268 F. Cazals, J. Giesen  Guarantees. No reconstruction guarantees are given.  Complexity. The running time of the Convection algorithm is dominated by the time needed to compute the Delaunay triangulation D(P ), i.e., it is Θ(n 2 ) where n is the size of P .  Extensions. One modification of the Convection algorithm is to keep an oriented facet f in S  if the same facet with the inverse orientation is also in S  . In doing so the convection algorithm can also reconstruct surfaces with boundaries. Sometimes the Convection algorithm stops too early, i.e., one would like to push the evolving surface even further. A heuristic to do so is provided. Another extension, geared towards large datasets, is presented in [18]. Borrowing the coarse-to-fine strategy from [57], the method first extracts a triangulated surface corresponding to a subset of the point cloud. This sur- face can be further refined by locally updating the Delaunay triangulation, and updating the reconstruction accordingly —a local process which does not require running the convection algorithm from scratch.  Comments. The Convection algorithm is dual to the Wrap algorithm (and the Flow complex) in the sense that the direction of “flow” is reversed. The Wrap algorithm retains the part of the Delaunay triangulation that does not “flow” to infinity whereas the Convection algorithm lets the convex hull of P “flow” towards the shape. 6.3.4 Empty Balls Methods A triangle reported in a reconstructed surface should be local in some sense. One way to specify locality is to use the empty ball property. Ball Pivoting Algorithm. Bernardini et al. [52]designed the ball pivoting algorithm to compute a surface subset of an α-shape of a sampling P in linear time and space.  Bottom-line and algorithm. Like in the definition of α-shapes a triangle pqr with vertices p, q, r ∈ P forms a triangle in the reconstruction if there is a ball of radius α that contains p, q and r in its boundary and no point from P in its interior. Starting with a seed triangle in the α-shape the ball pivoting algorithm pivots a ball around an edge of this seed triangle, i.e., it revolves around an edge while keeping the edge’s endpoints on its boundary, until it touches another point from P , forming another triangle in the α-shape. This process continues until all reachable edges have been processed.  Guarantees. No guarantees are given.  Complexity. Time and space complexity of the ball pivoting algorithm are linear under some assumptions, i.e., it is asymptotically faster than computing the Delaunay triangulation D(P)ofP . 6 Delaunay Triangulation Based Surface Reconstruction 269  Extensions. If not all points of P have been processed by the algorithm then one can restart it with a new seed triangle until all points in P have been considered. To accommodate non-uniform sampling the pivoting process can be re- peated with a larger value for α. Conformal α-shapes. Conformal α-shapes were introduced in [81] to circumvent the uniformity limitations inherent to α-shapes.  Bottom-line. In the context of surface reconstruction, the size of the small- est empty ball associated with a simplex does not have an absolute meaning: a ball of a given radius may be associated to neighbors on the surface at one location, but may connect points across the surface elsewhere. To get around this difficulty, conformal α-shapes re-scale the size of balls by taking into account the information provided by the poles.  Algorithm. Consider the Delaunay triangulation of the sample points, and the associated α-complex. The α values associated to the simplices incident to a sample p span the range [0,α + p ], with α + p the distance from p to its pole. In contrast to α-shapes, where α spans the real numbers, consider now a parameter ˆα ∈ [0, 1]. Placing around each sample point p a ball of radius α p (ˆα)=α + p ˆα, denote by C ˆα p the intersection of that ball with the Voronoi cell of p.Theconformal alpha shape is the Delaunay triangulation of the sample points P restricted to ∪ p∈P C ˆα p . Notice that the radii of the balls get scaled by a factor equal to the distance to the poles instead of using the same radius α for all balls.  Guarantees. For an ε-sample of a surface, it can be shown that the con- formal alpha shape contains the restricted Delaunay triangulation as soon as ˆα ≥ η, with η = ε/(1 − ε). It can also be shown that the conformal alpha shape does not contain large simplices for small values of ˆα. Note that such guarantees cannot be provided for ordinary α-shapes and are not known to hold for any method based on weighted α-shapes (an extension of α-shapes to weighted points, especially balls).  Complexity. Commuting the conformal α-shapes requires to compute the moment of appearance of the simplices as a function of ˆα. This is straightfor- ward from the α-values. Thus, the time complexity is the same as for ordinary α-shapes. Regular Interpolant. The regular interpolant was introduced in [281] by Petitjean and Boyer. Their work stresses the importance of Gabriel triangles for surface reconstruction, an observation also raised in [34]. 270 F. Cazals, J. Giesen  Bottom-line. The framework of ε-samples might not be the definitive set- up for solving practical problems. To bypass this difficulty, Petitjean and Boyer address the issue of finding an interpolant encoding the properties of the sampling P rather than those of an hypothetical smooth surface S.Tosee how, we first introduce the relevant notions. An interpolant O in R 3 is a 2-simplicial complex having P as vertex set. The interpolant is closed if each simplex bounds two distinct connected com- ponents of the ambient space. Given a sample point p ∈ P , its granularity g(p) is defined as the radius of the largest ball circumscribing a triangle incident to p. Now, given an interpolant, its associated discrete medial axis is the Voronoi diagram from which one removes the Voronoi cells dual to simplices of the interpolant. Notice that the process leaves Voronoi cells of dimensions from two to zero, and in particular all the Voronoi vertices. The discrete local feature size or local thickness t(p) at a sample point p is its least distance to the discrete medial axis with the convention t(p)=0 if p is on the boundary of a connected component of R 3 \O, which does not contain any piece of the discrete medial axis. Equipped with these notions, an interpolant is called regular is g(p) <t(p) for all sample points p. Getting back to the point cloud, P is said to be regular if it admits at least one regular interpolant. These notions are depicted in Fig. 6.25. Regular interpolants do not exist in general due to the presence of slivers, see Fig. 6.3. When such a tetrahedron is located near its equatorial plane, the granularity is indeed larger than the distance to the discrete medial axis, which contains at least the circumcenter of the tetrahedron.  Algorithm. For a regular interpolant, the triangles contributing to the interpolant are the Gabriel triangles minimizing the granularity at the vertices. They can be retrieved in an incremental fashion.  Guarantees. No guarantees are given.  Complexity. The Gabriel property must be checked for triangles incident to an edge. Using the Delaunay triangulation this can be done in time O(n 2 ).  Extensions. For non-regular point-sets, triangles are first selected so as to minimize the granularity, and are further decimated if they are not Gabriel. The interpolant built in this way is called a minimal interpolant.Itisnot manifold in general. A manifold extraction step can be applied, which consists of reporting groups of simplices that are simply connected, i.e., contractible to a point. 6 Delaunay Triangulation Based Surface Reconstruction 271 Fig. 6.25. A discrete version of the medial axis. Solid segments: interpolant; dotted segments: belonging to Delaunay but not the interpolant. The medial axis consists of the Voronoi segments dual to Delaunay edges, which do not contribute to the interpolant. The Voronoi edges not belonging to the medial axis are dotted too 6.4 Evaluating Surface Reconstruction Algorithms Evaluating surface reconstruction algorithms is a difficult task. Some of the algorithms presented in this chapter come with theoretical guarantees under certain conditions. But if these conditions are not met, their behavior is not specified. Thus it is an interesting question how reconstruction algorithms per- form on “real data”. In order to assess the performance of different algorithms on real data two surface reconstruction challenges have been organized. One challenge was organized within the Effective Computational Geometry project, a project funded by the European Union. The other challenge was organized within a DIMACS Workshop. For both challenges several data sets featuring the following difficulties were selected: undersampling, sharp features, thin parts, boundaries, high genus, noise. The reader is referred to www-sop.inria.fr/prisme/manifestations/ ECG02/SurfReconsTestbed.html and www.cse.ohio-state.edu/ dimacs-sr-challenge where the data sets used in the challenges are available. Some of these models are presented on Fig. 6.28, 6.29, 6.30, 6.27. 272 F. Cazals, J. Giesen 6.5 Software In this section, we provide information on the availability of implementa- tions of the different algorithms, and on the projects they have been used for. Whenever the information has been provided by the authors, we indicate so. Greedy [94]. Information provided by D. Cohen-Steiner. Algorithm Greedy has been mar- keted by the Geometry Factory, the company selling the Computational Geometry Algorithms Library library, and is also available through the web site cgal.inria.fr/Reconstruction. Cocone and variants [25, 123, 124]. Information provided by T. Dey. The suite of Cocone algorithms is available from www.cse.ohio-state.edu/ ∼ tamaldey/cocone.html. Depending on the constraints, users can choose from Cocone which reconstructs with boundaries, tight tcocone which returns a water-tight reconstruction, robust cocone which handles noise. Current implementations are based upon version 2.3 of the Computational Geometry Algorithms Library, www.cgal.org. Power crust algorithm [27, 26]. Information provided by N. Amenta. The power crust software was released in 2002 at www.cs.utexas.edu/users/amenta/powercrust/welcome.html. The software was ported into the Visual Toolkit VTK by Tim Hutton —see www.sq3.org.uk/powercrust. Unfortunately since powercrust was released under the GPL licence, it cannot be officially included in the VTK distribu- tion. Natural Neighbors [56, 58]. Information provided by F. Cazals and A. Lieutier. The surface reconstruction algorithm based on Natural Neighbors was purchased by Dassault Syt`emes, the editor of CAGD system CATIA, and has been integrated into the Digital Shape Editor of CATIA V5R6 since spring 2001. Wrap [132]. As pointed out in [132], Algorithm Wrap has been implemented in 1996 at Raindrop Geomagic, and successfully commercialized as geomagic Wrap.Itis also protected by the U.S patent No. 6,3777,865. 6 Delaunay Triangulation Based Surface Reconstruction 273 Ball pivoting algorithm [52]. Information provided by F. Bernardini. The algorithm is patented, US6968299: Method and apparatus for reconstructing a surface using a ball- pivoting algorithm. The code is copyright of IBM and not commercially available. The Ball Pivoting Algorithm has been used in two projects spon- sored by IBM Corporate Community Relations: Michelangelo’s Florence Pieta www.research.ibm.com/pieta, Eternal Egypt www.eternalegypt.org.The algorithm is part of a scanning system that IBM has made available to the Egyptian Center for Documentation of Cultural and Natural History (CULT- NAT) 6.6 Research Problems Exercise 5 (Independence from the Delaunay triangulation). It has been shown in [33] that the complexity of the Delaunay triangulation for rea- sonable point sets sampled from a smooth generic surface is O(n log n), which is better than the Θ(n 2 ) worst-case bound on the complexity of the Delaunay triangulation. Therefore, one challenge is to design a surface reconstruction algorithm whose running time is always independent of the size of the De- launay triangulation D(P ). Even better, the running time could be output sensitive in the size of the reconstructed surface. An example such an algo- rithm running in time O(n log n) is the modification of the Cocone algorithm by Funke and Ramos [174]. Exercise 6 (Boundaries). Define a meaningful sampling theory for a smooth surface with boundaries. Design an algorithm that comes with guar- antees in terms of your sampling theory not only for the sampled surface, but also for its bounding curves. Exercise 7 (ε-samples). The major drawback of the ε-sample framework is that the sufficient conditions of algorithms developed under its auspices cannot be checked as a pre-condition. Propose a more constructive framework. 274 F. Cazals, J. Giesen Fig. 6.26. The evolution and progress in the Cocone family of algorithms, ilustrated on the Stanford bunny, 36k points. From left to right: Cocone, Tight Cocone and Robust Cocone. Triangles featuring non manifold edges and vertices are colored Fig. 6.27. Pump, 47k points. Reconstructed with [94] 6 Delaunay Triangulation Based Surface Reconstruction 275 Fig. 6.28. Mechanical part, 12k points. Reconstructed with [56] Fig. 6.29. Vase, 2.7k points. Reconstructed with [56] 276 F. Cazals, J. Giesen Fig. 6.30. Plane engine, 11k points. Reconstructed with [82] 7 Computational Topology: An Introduction G¨unter Rote and Gert Vegter  7.1 Introduction Topology studies point sets and their invariants under continuous deforma- tions, invariants such as the number of connected components, holes, tunnels, or cavities. Metric properties such as the position of a point, the distance between points, or the curvature of a surface, are irrelevant to topology. Com- putational topology deals with the complexity of topological problems, and with the design of efficient algorithms for their solution, in case these prob- lems are tractable. These algorithms can deal only with spaces and maps that have a finite representation. To this end we restrict ourselves to simplicial complexes and maps. In particular we study algebraic invariants of topologi- cal spaces like Euler characteristics and Betti numbers, which are in general easier to compute than topological invariants. Many computational problems in topology are algorithmically undecid- able. The mathematical literature of the 20th century contains many (beauti- ful) topological algorithms, usually reducing to decision procedures, in many cases with exponential-time complexity. The quest for efficient algorithms for topological problems has started rather recently. The overviews by Dey, Edels- brunner and Guha [119], Edelsbrunner [133], Vegter [329], and the book by Zomorodian [351] provide further background on this fascinating area. This chapter provides a tutorial introduction to computational aspects of algebraic topology. It introduces the language of combinatorial topology, rele- vant for a rigorous mathematical description of geometric objects like meshes, arrangements and subdivisions appearing in other chapters of this book, and in the computational geometry literature in general. Computational methods are emphasized, so the main topological objects are simplicial complexes, combinatorial surfaces and submanifolds of some Euclidean space. These objects are introduced in Sect. 7.2. Here we also in- troduce the notions of homotopy and isotopy, which also feature in other  Chapter coordinator [...]... simplicial k-cycles The vector space Bk (K, Q) = im ∂k+1 is called vector space of sim- 7 Computational Topology: An Introduction 285 plicial k-boundaries Since the boundary of a boundary is 0, Bk (K, Q) is a subspace of Zk (K, Q) The quotient vector space Hk (K, Q) = Zk (K, Q)/Bk (K, Q) is the k-th homology vector space of K In particular, two k-cycles α and β are k-homologous if their difference is a k-boundary,... has dimension 5 Therefore, β1 (K, Q) = dim ker ∂1 = dim C1 (K, Q) − dim im ∂1 = 0 3 Betti numbers of the 2-sphere The simplicial complex K of Fig 7.7 is the boundary of a 3-simplex, consisting of four 2-simplices, six 1-simplices and four 7 Computational Topology: An Introduction 287 0-simplices For convenience it is shown flattened on the plane, after cutting the edges incident to 0-simplex v4 The underlying... Zero-homology of a connected simplicial complex Consider the connected simplicial complex K of Fig 7.5 The 0-chains α = v6 and β = v2 are 0-homologous since their difference is the boundary of the 1-chain γ = − v1 v2 + v1 v4 + v4 v6 , since ∂1 γ = −( v2 − v1 ) + ( v4 − v1 ) + ( v6 − v4 ) = α − β In the same way one shows that every 0-chain of v5 v2 v1 v3 v4 v6 Fig 7.5 Zero-homology of a graph the form... is homeomorphic to the unit d-sphere (Consider a d-simplex in Rd+1 The projection of its boundary from a fixed point in its interior onto its circumscribed d-sphere is a homeomorphism See Fig 7.2 The circumscribed d-sphere is homeomorphic to the unit d-sphere.) 280 G Rote, G Vegter p p Fig 7.2 The point p on the boundary of a 3-simplex is mapped onto the point p on the 2-sphere This mapping defines a... rational numbers), the set of all simplicial k-chains forms a vector space Ck (K, Q), called the vector space of simplicial k-chains of K The dimension of this vector space is equal to the number of k-simplices of K Therefore, the Euler characteristic of a ddimensional simplicial complex K can be expressed as an alternating sum of dimensions of the spaces of k-chains: d (−1)i dim Ck (K, Q) χ(K) = (7.1)... interior of a set X is the set of all points having an ε-neighborhood contained in X, for some ε > 0 The closure of a subset X of Rd is the set of points x in Rd every ε-neigborhood of which has non-empty intersection with X The boundary of a subset X is the set of points in the closure of X that are not interior points of X In particular, every ε-neighborhood of a point in the boundary of X has nonempty... number of i-simplices of K Examples of simplicial complexes are: 1 A graph is a 1-dimensional simplicial complex (think of a graph as being embedded in R3 ) The complete graph with n vertices is the 1-skeleton of an (n−1)-simplex 2 The Delaunay triangulation of a set of points in general position in Rd is a simplicial complex Combinatorial surfaces A Combinatorial closed surface is a finite two-dimensional... simplicial k-chain is a formal sum of the form j aj σj over the oriented k-simplices σj in K, with coefficients aj in the field Q of rational numbers In other words, it can be regarded as a rational vector whose entries are indexed by the oriented 284 G Rote, G Vegter k-simplices of K Furthermore, by definition, −σ = (−1)σ is the simplex obtained from σ by reversing its orientation With the obvious definition for. .. are a plane with a point removed, or a (solid or hollow) torus For example, on the hollow torus in Fig 7.3, the closed curve in the figure is not homotopic to its inverse If we require that γ(·, t) is a homeomorphism at all times during the deformation we arrive the stronger concept of isotopy For example, the smooth closed curves without self-intersections in the plane fall into two isotopy classes, according... In this chapter a topological space X (or space, for short) is a subset of some Euclidean space Rd , endowed with the induced topology of Rd In particular, an ε-neighborhood (ε > 0) of a point x in X is the set of all points in X within Euclidean distance ε from x A subset O of X is open if every point of O contains an ε-neighborhood contained in O, for some ε > 0 A subset of X is closed if its complement . accommodate non-uniform sampling the pivoting process can be re- peated with a larger value for α. Conformal α-shapes. Conformal α-shapes were introduced in [81] to circumvent the uniformity limitations. indicate so. Greedy [94]. Information provided by D. Cohen-Steiner. Algorithm Greedy has been mar- keted by the Geometry Factory, the company selling the Computational Geometry Algorithms Library. that the conformal alpha shape does not contain large simplices for small values of ˆα. Note that such guarantees cannot be provided for ordinary α-shapes and are not known to hold for any method