Elsevier

Computer Aided Geometric Design

Volumes 52–53, March–April 2017, Pages 135-153
Computer Aided Geometric Design

Implicit surface reconstruction with total variation regularization

https://doi.org/10.1016/j.cagd.2017.02.005Get rights and content

Abstract

Implicit representations have been widely used for surface reconstruction on account of their capability to describe shapes that exhibit complicated geometry and topology. However, extra zero-level sets or spurious sheets usually emerge in implicit algorithms and damage the reconstruction results. In this paper, we propose a reconstruction approach that involves the total variation (TV) of the implicit representation to minimize the occurrence of spurious sheets. Proof is given to show that the recovered shape has the simplest topology with respect to the input data. By using algebraic spline functions as the implicit representation, an efficient discretization is presented together with effective algorithms to solve it. Hierarchical structures with uniform subdivisions can be applied in the framework for fitting fine details. Numerical experiments demonstrate that our algorithm achieves high quality reconstruction results while reducing the existence of extra sheets.

Introduction

Creating 3D digital representations of real world objects has been an attractive task with a wide range of applications in computer graphics and geometry processing. Many scanning techniques have been developed for data acquisition, such as laser-based range scanners, structured light scanners, multiview stereo camera systems, and depth cameras. When data are obtained from different scanning systems, the quality of the input point cloud for surface reconstruction varies a lot. The presence of missing data, noise, outliers, or non-uniform sampling becomes quite common. Combined with potentially complicated topology, regional geometric details, and large-scale data, surface reconstruction becomes an even more challenging task. A comprehensive survey on the recent progress is provided by Berger et al. (2014).

Existing surface reconstruction methods can be roughly classified into two categories: polygonal mesh approaches and implicit surface representations (Kazhdan and Hoppe, 2013). Polygonal mesh approaches are mainly based on the Voronoi diagram and its dual, the Delaunay triangulation (Boissonnat, 1984, Amenta et al., 1998). They produce mesh representations by interpolating a subset of the input data as vertices (Cazals and Giesen, 2006, Dey, 2006). However, polygonal methods are usually sensitive to noise, and they are not very robust on incomplete or non-uniform data. These issues are addressed in Dey and Goswami (2004), Hornung and Kobbelt (2006), Chazal and Lieutier (2006).

Implicit approaches describe the underlying surface by the zero level-set of an implicit function or an indicator function. Compared with polygonal mesh approaches, implicit methods are more suitable for representing surfaces with complicated topology and geometry (Bloomenthal and Bajaj, 1997, Gomes et al., 2009), and they are less sensitive to noise and sampling strategy of the data. Consequently, many implicit approaches has been proposed for surface reconstruction, for example the Blobby model, signed distance fields, algebraic surfaces, moving least squares (MLS) surfaces, radial basis functions (RBFs), multilevel partition of unity (MPU), and so on.

However, extra zero-level sets can emerge in the reconstruction results of implicit approaches (Moore and Warren, 1991, Ohtake et al., 2003, Ohtake et al., 2005a) and produce redundant mesh components in iso-surface extraction processes. A clean mesh is important in almost every application of surface reconstruction. Extra spurious sheets outside the surface surely affect the aesthetic appearance of the mesh object. In manufacturing, extra sheets inside a surface can damage the mechanical properties of the products. With the rapid progress of 3D printing techniques, implicit representations show great advantages in the field of manufacturing (Huang et al., 2013, Steuben et al., 2016). Attaining the correct topology for implicit representations is very important in such applications; yet, when the surface shape contains complicated features, there can be quite a lot of redundant components. Thus, intensive and massive tedious manual handling is usually required to obtain a clean mesh, especially for the inside part of the object.

The main cause of spurious sheets is that the value of the implicit function is only constrained near the sample points (Kazhdan et al., 2006). Moore and Warren (1991) avoided these extra sheets by using a distance fit, which approximates the discrete distance field on the whole domain using a regular grid. Similar strategies are also used by Carr et al. (2001), Ohtake et al. (2003). Jüttler and Felis (2002) dealt with this problem by adding tension to the surface and making the shape simple. Kazhdan et al. (2006) constrained the gradient of the implicit function near the input data to get rid of the extra sheets.

The strategies applied in the existing reconstruction methods generally fall into two categories. The first one is to construct an approximation of the signed distance field over the whole bounding box of the data. The other is to make use of the information near the data points, like normal vectors, and try to alleviate the emergence of extra sheets far from the surface. In the second case, there is no guarantee that no extra sheets will arise in the far region. While in the first case, a discrete approximation of the signed distance field usually has to be computed over quite a dense grid on the whole domain, which can be greatly affected by the accuracy of the approximation, especially if there is complicated geometry and noisy data.

In this paper, we present an algorithm to handle the extra zero-level sets, reducing them to the minimum possible occurrence. Given a set of 3D points, a multilevel algebraic spline function is constructed on the bounding box, whose zero-level set gives an approximation of the underlying surface. Besides the constraints close to the input data, the total variation (TV) of the implicit function is also simultaneously incorporated. The TV energy prevents the function from oscillating in regions far from the surface. With this regularization, only a set of off-surface points, or normals of the input data, are needed to indicate the orientation of the surface. Auxiliary data, like a dense grid on the whole domain, is not required in our approach. An accurate approximation of the distance field is not necessary either.

The main contributions of our work are summarized as follows:

  • We propose a novel approach that incorporates TV regularization to avoid the occurrence of spurious sheets in the implicit surface reconstruction from point clouds.

  • The need for auxiliary data structures is greatly reduced. Neither a grid on the whole domain nor an accurate approximation of distance field is required.

  • An effective algorithm is presented to solve the problem.

The remainder of this paper is organized as follows. In section 2, we review some related work. Section 3 presents preliminary knowledge on implicit representation, and we introduce the problem rigorously. Our approach and its properties are presented in sections 4 and 5. Experimental results are demonstrated in section 6, together with comparisons and discussions. And finally, we conclude the paper and propose possible future work.

Section snippets

Related work

The problem of surface reconstruction from point clouds has been extensively explored over the past few decades, and a substantial amount of work has been proposed in the categories of polygonal and implicit representations. Berger et al. (2014) provides a comprehensive overview of these methods. Here, we mainly refer to the implicit approaches that are most closely related to ours.

Preliminaries and problem statement

In this section, we give some preliminary information regarding the concept of implicit representation and describe the reconstruction problem in detail.

Surface reconstruction with total variation minimization

To avoid generating spurious sheets in the reconstruction process, constraints are needed in the whole domain of the implicit representation. From the definition of the implicit surface, we can see that the essential target is to avoid unnecessary intersections of the implicit function with the zero plane. If we can prevent the function from oscillating in regions far from the surface, or make it flatter, the emergence of spurious sheets can be greatly reduced. Since the concept of total

Implementation

In this section, we discuss the implementation details.

Experiments and discussion

A series of experiments have been carried out to evaluate the performance of our method. In this section, we present some of the results and discussions. Comparisons with other methods are also provided. All experiments are performed on a PC with a quad-core Intel i5 @ 2.8 GHz processor and 8 GB of RAM.

Conclusion and future work

In this paper, we presented a novel approach to deal with the problem of spurious sheets in implicit surface reconstruction. The extra sheets are reduced to the minimum possible occurrence by incorporating the total variation of the implicit representation during the reconstruction process. The reconstructed surfaces are guaranteed to have the simplest topology among all possible reconstructions. An effective discretization is proposed with the multi-resolution algebraic spline function used as

Acknowledgements

We would like to thank the anonymous reviewers for their constructive and valuable comments. The work is supported by the NSF of China (Nos. 11626253, 11371341), and the Fundamental Research Funds for the Central Universities.

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