The running time of the Convection algorithm is dominated by the time needed to compute the Delaunay triangulation DP , i.e., it is Θn2 where n is the size of P.. [52]designed the ball p
Trang 1268 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(n2) 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 withboundaries
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 face can be further refined by locally updating the Delaunay triangulation,and updating the reconstruction accordingly —a local process which does notrequire running the convection algorithm from scratch
sur-" Comments The Convection algorithm is dual to the Wrap algorithm (and
the Flow complex) in the sense that the direction of “flow” is reversed TheWrap 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 revolvesaround 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 ) of P
Trang 26 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 atone 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 the intersection of that ball with the Voronoi cell
of p The conformal alpha shape is the Delaunay triangulation of the sample points P restricted to ∪ p ∈P C p αˆ 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 asˆ
α ≥ 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 ˆ α 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 ward from the α-values Thus, the time complexity is the same as for ordinary α-shapes.
Trang 3270 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 andBoyer address the issue of finding an interpolant encoding the properties of
the sampling P rather than those of an hypothetical smooth surface S To see
how, we first introduce the relevant notions
An interpolant O inR3 is a 2-simplicial complex having P as vertex set.
The interpolant is closed if each simplex bounds two distinct connected ponents of the ambient space
com-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 theinterpolant Notice that the process leaves Voronoi cells of dimensions fromtwo 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 R3\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 inFig 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(n2)
" 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 It is not
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
Trang 46 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 theinterpolant 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 thealgorithms presented in this chapter come with theoretical guarantees undercertain conditions But if these conditions are not met, their behavior is notspecified 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 organizedwithin a DIMACS Workshop For both challenges several data sets featuringthe following difficulties were selected: undersampling, sharp features, thinparts, 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 areavailable Some of these models are presented on Fig 6.28, 6.29, 6.30, 6.27
Trang 5272 F Cazals, J Giesen
6.5 Software
In this section, we provide information on the availability of 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
implementa-Greedy [94].
Information provided by D Cohen-Steiner Algorithm Greedy has been
mar-keted by the Geometry Factory, the company selling the ComputationalGeometry Algorithms Library library, and is also available through the website 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 theconstraints, users can choose from Cocone which reconstructs with boundaries,tight tcocone which returns a water-tight reconstruction, robust cocone whichhandles 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 —seewww.sq3.org.uk/powercrust Unfortunately since powercrust was releasedunder 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 atRaindrop Geomagic, and successfully commercialized as geomagic Wrap It isalso protected by the U.S patent No 6,3777,865
Trang 66 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 pivoting algorithm The code is copyright of IBM and not commerciallyavailable The Ball Pivoting Algorithm has been used in two projects spon-sored by IBM Corporate Community Relations: Michelangelo’s Florence Pietawww.research.ibm.com/pieta, Eternal Egypt www.eternalegypt.org Thealgorithm is part of a scanning system that IBM has made available to theEgyptian Center for Documentation of Cultural and Natural History (CULT-NAT)
ball-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 sonable point sets sampled from a smooth generic surface is O(n log n), which
rea-is better than the Θ(n2) worst-case bound on the complexity of the Delaunaytriangulation Therefore, one challenge is to design a surface reconstructionalgorithm 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 algoalgo-rithm
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 antees in terms of your sampling theory not only for the sampled surface, butalso for its bounding curves
guar-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
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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 andRobust Cocone Triangles featuring non manifold edges and vertices are colored
Fig 6.27 Pump, 47k points Reconstructed with [94]
Trang 86 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]
Trang 9276 F Cazals, J Giesen
Fig 6.30 Plane engine, 11k points Reconstructed with [82]
Trang 10Many computational problems in topology are algorithmically able The mathematical literature of the 20th century contains many (beauti-ful) topological algorithms, usually reducing to decision procedures, in manycases with exponential-time complexity The quest for efficient algorithms fortopological problems has started rather recently The overviews by Dey, Edels-brunner and Guha [119], Edelsbrunner [133], Vegter [329], and the book byZomorodian [351] provide further background on this fascinating area.This chapter provides a tutorial introduction to computational aspects ofalgebraic 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
undecid-in the computational geometry literature undecid-in general
Computational methods are emphasized, so the main topological objectsare simplicial complexes, combinatorial surfaces and submanifolds of someEuclidean 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
Trang 11278 G Rote, G Vegter
parts of this book, like Chapter 5 Most of the computational techniques areintroduced in Sect 7.3 Topological invariants, like Betti numbers and Eulercharacteristic, are introduced and methods for computing such invariants arepresented Morse theory plays an important role in many recent advances incomputational geometry and topology See, e.g., Sect 5.5.2 This theory isintroduced in Sect 7.4
Given our focus on computational aspects, topological invariants like Bettinumbers are defined using simplicial homology, even though a more advancedstudy of deeper mathematical aspects of algebraic topology could better bebased on singular homology, introduced in most modern textbooks on alge-braic topology Other topological invariants, like homotopy groups, are harder
to compute in general; These are not discussed in this chapter
The chapter is far from a complete overview of computational algebraictopology, and it does not discuss recent advances in this field However, readingthis chapter paves the way for studying recent books and papers on compu-tational topology Topological algorithms are currently being used in appliedfields, like image processing and scattered data interpolation Most of theseapplications use some of the tools presented in this chapter
7.2 Simplicial complexes
Topological spaces.
In this chapter a topological space X (or space, for short) is a subset of some
Euclidean spaceRd, endowed with the induced topology ofRd 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 in X is open The 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 ofRd is the set of points x inRd 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 non- empty intersection with both X and the complement of X See [28, Sect 2.1]
for a more complete introduction of the basic concepts and properties of pointset topology
The spaceRd is called the ambient space of X Examples of topological
Trang 127 Computational Topology: An Introduction 279
4 The unit d-sphereSd ={(x1, , x d+1)∈ R d+1 | x2+· · · + x2
d+1 = 1} (the
boundary of the (d+1)-ball);
5 A d-simplex, i.e., the convex hull of d + 1 affinely independent points in
some Euclidean space (obviously, the dimension of the Euclidean space
cannot be smaller than d) The number d is called the dimension of the
simplex Fig 7.1 shows simplices of dimensions up to and including three
Fig 7.1 Simplices of dimension zero, one, two and three
Homeomorphisms.
A homeomorphism is a 1–1 map h : X → Y from a space X to a space Y with a continuous inverse (In this chapter a map is always continuous by definition.) In this case we say that X is homeomorphic to Y , or, simply, that
X and Y are homeomorphic.
1 The unit d-sphere is homeomorphic to the subset Σ of Rm defined by
Σ = {(x1, , x d+1 , 0, , 0) ∈ R m | x2+· · ·+x2
d+1= 1} (m > d) Indeed,
the map h :Sd → Σ, defined by h(x1, , x d+1 ) = (x1, , x d+1 , 0, , 0),
is a homeomorphism Loosely speaking, the ambient space does not matterfrom a topological point of view
2 The map h : Rk → R m , m > k, defined by
sim-5 The boundary of a d-simplex is homeomorphic to the unit d-sphere sider a d-simplex in Rd+1 The projection of its boundary from a fixed
(Con-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.)