Undersampling and Oversampling in Sample Based Shape Modeling Tamal K. Dey Joachim Giesen Samrat Goswami James Hudson Rephael Wenger Wulue Zhao Ohio State University Columbus, OH 43210 Abstract early paper on the problem was by Boissonat [11] who pro- posed a ‘sculpting’ of the Delaunay triangulation for recon- Shape modeling is an integral part of many visualization struction. A more refined sculpting strategy was designed problems. Recent advances in scanning technology and a by Edelsbrunner and Muck¨ e [16] in their -shape algorithm. number of surface reconstruction algorithms have opened up Bajaj, Bernardini and Xu [9] used -shapes for reconstruct- a new paradigm for modeling shapes from samples. Many of ing scalar fields and 3D CAD models. In [15] Edelsbrun- the problems currently faced in this modeling paradigm can ner reported the design of a commercial software WRAP be traced back to two anomalies in sampling, namely under- that eliminated the need for uniform samples in -shapes. sampling and oversampling. Boundaries, non-smoothness Hoppe et al. [24] reconstructed the surface using the zero and small features create undersampling problems, whereas level set of a distance function defined over the samples. oversampling leads to too many triangles. We use Voronoi Curless and Levoy [14] used a distance function to con- cell geometry as a unified guide to detect undersampling and struct an implicit surface from multiple range scans. Turk oversampling. We apply these detections in surface recon- and Levoy [31] devised an incremental algorithm that itera- struction and model simplification. Guarantees of the algo- tively improves a reconstruction by erosion and zippering. A rithms can be proved. In this paper we show the success of cluster based strategy was used by Heckel et al. for recon- the algorithms empirically on a number of interesting data structions in [22]. Gopi, Krishnan and Silva [21] projected sets. sample points with their neighbors on a plane and lifted the Keywords: Computational Geometry, Surface Reconstruction, local 2D Delaunay triangulations to reconstruct the surface. Geometric Modeling, Mesh Generation, Polygonal Mesh Reduc- Bernardini et al. [10] proposed a ball pivoting algorithm that tion, Polygonal Modeling, Shape Recognition. reconstructs the surface incrementally by rolling a ball over the sample points. Kobbelt and Botsch [25] used hardware projections to reconstruct surfaces from large data. Attali 1 Introduction [8] introduced normalized meshes to reconstruct surfaces. Very recently Amenta, Bern and Kamvysselis [2] proposed Visualizations of shapes with their models are integral part of a Voronoi based surface reconstruction called CRUST and many scientific computations. Surface reconstruction which proved its theoretical guarantees. This algorithm was later builds a piecewise linear approximation of a surface from its improved by the COCONE algorithm in [3] and the POWER samples provides a powerful paradigm for modeling shapes. CRUST in [4]. Boissonnat and Cazals [12] showed how nat- We call this paradigm Sample Based Shape Modeling, or ural neighbors can aid surface reconstructions. See the sur- SBSM in short. For many applications SBSM can provide veys by Mencl and Muller¨ [26] for other surface reconstruc- an initial mesh for the model which can be processed fur- tion algorithms and by Uselton [32] for contour based sur- ther according to the need. For example, the traditional spline face reconstructions. based surface modeling can benefit from SBSM in generat- ing an initial control mesh and then smoothing it out with 1.2 Undersampling and oversampling techniques such as surface fairing [29] or subdivision meth- ods [33]. Although a considerable success has been made by recent surface reconstruction algorithms, some of the major prob- lems in SBSM that still remain can be traced back to two 1.1 Background anomalies in sampling, namely undersampling and oversam- pling. In this paper we concentrate on detecting these two A vast literature has built up on the problem of surface re- anomalies using Voronoi structures and show how they can construction in recent years. Various algorithms with differ- aid surface reconstructions. ent capabilities and guarantees have been proposed. A very Undersampling happens when a surface has small features such as high curvatures that are sampled inadequately. It can- not be avoided when a surface is not smooth. In this case no finite sampling is dense enough for sharp edges or cor- ners. Even the presence of boundaries in the surface can be thought of as a consequence of undersampling. The sampling ficiently ‘fat’. This strategy decimates the samples up to a in this case is intentionally stopped to introduce the bound- level determined by an input parameter. aries. In all these cases, the reconstructed surface often has undesirable holes and triangles and may even not be a man- ifold with boundary. Detection of undersampling can help 2 Voronoi cell geometry mend these surfaces either by resampling manually, or by stitching the holes algorithmically. Other than this, it has di- The main tool we use to detect undersampling and oversam- rect application in reconstructing surfaces with boundaries pling is the geometry of the Voronoi cell structure. Our rea- and determining features such as discontinuities in scalar soning is based on a density condition called -sampling as fields [20]. We also show that nonsmooth surface reconstruc- introduced in [2]. This definition builds on the medial axis tion can also benefit from this detection. of a surface and the related term of local feature size. The 3 Not only undersampling poses challenge to SBSM, but medial axis of a surface S ¡£¢ is defined as the locus of also its counterpart, oversampling, causes difficulties, partic- all points that have more than one closest point on S. The ularly in post-processing. A surface, sometimes, is sampled local feature size at a point x ¤ S, denoted as f (x), is the 3 with unnecessarily high density. Surfaces reconstructed from least distance of x to the medial axis. Let P ¡¥¢ be a set of an unnecessarily dense sample contain large numbers of tri- sample points on a surface S. We say P is an -sample of S angles and thus become unwieldy for further processing such if each point x ¤ S has a sample within f (x) distance. It has as graphic rendering or finite element analysis. A variety of been observed that typically §¦ 0. 4 gives a dense enough algorithms have been proposed to simplify a piecewise lin- sampling for surface reconstruction. For a dense sample each ear surface [13, 17, 19, 23, 28, 30]. Most of these algorithms Voronoi cell is long and thin as shown in Figure 1. We in- choose a subset of vertices, edges, or triangles for deletion so troduce a measure of this structural property that is easily that the overall shape of the space delimited by the surface is computable. We need the following definitions for further maintained. The major concerns in such simplification strate- expositions. gies are to preserve topological and geometric features of the surface and avoid self intersections. Oversampling detection Definitions can directly benefit the simplification process by throwing away unnecessary sample points before reconstruction. Such Let VP and DP denote the Voronoi diagram and Delaunay tri- 3 sample decimations have been proposed earlier by [2] which angulation of the sample P ¡¨¢ respectively. For a sample we refine with new concepts and a guarantee in surface re- p ¤ P, let Vp be its Voronoi cell. For convenience we use the construction. This alternative approach of sample decimation notation © (v, w) to denote the acute angle between the lines in place of model simplification has several advantages. First supporting two vectors v and w. + of all, one does not need to worry about preserving features Poles: The farthest Voronoi vertex p in Vp is called the or avoiding self intersections. If the decimation guarantees a positive pole of p. The negative pole of p is the farthest point + sample density that is still appropriate for surface reconstruc- p ¤ Vp from p such that two vectors from p to p and p + tion, the reconstructed model retains all features without any make an angle more than 2 . We call vp = p p, the pole + self intersection. Furthermore, sample decimation at differ- vector for p. If Vp is unbounded, p is taken at infinity, and ent levels allows an alternative approach to multiresolution the direction of vp is taken as the average of all directions meshing [18, 27]. given by unbounded Voronoi edges. Cocone: The set Cp( , v) = y Vp : ((y p), v) is called the cocone of p with axis v and angle . In 1.3 Our approach 2 words, Cp( , v) is the complement of a double cone (clipped We use the structure of Voronoi diagrams as the key ingredi- within Vp) centered at p with an opening angle around 2 ent for undersampling and oversampling detection. A result the axis aligned with v. Usually, we take v = vp, the pole in [2] says that the Voronoi cells for a dense sample of a vector and small. See Figure 1 for an example of a 8 surface are long and thin along the direction of normals at cocone. the sample points. We use this structural information to de- Cocone neighbor: A sample q is called a cocone neighbor tect samples in the vicinity of undersampled regions, which of another sample p if Vq overlaps with the cocone of p. Thus, " we call boundary samples. The information about boundary the set of cocone neighbors denoted Np is given by Np = ! q # % & ')( samples can be exploited in many surface reconstruction al- P Vq $ Cp( , vp) = . gorithms simply by disallowing triangles that are incident to Let dp denote the maximum distance of any point from p the boundary samples. that has p as its nearest sample on S.