In the previous chapter we learned that the restricted Delaunay triangulation is a good approximation of a densely sampled surface Σ from both topological and geometric view point. Unfortunately, we cannot compute this triangulation because the restricted Voronoi diagram Vor P|Σ cannot be computed without knowing Σ. As a remedy we approximate the restricted Voronoi diagram and compute a set of triangles that is a superset of all restricted Delaunay triangles. This set is pruned to extract a manifold surface which is output as an approximation to the sampled surface Σ.

Algorithm

First, we observe that each restricted Voronoi cell Vp|Σ = Vp ∩ Σ is almost flat if the sample is sufficiently dense. This follows from the Normal Variation Lemma 3.3 as the points in Vp|Σ cannot be far apart if ε is small. In particular, Vp|Σ lies within a thin neighborhood of the tangent plane τp at p. So, we need two approximations: (i) an approximation to τp or equivalently to np and (ii) an approximation to Vp|Σ based on the approximation to np. The following definitions of poles and cocones are used for these two approximations.

Poles and Cocones

Definition 4.1 (Poles). The farthest Voronoi vertex, denoted p+, in Vpis called the positive pole of p.

The algorithms for surface reconstruction in previous chapters assume that the input is noise-free. Although in practice all of them can handle some amount of displacements of the points away from the surface, they are not designed in principle to handle such data sets. As a result when the points are scattered around the sampled surface, these algorithms are likely to fail. In this chapter we describe an algorithm that is designed to tolerate noise in data.

The algorithm works with the Delaunay/Voronoi diagrams of the input points and draws upon some of the principles of the power crust algorithm. The power crust algorithm exploits the fact that the union of the polar balls approximates the solid bounded by the sampled surface. Obviously, this property does not hold in the presence of noise. Nevertheless, we have observed in Chapter 7 that, under some reasonable noise model, some of the Delaunay balls remain relatively big and can play the role of the polar balls. These balls are identified and partitioned into inner and outer balls. We show that the boundary of the union of the outer (or inner) big Delaunay balls is homeomorphic to the sampled surface. This immediately gives a homeomorphic surface reconstruction though the reconstructed surface may not interpolate the sample points.

Simply stated, the problem we study in this book is: how to approximate a shape from the coordinates of a given set of points from the shape. The set of points is called a point sample, or simply a sample of the shape. The specific shape that we will deal with are curves in two dimensions and surfaces in three dimensions. The problem is motivated by the availability of modern scanning devices that can generate a point sample from the surface of a geometric object. For example, a range scanner can provide the depth values of the sampled points on a surface from which the three-dimensional coordinates can be extracted. Advanced hand held laser scanners can scan a machine or a body part to provide a dense sample of the surfaces. A number of applications in computer-aided design, medical imaging, geographic data processing, and drug designs, to name a few, can take advantage of the scanning technology to produce samples and then compute a digital model of a geometric shape with reconstruction algorithms. Figure 1.1 shows such an example for a sample on a surface which is approximated by a triangulated surface interpolating the input points.

The reconstruction algorithms described in this book produce a piecewise linear approximation of the sampled curves and surfaces.

In the previous chapters we have assumed that the input points lie exactly on the sampled surface. Unfortunately, in practice, the input sample often does not satisfy this constraint. Noise introduced by measurement errors scatters the sample points away from the surface. Consequently, all analysis as presented in the previous chapters becomes invalid for such input points. In this chapter we develop a noise model that accounts for the scatter of the inputs and then analyze noisy samples based on this model. We will see that, as in the noise-free case, some key properties of the sampled surface can be computed from the Delaunay triangulation of a noisy sample. Specifically, we show that normals of the sampled surface can still be estimated from the Delaunay/Voronoi diagrams. Furthermore, the medial axis and hence the local feature sizes of the sampled surface can also be estimated from these diagrams. These results will be used in Chapters 8 and 9 where we present algorithms to reconstruct surfaces from noisy samples.

Noise Model

In the noise-free case ε-sampling requires each point on the surface have a sample point within a distance of ε times the local feature size. When noise is allowed, the sample points need not lie exactly on the surface and may scatter around it.

Most of the surface reconstruction algorithms face a difficulty when dealing with undersampled surfaces and noise. While the algorithm described in Chapter 5 can detect undersampling, it leaves holes in the surface near the undersampled regions. Although this may be desirable for reconstructing surfaces with boundaries, many applications such as CAD designs require that the output surface be watertight, that is, a surface that bounds a solid. Ideally, this means that the watertight surface should be a compact 2-manifold without any boundary. The two algorithms that are described in this chapter produce these types of surfaces when the input sample is sufficiently dense. However, the algorithms are designed keeping in mind that the sample may not be sufficiently dense everywhere. So, in practice, the algorithms may not produce a perfect manifold surface but their output is watertight in the following sense:

Notice that the above definition allows the watertight surface to be nonmanifold. The closure of a 3-manifold can indeed introduce nonmanifold property; for example, a surface pinched at a point can be in the closure of a 3-manifold.

Power Crust

In Chapter 4, we have seen that the poles for a dense point sample lie quite far away from all samples (proof of the Pole Lemma 4.1) and hence from the surface. Indeed, they lie close to the medial axis.