Phase-field guided surface reconstruction based on implicit hierarchical B-splines

Highlights

• A phase-field guided implicit surface reconstruction method is proposed.

• The reconstructed surface is represented by a hierarchical B-spline.

• Our mathematical model avoids the use of the normal information of the input point cloud.

• Our approach can greatly reduce the storage space of the reconstructed implicit surface.

Abstract

Constructing smooth surface representations from point clouds is a fundamental problem in geometric modeling and computer graphics, and a wealthy of literature has focused on this problem. Among the many approaches, implicit surface reconstruction has been a central topic in the past two decades due to its ability to represent objects with complicated geometry and topology. Recently, the problem of reducing the storage requirement for implicit representations has attracted much attention. In this paper, we propose a phase-field guided implicit surface reconstruction method to tackle this problem. The implicit function of our method behaves like the phase-field of a binary system, in which it takes distinct values (i.e., −1 and 1) in each of the phases with a smooth transition between them. Given an unorganized point cloud, we present a method to construct a phase-filed function represented by a hierarchical B-spline whose zero level set approximates the point cloud as much as possible. Unlike previous approaches, our mathematical model avoids the use of the normal information of the point cloud. Furthermore, as demonstrated by experimental results, our method can achieve very compact representation since we mainly need to save the coefficients of the hierarchical B-spline function within a narrow band near the point cloud. The ability of our method to produce reconstruction results with high quality is also validated by experiments.

Introduction

Surface reconstruction prevails in creating digital models of real world objects. Owing to the recent development in 3D scanning technology, and consumer-level depth sensors such as Microsoft Kinect, the acquisition cost of 3D objects decreases dramatically. As the diversity, ease of use, and popularity of 3D acquisition methods continue to increase, so does the need for the development of new surface reconstruction techniques. A recent survey on the development of surface reconstruction can be found in Berger et al. (2014).

Among various surface reconstruction methods, implicit surfaces are the predominant representations. The underlying reason is that implicit surfaces are more suitable for reconstructing surfaces from data sets that are noisy, incomplete or non-uniformly distributed than other surface representations such as polygonal meshes, parametric surfaces and subdivision surfaces. In addition, implicit representations greatly facilitate the classification problem of whether a given point is on, inside or outside a surface, and are able to represent shapes with complicated topology and geometry, even with dynamic topology (Bloomenthal and Wyvill, 1997, Gomes et al., 2009). Consequently, many representations of implicit surfaces have been proposed, e.g. Blobby models (Muraki, 1991), signed distance fields (Hoppe et al., 1992), radial basis functions (RBFs) (Carr et al., 2001), moving least squares (MLS) surfaces (Levin, 2003, Shen et al., 2004, Feng et al., 2014), algebraic spline (AS) surfaces (Jüttler and Felis, 2002), multilevel partition of unity (MPU) (Ohtake et al., 2003) and so on.

An implicit surface $\mathcal{S}$ is identified as the zero level-set of a function $f(\mathbf{p}):\mathrm{\Omega }\subset {\mathbb{R}}^3\to \mathbb{R}$. That is

$$\mathcal{S}=f^{-1}(0)=\{\mathbf{x}\in \mathrm{\Omega }|f(\mathbf{x})=0\}.$$

Implicit surfaces are a volume-based representation that enables many operations to be performed easily, such as inside/outside test, offset, blending and boolean operations. However, the function information at the position far from $\mathcal{S}$ is usually less useful. Moreover, the Jordan–Brouwer separation theorem states that an implicit surface $\mathcal{S}\in {\mathbb{R}}^3$ separates ${\mathbb{R}}^3$ into the surface itself and two connected open sets, i.e., the outside and the inside. Therefore, the most compact implicit function is the characteristic function of $\mathcal{S}$ that is defined as

$$\chi (\mathbf{p})=\{\begin{array}{cc}-1\hfill & \text{if}\mathbf{p}\in {\mathrm{\Omega }}_{-},\hfill \\ 0\hfill & \text{if}\mathbf{p}\in \mathcal{S},\hfill \\ +1\hfill & \text{if}\mathbf{p}\in {\mathrm{\Omega }}_{+},\hfill \end{array}$$

where $\mathrm{\Omega }={\mathrm{\Omega }}_{-}\cup \mathcal{S}\cup {\mathrm{\Omega }}_{+}$, with ${\mathrm{\Omega }}_{-}$ and ${\mathrm{\Omega }}_{+}$ being the inside and outside of $\mathcal{S}$ respectively. In order to have a smooth implicit surface $\mathcal{S}$, we require that the implicit function $f(\mathbf{p})$ meet some continuity criterions, e.g. $f(\mathbf{p})\in C^1$ and $\mathrm{\nabla }f(\mathbf{p})\ne \mathbf{0}$ for all $\mathbf{p}\in \mathrm{\Omega }$, here ∇f is the gradient of f. So, the idealized implicit function should behave like the phase-field of a binary system as follows

$$\varphi (\mathbf{p})=\{\begin{array}{cc}-1\hfill & \text{if}\mathbf{p}\in {\mathrm{\Omega }}^{-},\hfill \\ [-1,+1]\hfill & \text{if}\mathbf{p}\in {\mathrm{\Omega }}^{NB}\supset \mathcal{S},\hfill \\ +1\hfill & \text{if}\mathbf{p}\in {\mathrm{\Omega }}^{+},\hfill \end{array}$$

where Ω is the disjoint union of ${\mathrm{\Omega }}^{-},{\mathrm{\Omega }}^{NB}$ and ${\mathrm{\Omega }}^{+}$, ${\mathrm{\Omega }}^{-}\subset {\mathrm{\Omega }}_{-}$, ${\mathrm{\Omega }}^{+}\subset {\mathrm{\Omega }}_{+}$, and ${\mathrm{\Omega }}^{NB}$ is a narrow band near the implicit surface $\mathcal{S}$. The value of function $\varphi (\mathbf{p})$ changes smoothly from −1 to +1 over the region ${\mathrm{\Omega }}^{NB}$, and is constant (i.e., 1 or −1) otherwise as demonstrated in Fig. 1 for one and two dimensional cases. It is obvious that the idealized implicit function is more suitable for compression than other implicit representations, since it always takes constant values in the region ${\mathrm{\Omega }}^{-}\cup {\mathrm{\Omega }}^{+}$.

In this paper, we propose a phase-field guided surface reconstruction method based on the above observation. Given a point cloud, we present a method to construct an implicit function which approximates the phase-filed of the point cloud. The implicit function takes values −1 inside, +1 outside and a hierarchical B-spline in a narrow band of the point cloud. Such representation is very compact and uses much less storage space than other representations, and thus it may save computational cost in subsequent operations (e.g., function evaluation) and reduce the transmission time of the information of the implicit representation on internet. Furthermore, the hierarchical B-spline representation in the narrow band can capture the geometric details of the reconstructed surface, due to the adaptivity, local refinement and nonnegativity of their basis functions. When compared with some global reconstruction methods, our method only needs to determine the unknown coefficients of hierarchical B-spline basis functions within the narrow band, which enables us to cut down the cost of computation. On the other hand, our method generates a global representation that is still able to perform inside/outside test, boolean operations and so on, while the existing narrow band methods cannot.

The organization of this article is as follows. In Section 2, we review related work. Section 3 introduces some preliminary knowledge about hierarchical B-splines. Section 4 presents the mathematical model and the algorithm for our phase-field guided surface reconstruction method. Experimental results and performance of our algorithm are shown in Section 5. Comparisons with the state-of-the-art methods are also provided. Section 6 concludes the paper with proposals for future work.