Abstract
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.
1. 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 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].
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 [11–14]. 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.
2. 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 in the two-dimensional space or in the three-dimensional space, denote the point cloud in the two- or the three-dimensional space, respectively. The geometric active contour model based on the mean curvature motion is given by the following evolution equation [20]:where is an edge indicator function, , 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 . Let be the uniform mesh size, where and are the number of grid points. Let be the discrete domain. Let be approximations of , where is the time step. Let be the distance to the data , where . In fact, we will use the distance function as an edge indicator function, . In practice, for , we define a local domain which embeds the point and set the minimum value at the grid point between the point and the grid point. For example, is a grid. Then, the computational narrow band domain is defined asOutside the narrow band domain, we set a large value to the edge indicator function, see Figure 1 for the procedure.
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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 inside on the narrow band domain and set outside the narrow band domain.
Given , we solve (3) on the narrow band domain 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 act as Dirichlet boundary condition in computing the discrete Laplace operator, .
Then, we analytically solve (4) by the method of separation of variables [21, 22]. That is, by analytically solvingThe analytic solution is given asTherefore, (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.
3. Computational Experiments
3.1. Two-Dimensional Experiments
3.1.1. Motion by Mean Curvature
To test the proposed numerical scheme, we perform a numerical experiment. The test is motion by mean curvature. If we set and , then the governing equation (1) becomes the original Allen-Cahn equation [23], which is a reaction-diffusion equation describing the process of phase separation in a binary alloy mixture. In the two-dimensional case, as approaches zero, the zero level set of evolves with the following velocity: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 with . Therefore, analytic solution is given as .
On the computational domain , we investigate the motion by mean curvature of the circle in the annulus narrow band domain: as shown in Figure 2(a). We define an initial condition asIn this numerical simulation, we use the following parameters: , , , , and . Figures 2(a) and 2(b) show the temporal evolution of the initial circle and its radius with respect to time, respectively. For verification of our numerical results, we include the results of the analytic solution. As shown in Figure 2, the initial circle shrinks under the motion by mean curvature.
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3.1.2. The Basic Working Mechanism of the Algorithm
The edge indicator function 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), 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: , , and .
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3.2. Three-Dimensional Experiments
Next, we discretize (1) in the three-dimensional space, that is, . Let be the uniform mesh size, where , , and are the total number of grid points. Let be the discrete domain. We define as approximations of , where is the time step size. Let be the distance to the data , where .
Figure 4 represents construction of the three-dimensional distance function 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 of single point . Here, we calculate the distance between the given point and the grid points on the local mesh . The distance function is defined by the shortest one among the distance. Then, we obtain the narrow band domain which is determined by the distance function.
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Now, we can straightforwardly extend the two-dimensional numerical solutions (5) and (7) to the following three-dimensional solutions:
3.2.1. Reconstruction from Various Point Clouds
First, we reconstruct volume of Happy Buddha from the given scattered points () as shown in Figure 5(a) [25]. For numerical test, we use the following parameters: , , , , , , , and . In the first and second rows in Figure 5, we can see the front and back views of Happy Buddha. By the proposed scheme, we obtain the numerical solution after 200 iterations (see Figure 5(c)) with the initial condition in Figure 5(b).
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Next, we reconstruct volume of Armadillo model from the given scattered points () as shown in Figure 6(a) [25]. For numerical test, we use the following parameters: , , , , , , , and . 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).
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As the final example, we reconstruct volume of Stanford Dragon from the given scattered points () as shown in Figure 7(a) [25]. For numerical test, we use the following parameters: , , , , , , , and . In the first and second rows in Figure 7, we can see the front and back views of Stanford Dragon. By the proposed scheme, we obtain the numerical solution after 400 iterations (see Figure 7(c)) with the initial condition in Figure 7(b).
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3.2.2. Effect of
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.
4. 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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The first author (D. Jeong) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIP) (NRF-2017R1E1A1A03070953). The corresponding author (Junseok Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03933243). This work was supported by the BK21 PLUS program.