Topologically Correct Surface Reconstruction Using Alpha Shapes and Relations to Ball-Pivoting Peer Stelldinger University of Hamburg, Germany
[email protected]
Abstract The problem to reconstruct a surface given a finite set of boundary points is of growing interest, e.g. in the context of laser range images. While a lot of heuristic methods have been published in this context (e.g. the ball-pivoting algorithm), there exist only a few algorithms which guarantee the reconstruction to be homeomorphic to the original surface if a certain sampling density is reached. However, the sampling density mentioned is in most cases much higher than what seems to be sufficient on real data. In this paper we show how recently proved results about homology extraction from surface samples can be adopted to surface reconstruction and we significantly improve the bounds on the sampling density in case of noise-free samplings. This allows us to prove for the first time that the ball-pivoting algorithm reconstructs certain object surfaces without any topological changes and we can give bounds on the reconstruction error regarding both position and normal direction of the boundary.
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of sampling points is far away to be practically useful, since one only needs 4 sampling points to construct a tetrahedron whose surface is of the same topology as a sphere. By using a global criterion for the sampling density, Niyogi, Smale and Weinberger [9] recently derived much better bounds for reconstructing the topology of the boundary from its sampling points. They only need 22 sampling points on a sphere in the noisefree case and 168 points in case of noisy samplings. Unfortunately they do not give an algorithm for reconstructing a thin boundary but only consider a thick representation as union of spheres. Bernardini and Bajaj even conjecture, that any point on a unit sphere only needs to have a distance of less than 1 to the nearest sampling point [3], which requires at least 6 sampling points. We will show that their conjectured bound is true although one has to use a different reconstruction method. Finally there exist several surface reconstruction algorithms for which no proof of correct behavior is known. One of the most cited methods is the ballpivoting algorithm, which is extremely fast, can be applied to huge datasets and which behaves good in practical applications [4].
Introduction
The process of surface reconstruction from a finite set of sampling points on the boundary of an object is of increasing interest with the growing use of laser range scanners and the ability of modern computers to deal with huge 3D data sets. A number of algorithms gives theoretical guarantees for the topological correctness if the sampling density is proportional to the so-called local feature size lf s(x), which is the shortest distance of surface a point to the medial axis. Two well known examples are the crust algorithm [1], the co-cone algorithm [2]. They are proven to reconstruct topologically correct surfaces if the sampling density is at least 0.1lf s(x) [1], resp. 0.06lf s(x) [2]. One can show, that this requires at least 484, resp. 1343 sampling points on a sphere, see [7]. Thus the required number
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Preliminaries
We consider the task of reconstructing a 3D object from a sampled representation of its boundary. The boundary sampling is defined as follows: Definition 1 A finite set of sampling points S = {si ∈ R3 } is called a (p, q)-sampling of ∂A (with ∂X being the boundary of a set X) when the distance of every boundary point a ∈ ∂A to the nearest point in S is at most p, and the distance of sampling point point s ∈ S to the nearest boundary point in a ∈ ∂A is at most q. ♦ In [3] it is conjectured for the 3D case (and proved for the 2D case) that the boundary of an object can be reconstructed without any topological changes, if the ob-
ject and the boundary sampling points satisfy the following sampling requirements (see conjecture 4.1 in [3]): Conjecture 2 (Bernardini and Bajaj) Let ∂A ⊂ R3 be a compact 2-manifold without boundary, and S ⊂ ∂A a finite point set. If (1) for any closed ball Bα ⊂ R3 of radius α, ∂A ∩ Bα is either (a) empty, (b) a single point p (then p ∈ ∂Bα ) or (c) homeomorphic to a closed disc D, such that Bα0 ∩∂A = D0 (with X 0 denoting the opening of a set X), and if (2) an open ball of radius α centered on ∂A contains at least one point of S, then the α-shape |Dα | is homeomorphic to ∂A. This conjecture uses the concept of α-shapes for reconstruction. This well-known concept is also based on the Delaunay complex and has been introduced by Edelsbrunner in [5]: Definition 3 The α-complex Dα of a set of points S is defined as the subcomplex of the Delaunay complex D of S which contains all cells c such that the radius of the smallest open circumsphere of c is smaller than α, and the corresponding open ball contains no point of S, or an incident cell c0 with higher dimension is in Dα . The polytope |Dα |, i.e. the union of all elements of Dα , is called α-shape. ♦ In [10] it is shown that r-regular sets can be characterized by the property that the intersection of an r-regular set A with a ball B of radius smaller than r is always topologically simple. Thus part 1 of the conjecture simply means that A is r-regular with r > α. Part 2 says that the distance from any boundary point to the nearest sampling point is at most α. It follows that the conjecture can be rewritten as: Let A ⊂ R3 be an r-regular set and S be an (α, 0)sampling of ∂A with α < r. Then the α-shape |Dα | is homeomorphic to ∂A. Unfortunately this conjecture is not true, as can be seen in the following way: Given an r-regular object A being different from a ball. Then there exists a boundary point s ⊂ ∂A with two different principal curvatures on the surface (e.g. a concave part of the surface). Then we can place four non-coplanar sampling points s1 , s2 , s3 , s4 on ∂A all with the same distance smaller than α to s (see Figure 1(a)). By adding further sampling points on the boundary we can construct an (α, 0)sampling with s1 , s2 , s3 , s4 spanning a Delaunay tetrahedron which is in the α-complex (see Figure 1(b)). Thus the α-complex is not everywhere thin and cannot be homeomorphic to the thin boundary ∂A, see Figure 1.
(a)
(b)
(c)
Figure 1. (a) The shown tetrahedron must be part of the α-shape. (b) by adding more points one gets an α-sampling of the surface. (c) The outer boundary of the αshape is a correct surface reconstruction.
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Topologically Correct Surface Reconstruction
Although the conjecture given above is not true, the method used is quite promising. First, α-shapes are a piecewise linear interpolant (which may locally be thick) of the surface, that converges to the surface with increasing sampling density. Second, α-shapes are easy to compute without knowledge of the exact surface position. And finally, although the reconstruction is not guaranteed to be homeomorphic to the original surface due to thick parts, experiments show, that it seems to separate different regions correctly, e.g. see [4]. In the following we will prove this fact and show how one can derive a homeomorphic reconstruction based on αshapes. Therefore we need to have a closer look at the properties of α-shapes in general and α-shapes on rregular sets in special. One of the most important properties of α-shapes is given in [6]: Theorem 4 (Edelsbrunner) The union of closed balls of radius α centered at the points si ∈ S covers the polytope |Dα |, and the two sets are of the same homotopy type. Consequently, the polytope |Dα | is of the same homotopy type as the boundary ∂A of an r-regular set, if and only if the dilation of the sampling points with α-balls is of the same homotopy type as the boundary of the partition. Comparing the homotopy types has the great advantage that the thin boundary can have the same homotopy type as a partially thick reconstruction. Now we only have to combine Theorem 4 with the results given in [9]. There is was shown that the union of α-balls is of the same homology as the original surface it is sampled by a (p, q)-sampling with √ if √ p, q < 9 − 8 ≈ 0.17157 and if α = p+1 2 in case of noisy samplig points. In case of a noise-free (p, 0)-
sampling they prove that for any p < 0.48 the homology is correct. By using Theorem 4 it now directly follows that these bounds also guaranty the α-shape to be topologically equivalent to the original surface. In the following we will show that in the noise-free case these bounds can further be improved. In [8] it is shown that an r-regular set A is homeomorphic to both its α-erosion and α-dilation for any α < r. Further it is shown that any two normals of ∂A do not intersect inside ∂A ⊕ Bα . This means that the function which maps a point of ∂(A Bα ) to the point of ∂(A ⊕ Bα ) which lies on the same normal, is a well-defined homeomorphism. Note, that the normals connect any point with distance to the boundary of at most α with its unique nearest boundary point. One can even show that all triangle faces of Dα approximate the surface tangents. The smaller we choose α relatively to r, the better is the approximation of the tangents: Lemma 5 Let A be an r-regular set and S be an (α, 0)sampling of its boundary ∂A such that α ≤ r. Further let T be a face of Dα and let T 0 be the set containing every point in ∂A which is the nearest boundary point to one point in T (i.e. T 0 is the projection of T along the normals onto ∂A). Then the angle between the normal of T and√any surface normal at a point in T 0 is at most r2 − α2 /r . E.g. for 2α ≤ r the angle is at cos−1 most π6 . Proof: Any point in T 0 is a boundary point whose normal intersects the triangle T in a point with distance of at most α to ∂A. Now suppose there exists a point t0 ∈ T 0 , such that the angle to the surface normal of T is greater than π6 . Let t ∈ T be the corresponding triangle point which lies on the normal of ∂A in t0 . Further let P be the plane which contains T . Due to r-regularity of A there exists an inside and an outside tangent r-ball in t. Now t lies either in the outside tangent ball or in the inside tangent ball or it lies on the boundary of both balls. In the last case t is equal to t0 . In the first two cases let B be the ball containing t. B intersects P in a circle C1 being inside the circumcircle C2 of the triangle. The radius r1 of C1 depends on the angle γ between the normal of T and the surface normal in t0 and on the position of t on the surface normal, which is uniquely determined by 0 . One can show that the distancedpbetween t and t −1 2 2 r − r1 /(r − d) . For fixed r, r1 , this γ = cos p is maximal at d = 0, i.e. γ ≤ cos−1 r2 − r12 /r . Since every face of an α-shape has a circumradius√r1 of at most α, it follows for 2α ≤ r that γ ≤ cos−1 23 = π 6.
This implies that for any α < r the α-shape contains no triangle face which is collinear to a touching normal. With increasing sampling density the error of the reconstructed normal approaches zero. In computer graphic applications this can be used to compute a maximal approximation error of the phong shading approach. We can also show that any edge of the α-shape has at least two neighboring faces in the α-shape: Lemma 6 Let A be an r-regular set and S be an (α, 0)sampling of its boundary ∂A such that α < r. Further let e be an edge of Dα . Then there exist at least two triangular faces T1 , T2 ∈ Dα with e ⊂ T1 and e ⊂ T2 . Proof: Let C be the circle of all points having a distance of α to both endpoints of e. Due to r-regularity is C separated by ∂A into two parts, one lying inside and one lying outside A. If there would be at most one face in Dα touching e, one could pivot an α-ball with center in C along the circle from the outside to the inside of A without touching any further sampling point, which is in contradiction to the (α, 0)-sampling. This implies that also no edge can be collinear with a touching normal of ∂A, since otherwise there would exist a triangle being also collinear. Moreover this lemma shows the relation to the wellknown ball-pivoting algorithm [4]. The ball-pivoting algorithm starts with a triangle lying on the boundary of the α-shape, finds an α-ball which touches all three triangle points without containing any other sampling point and adds iteratively new triangles by pivoting the ball around a triangle edge until it touches a new sampling point. Thus the ball propagates along the whole surface component. Since an α-ball cannot pass the surface ∂A without touching a sampling point, the pivoting ball always stays outside of the object if it started outside of it. Thus it scans iteratively the outer boundary of |Dα |, e.g. see Fig. 2. Formally the inner and the outer boundary are defined as follows: Definition 7 Let A be an r-regular set and S ⊂ ∂A be a finite set of sampling points. Further let Dα be the α-complex of S. Then any triangle of Dα which is face of a Delaunay tetrahedron with circumradius of at least α (e.g. any triangle of the complex which lies in ∂|Dα |) is an outer triangle of |Dα | if the center point of the circumsphere lies outside of A and it is an inner triangle of |Dα | if the center point of the circumsphere lies inside of A. The outer (inner) boundary is the union of all outer (inner) triangles, see Figure 1. ♦ Lemma 6 simply states that this pivoting can always be done without penetrating the surface and that a partially traced surface can always be further traced until one gets a closed surface reconstruction. Thus it proves
of noisy samplings or extremely sparse noise-free samplings.
References [1] N. Amenta, M. Bern, M. Kamvysselis: A new Voronoi-based Surface Reconstruction Algorithm, Proc. Conference on Computer Graphics and Interactive Techniques, pp. 415–421, 1998. Figure 2. Outer boundary of the Stanford bunny dataset.
the correctness of the ball-pivoting algorithm in case of an (α, 0)-sampling of an r-regular set with α < r. This implies that in case of a sphere we only need 6 sampling points in order to be able to reconstruct the surface topologically correctly. Theorem 8 Let A be an r-regular set and S be an (α, 0)-sampling of its boundary ∂A such that α < r. Then the outer and the inner boundary of |Dα | are homeomorphic to ∂A. Proof: Follows directly from Lemma 6. Note, that in case of α < r the polytope |Dα | must not be homotopy equivalent to ∂A. It could be that it contains cavities (this is not possible in case of α < 0.48r as shown in [9]). But since these can not be found by the ball-pivoting algorithm, such cavities can not cause any problems. The ball pivoting algorithm reconstructs the outer boundary of |Dα |, which we proved to be homeomorphic to the original surface. Thus for the first time we are able to explain the good behavior of the algorithm in practice and we can predict when it will produce a good reconstruction and under which circumstances this is not the case.
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Conclusions
This paper addresses the well-known problem of correct surface reconstruction from unorganized point clouds. Previously known solutions guarantee topological correctness only for unnecessarily dense surface samplings or are not proved to be correct at all. Other results give good bounds but no reconstruction algorithm. We showed, that the well-known ball-pivoting algorithm, whose theoretical behavior up to now had not been proved to be correct, can be used as such a reconstruction algorithm and that it guarantees to reconstruct the topology of the surface correctly, even in case
[2] N. Amenta,S. Choi, T.K. Dey, N. Leekha: A Simple Algorithm for Homeomorphic Surface Reconstruction, Proc. Symposium on Computational Geometry, pp. 213–222, 2000. [3] F. Bernardini, C.L. Bajaj: Sampling and Reconstructing Manifolds Using Alpha-Shapes,Proc. Canadian Conference on Computational Geometry, pp. 193–198, 1997. [4] F. Bernardini, J. Mittleman, H. Rushmeier, C. Silva, G. Taubin: The Ball-Pivoting Algorithm for Surface Reconstruction, IEEE Transactions on Visualization and Computer Graphics, Vol. 5, No. 4, pp. 349–359, 1999. [5] H. Edelsbrunner, E.P. M¨ucke: Three-dimensional alpha shapes, ACM Transactions on Graphics, Vol. 13, pp. 43–72, 1994. [6] H. Edelsbrunner: The union of balls and its dual shape, Discrete Computational Geometry, Vol. 13, pp. 415–440, 1995. ¨ [7] L. Fejes: Uber einige Extremaleigenschaften der regul¨aren Polyeder und des gleichseitigen Dreiecksgitters, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze S´er. 2, Vol. 13, No. 1-4, 1948. [8] L.J. Latecki, C. Conrad, A. Gross: Preserving Topology by a Digitization Process, Journal of Mathematical Imaging and Vision, Vol. 8, pp. 131–159, 1998. [9] P. Niyogi, S. Smale, S. Weinberger: Finding the Homology of Submanifolds with High Confidence from Random Samples, Discrete and Computational Geometry, Vol. 39, No. 1, 2008. [10] P. Stelldinger, L.J. Latecki, M. Siqueira: Topological Equivalence between a 3D Object and the Reconstruction of its Digital Image, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 29, No. 1, 2007.
