Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 7090186, 9 pages

https://doi.org/10.1155/2018/7090186

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

Correspondence should be addressed to Junseok Kim

Received 22 November 2017; Accepted 1 January 2018; Published 6 February 2018

Academic Editor: Costică Moroșanu

Copyright © 2018 Darae Jeong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.