature motion is given by the following evolution equation [20]: where (x) is an edge indicator function, () = 0.25( 2 − 1) 2 , and  is a constant which is related to the phase transition width.Note that here we use a different edge indicator function and efficient explicit numerical algorithm.
For simplicity of exposition, we first discretize (1) in the two-dimensional space Ω = (, ) × (, ).Let ℎ = ( − )/(  − 1) = ( − )/(  − 1) be the uniform mesh size, where   and   are the number of grid points.Let be approximations of (  ,   , Δ), where Δ is the time step.Let (x) = dist(x, ) = min 1≤≤ |X  − x| be the distance to the data , where X  = (  ,   ).In fact, we will use the distance function as an edge indicator function, (x).In practice, for  = 1, . . ., , we define a local domain Ω  which embeds the point X  and set the minimum value at the grid point between the point and the grid point.For example, Ω  is a 3×3 grid.Then, the computational narrow band domain is defined as Outside the narrow band domain, we set a large value to the edge indicator function, see Figure 1 for the procedure.
In this study, we apply the simplest sequential splitting procedure.We split (1) into two equations by using the operator splitting method: For a good initial configuration, we set  0  = 1 inside on the narrow band domain and set  0  = −1 outside the narrow band domain.
Given   , we solve (3) on the narrow band domain Ω nb by using the explicit Euler method: where  *  is the intermediate value which is defined at point (  ,   ).The initial values at inside and outside region of the narrow band domain Ω nb act as Dirichlet boundary condition in computing the discrete Laplace operator, Δ  .
Then, we analytically solve (4) by the method of separation of variables [21,22].That is,  +1  = (Δ) by analytically solving with the initial condition  (0) =  *  . ( The analytic solution is given as Therefore, ( 5) and ( 7) consist of an efficient and robust algorithm for volume reconstruction from a point cloud.We should note that the proposed numerical solution algorithm is fully explicit.Therefore, we do not need an iterative method such as multigrid method to solve the governing equation.Also, the implementation of the algorithm is straightforward.mixture.In the two-dimensional case, as  approaches zero, the zero level set of  evolves with the following velocity:

Computational Experiments
where  is the normal velocity,  is the curvature, and  is the radius of curvature at the point of the zero level set [24].Then, ( 8) is rewritten by ()/ = −1/() with (0) =  0 .

The Basic Working Mechanism of the Algorithm.
The edge indicator function (x) is close to zero where the point cloud exists.Therefore, the evolution will stop or slow down in the neighborhood of the point cloud.In (1), / = −  ()/ 2 + Δ makes the phase-field shrink until it reaches the point cloud by the mean curvature flow.If the geometry of the point cloud is not convex, then the term () makes the level set of the phase-field further shrink.For more details, please refer to [20].To confirm the working mechanism of the algorithm, the temporal evolution of the interface in the two-dimensional space is shown in Figure 3. Here, we use the following parameters:  = 0.0096, ℎ = 1/200, and Δ = 2.8284 − 5.
Figure 4 represents construction of the three-dimensional distance function (x) and the narrow band domain.In Figures 4(a) and 4(b), we can see the given point cloud data on computational grid and a local mesh Ω  of single point X  .Here, we calculate the distance between the given point X  and the grid points x on the local mesh Ω  .The distance function (x) is defined by the shortest one among the distance.Then, we obtain the narrow band domain which is determined by the distance function.Now, we can straightforwardly extend the two-dimensional numerical solutions ( 5) and ( 7 Next, we reconstruct volume of Armadillo model from the given scattered points ( = 129732) as shown in Figure 6(a) [25].For numerical test, we use the following parameters:  = 0.0127, ℎ = 0.0088, Δ = ℎ/(200 √ 3),   = 220,   = 250,   = 200, Ω = (0, 2.2) × (0, 2.5) × (0, 2), and  = 1000.In the first and second rows in Figure 6, we can see the front and back views of Armadillo model.By the proposed scheme, we obtain the numerical solution  after 400 iterations (see Figure 6(c)) with the initial condition in Figure 6(b).
As the final example, we reconstruct volume of Stanford Dragon from the given scattered points ( = 656469) as shown in Figure 7(a) [25].For numerical test, we use the

Effect of 𝜆.
In this section, we investigate the effect of  parameter on the three-dimensional volume reconstruction.
The parameter makes the level set of  shrink to the given points.We use the same parameters in Figure 7 except for the  value.As shown in Figure 8, if the value of  is small, then the surface is oversmoothed by the motion by mean curvature.On the other hand, if it is too large, then the surface is rough.

Conclusions
In this article, we developed an explicit hybrid numerical algorithm for the efficient 3D volume reconstruction from unorganized point clouds using a modified Allen-Cahn equation.The 3D volume reconstruction algorithm is based on the 3D binary image segmentation method.The proposed algorithm has potential to be used in various practical industry such as 3D model printing from scattered scanned data.The computational results confirmed that the algorithm is very efficient and robust in reconstructing 3D volume from point clouds.

Figure 1 :
Figure 1: Schematic of defined distance function.(a) Point cloud data, (b) mesh grid covering the point cloud and local mesh grid, (c) local mesh grid Ω  embedding a point X  , (d) narrow band domain, and (e) edge indicator function, (x).

3. 1 .Figure 2 :Figure 3 :
Figure 2: Temporal evolutions of the radius with Δ = 0.15ℎ 2 up to  = 5000Δ in the two-dimensional space.(a) Zero level contour and (b) radius () of circle with respect to time.

Figure 4 :
Figure 4: Construction of the three-dimensional distance function (x) and the narrow band domain: (a) point cloud data and mesh, (b) cross section of point cloud and a local mesh Ω  of single point X  , (c) local mesh grid Ω  embedding a point X  , and (d) narrow band domain which is determined by the distance function.

Figure 5 :
Figure 5: Front and back views of Happy Buddha: (a) point clouds, (b) initial condition of , and (c) numerical solution  after 200 iterations.

Figure 6 :Figure 7 :
Figure 6: Front and back views of Armadillo model: (a) initial point clouds, (b) initial condition of , and (c) numerical solution  after 400 iterations.
et al. proposed a 3D reconstruction technique from nonuniform point clouds via local hierarchical clustering [6].Zhao et al. developed a fast sweeping level set and tagging methods [7].Yezzi Jr. et al. proposed a new medical image segmentation based on feature-based metrics on a given image [8].