This distance function is smooth every-where besides at the points in P and on the lower dimensional Voronoi faces, i.e., on the facets, edges and vertices.. It was was observed by Edels
Trang 1b
c d
Induced Distance Function.
Voronoi diagrams of a sample P are closely related to the distance function
h :R3→ R, x → min
induced by the set of sample points This distance function is smooth
every-where besides at the points in P and on the lower dimensional Voronoi faces,
i.e., on the facets, edges and vertices
At every point x inside a Voronoi region, the gradient of h is the unit
vector pointed away from the center of the region Interestingly, for points
x on lower dimensional Voronoi faces, one can define a generalized gradient,
as depicted in Fig 6.10 Let x be a point and denote by C(x) the set of closest points in P to x, i.e., C(x) consists of the vertices dual to the (lowest dimensional) Voronoi face containing x If x does not belong to the convex hull of C(x), then the generalized gradient at x is the unit vector that points from the closest neighbor of x in the convex hull of C(x) to x Otherwise, i.e.,
x is contained int eh convex hull of C(x), the point x is called a critical point.
It was was observed by Edelsbrunner [132] and later proved by Giesenand John [184] that the critical points of the distance function, i.e., the localextrema and the saddle points, can be characterized in terms of Delaunaysimplices and Voronoi faces
Theorem 3. The critical points of h are the intersection points of Voronoi faces and their dual Delaunay simplices The local maxima are Voronoi ver- tices contained in their dual Delaunay cell The saddle points are intersection
Trang 2points of Voronoi facets and their dual Delaunay edges and intersection points
of Voronoi edges and their dual Delaunay triangles All sample points are ima.
min-The index of a critical point is the dimension of the Delaunay simplex
involved in its definition See Fig 6.2.1 for an example in two dimensions
Induced Flow and Stable Manifolds.
As in the case of smooth functions there is a unique direction of steepest
ascent of h at every non-critical point of h Assigning to the critical points
exists and is equal to the unique unit vector of steepest ascent at x The
flow basically tells how a point moved if it would always follow the steepest
ascent of the distance function h The curve that a point x follows is given
example orbits in two dimensions
Given a critical point x of h the set of all points whose orbit ends in x, i.e., the set of all points that flow into x, is called the stable manifold of x The collection of all stable manifolds forms a cell complex which is called flow
complex See Fig 6.2.1 for examples of stable manifolds in two dimensions.
6.2.2 Medial Axis and Derived Concepts
Medial Axis.
R3\S having two or more nearest points on S In a way the medial axis
generalizes the concept of the Voronoi diagram of a point set We have seenwhen discussing the empty ball property that the Voronoi faces of dimension
k with k = 0, , 2 consist of all points equidistant from 3 − k + 1 sample
points
Smooth surfaces S play a special role in reconstruction since for their
reconstruction several guarantees can be provided under a certain sampling
condition This sampling condition is based on the medial axis of S That is
the reason why we here provide some more details on the structure of the
medial axis of a smooth surface S.
Trang 3Fig 6.11 From the left: 1) The local minima, saddle points and local maxima
⊕ of the distance function induced by the sample points (local minima) 2) Some
orbits of the flow induced by the sample points 3) The stable manifolds of the saddlepoints 4) The stable manifolds of the local maxima
Structure of the Medial Axis of a Smooth Surface.
The medial axis of a smooth surface S shares another structural property with
the Voronoi diagram of a finite point set, namely, it has a stratified structure.For the Voronoi diagram this structure means that a Voronoi facet is thecommon intersection of two Voronoi regions, a Voronoi edge is the commonintersection of three Voronoi facets and a Voronoi vertex is the common inter-section of four Voronoi edges To precisely describe the stratified structure of
M (S) one needs the notion of contact between a sphere and the surface
Infor-mally, the contact of a sphere at a point p of S tells how much the sphere and
extreme along the corresponding line of curvature Focusing on the centers ofthe contact spheres rather than the contact points themselves, and denoting
the structure of the M (S) is described by the following theorem [347, 181]
which is illustrated by an example in Fig 6.12
Theorem 4 The medial axis of a smooth surface S inR3 is a stratified ety containing sheets, curves and points The sheets correspond to A2contacts,
Trang 4vari-the curves to A3 and A3 contacts, and the points to A4 and A3A1 contacts Moreover, one has the following incidences At an A4 point, six A1sheets and four A3 curves meet Along an A3 curve, three A2 sheets meet A3 curves bound A2 sheets At last, the point where an A2 sheet vanishes is an A3A1
point.
Fig 6.12 The stratified structure of the medial axis of a smooth surface
Medial Axis Transform.
skeleton of R3\S, which consists of the centers of maximal spheres included
in R3\S Here maximal is meant with respect to inclusion among spheres.
For a smooth surface S the closure of the medial axis is actually equal to the
of the skeleton and the medial axis, namely, the medial axis transform is the
collection of maximal empty balls centered at the medial axis of S It can be
Trang 5shown that a smooth surface S can be recovered as the envelope of its medial
axis transform
Tubular Neighborhoods.
A natural tool involved in the analysis of several reconstruction algorithms
is that of tubular neighborhood or tube of a surface S As indicated by the
name, a tube of a surface is a thickening of the surface such that within the
volume of the thickening, the projection of a point x to the nearest point
π(x) on S remains well defined Following our discussion of the medial axis,
a surface can always be thickened provided the thickening avoids the medial
axis Moreover, it is easily checked that the projection onto S proceeds along the normal at the projection point This property provides a way to retract
the neighborhood onto the surface
Local Feature Size.
to each point in S its least distance to the medial axis of S An immediate
consequence of the triangle inequality is that the local feature size of a smoothsurface is Lipschitz continuous with Lipschitz constant 1, see Fig 6.13 for anillustration The local feature size can be used to establish another quanti-tative connection of a surface and its medial axis [59] by using the followingtheorem
Theorem 5 Let B be a ball centered at x ∈ R3 with radius r that intersects the surface S If this intersection is not a topological ball then B contains a point of the medial axis of S.
From this theorem we can conclude that any ball centered at any point
p ∈ S whose radius is smaller then the local feature size lfs(p) at p intersects
S in a topological disk.
Fig 6.13 The local feature size is 1-Lipschitz
Trang 6Fig 6.14 For a non-smooth curve, some Voronoi centers may not converge to the
medial axis
ε-sample.
Amenta and Bern [23, 22] introduced a non-uniform measure of sampling
density using the local feature size For ε > 0 a sample P of a surface S is called an ε-sample of S if every point x on S has a point of P in distance at most ε lfs(x).
We next provide three theorems that involve ε-samples The first theorem
is concerned with the topological equivalence of the restricted Delaunay
is concerned with the convergence of Voronoi vertices of the Voronoi diagram
of an ε-sample of a smooth surface S towards the medial axis M (S) of S The last theorem provides a good approximation of the normal of S at some sample point in an ε-sample P
Amenta and Bern [22] stated the following theorem, which provides a
proven in [88] In the context of surface reconstruction, this theorem should
be put in perspective with respect to Theorem 1:
Theorem 6 If P is an ε-sample of S such that ε satisfies
cos
arcsin
then V S (P ) has the topological ball property.
It can be shown that the Voronoi vertices of a dense sample of a planarsmooth curve lie close to the medial axis of the curve This result is false ingeneral for non smooth curves, as illustrated in Fig 6.14 It is also false ingeneral for dense samples of smooth surfaces In fact for almost any point
x ∈ R3\S, there exists an arbitrarily dense sample P of S such that x is a
Voronoi vertex of V (P ) provided some non-degeneracy holds To see this grow
a ball around x until it touches S Now grow it a little bit further and put four sample points on the intersection of S with the boundary of the ball Then x
is shared by the Voronoi cells of the four points, i.e., it is a Voronoi vertex ifthe four points are in general position
Trang 7Fortunately, it was observed by Amenta and Bern [22] that the poles ofthe Voronoi diagram of a sample of a smooth surface converge to the medialaxis.
Theorem 7 Let P be an ε-sample of a smooth surface S The poles of the
Voronoi diagram V (P ) converge to the medial axis M (S) of S as ε goes to zero.
Fig 6.15 In 2D, all Voronoi vertices converge to the medial axis In 3D, some
Voronoi vertices may be far from the medial axis but poles are guaranteed to verge to the medial axis
con-Finally, also the following theorem is due to Amenta and Bern [22] Itfollows from Theorem 7
Theorem 8 Let P be an ε-sample of a smooth surface S For any sample
point p ∈ P let p+ be the positive pole of the Voronoi cell V p The angle between the normal of S at p and the vector p − p+ if oriented properly can
6.2.3 Topological and Geometric Equivalences
To assess the quality of a reconstruction we need topological and ric concepts Our presentation of these concepts is informal, and the reader
geomet-is referred to [208] for an exposition involving the apparatus of differentialtopology
Topological Concepts.
Homeomorphy Two surfaces are called homeomorphic if there is a
homeo-morphism between them A homeohomeo-morphism is a continuous bijection of one
Trang 8surface onto the other, such that the inverse is also continuous Two omorphic surfaces have the same properties regarding open and closed sets,and also neighborhoods Note that homeomorphy is an equivalence relation.
to homeomorphy by their genus, i.e., the number of holes For example thetorus of revolution, i.e., a doughnut, and a “knotted” torus are homeomorphicsince both have genus 1 This example shows that homeomorphy is a weak
account This is done by the concept of isotopy which for example accounts
for the knottedness of a torus
Isotopy Two surfaces are isotopic if there exists a one-parameter family of
one Isotopy is also an equivalence relation Note the knotted torus can not
be deformed continuously into the unknotted one Any transformation thatdeforms the knotted torus in the unknotted one has to tear the torus at somepoint Thus it cannot be continuous
Homotopy equivalence If we want to topologically compare the medial
axes of two surfaces even the concept of homeomorphy (which can be extended
to more complex faces than surfaces) seems too strong since the medial axis
of a surface is a more complicated object than the surface itself For
compar-ing medial axes the concept of homotopy equivalence seems to be appropriate Intuitively, the homotopy type of a space encodes its system of internal closed
paths, regardless of size, shape and dimension For example, an annulus hasthe homotopy type of a circle Two topological spaces are homotopy equiva-lent if they have the same homotopy type Homotopy equivalence is anotherequivalence relation on topological spaces
Fig 6.16 Two homeomorphic topological spaces
Trang 9Fig 6.17 The first two figures have the same homotopy type, but are not
homeo-morphic The third one has a different homotopy type
Geometric Concepts.
Hausdorff distance The Hausdorff distance is a measure for the distance
of two subsets of some metric space We are interested in the case where these
h(X, Y ) is defined as h(X, Y ) = max x∈Xminy∈Y x − y The one-sided
Hausdorff distance is not a distance measure since in general it is not
symmetric Symmetrizing h yields the Hausdorff distance as H(X, Y ) =
max{h(X, Y ), h(Y, X)} See also Fig 6.18.
Intuitively, the Hausdorff distance is the smallest thickening such that the
tubular neighborhood of X contains Y and the tubular neighborhood of Y contains X.
Fig 6.18 The one-sided Hausdorff distance is not symmetric
Normals and tangent planes Given two surfaces, their Hausdorff
dis-tance just takes into account their relative positions In the context of surfacereconstruction, we are also be interested in differential properties of the recon-structed surface with respect to the sampled surface At the first order, such
Trang 10a measure is provided by the tangent planes (or the normals) to the surfaces,
a quantity known to play a key role in the definition of metric properties ofsurfaces [259]
6.2.4 Exercises
The following exercises are meant to make sure the important notions havebeen understood We also provide selected references to further investigatethe problems addressed
Exercise 1 (Sampling conditions and reasonable reconstructions).
Consider the one-parameter family of curves that are indicated in Fig 6.19
components with boundaries
reconstruc-tion would be depending on d.
Exercise 2 (Sorting Gabriel edges) Consider a curve bounding a
sta-dium, i.e., two line-segments joined by two half-circles Also consider a dense
sampling of the curve, that is the distance between to samples along the curve
is much smaller than the radius of the circles and the distance between thetwo line-segments
Plot the Delaunay triangulation of the samples, and report the Gabrieledges Explain which of the Gabriel edges are relevant for the reconstruction
of the curve, and which are not Now, perturb slightly the boundary of the
stadium, pick samples on the new curve, and answer the same questions
Exercise 3 (Local geometry of points on the Medial Axis) Specifying
four degrees of freedom of the sphere Explain why
By matching the number of constraints and the number of degrees offreedom, show that
the medial axis, and
Exercise 4 (Using poles) Let S be a closed surface which is in C0but not
not continuous Describe the geometry of the Voronoi cell of a sample point
Trang 11on such a curve Which problem may arise in using the poles associated to
such a sample point? Answer the same question assuming the surface S is
smooth but has boundaries
Fig 6.19 A one parameter family of shapes
Further reading
• Exercise 1 For curve reconstruction, see [24, 125, 127].
• Exercise 2 The separation of critical points of the distance functions to
an ε-sample a smooth surface is studied in [122].
• Exercise 3 An intuitive presentation of the local geometry of the medial
axis is provided in [181]
• Exercise 4 To learn more on the geometry of Voronoi cells, refer to [21]
and [120]
6.3 Overview of the Algorithms
6.3.1 Tangent Plane Based Methods
We assume the sampled surface S is smooth, i.e., there exists a well defined
tangent plane at each point of the surface Since we only deal with surfaces
plane at some point of the surface is equivalent to approximating the normal
at this point Thus here tangent plane based methods include normal basedmethods The first algorithm based on the tangent planes at the sample points
is Boissonnat’s [54] algorithm Boissonnat’s paper probably is the first ence to the surface reconstruction problem at all Simply put, his algorithmreduces the reconstruction problem to the computation of local reconstruc-tions in the tangent planes at the sample points These local reconstructionshave to be pasted together in the end
refer-Lower Dimensional Localized Delaunay Triangulation.
Gopi, Krishan and Silva [188] designed an algorithm that is very similar innature to Boissonnat’s early algorithm
Trang 12" Bottom-line This algorithm has three major steps First, normal and
tangent plane approximation at the sample points Second, selection of aneighborhood of sample points for each sample point Third, projection ofthe neighborhood of a sample point on its tangent plane and computation ofthe Delaunay neighborhood of the sample point in its projected neighborhood
mutually contained in their Delaunay neighborhoods
" Algorithm The normal and tangent plane approximation at the sample
points is done using the eigenvectors of the covariance matrix of the k nearest
C =
i
(q i − ˆp)(q i − ˆp) T
centroid of these neighbors The eigenvector corresponding to the smallest
eigenvalue of the positive definite, symmetric matrix C is taken as the proximate normal at p The remaining two eigenvectors span an approximate tangent plane at p The approximate normals at the sample points are con-
ap-sistently oriented by propagating the orientation at some seed sample point
along the edges of the Euclidean minimum spanning tree of P
From the approximated normals at the sample points the directional mal variations and even the principal curvatures at the sample points can
nor-be estimated using again the k nearest neighbors of the sample points The
are used to locally approximate the unknown surface S by a height function
The neighbors of a sample point p are projected onto the approximated tangent plane at p by rotating the all the vectors from p to its neighbors into the tangent plane In the tangent plane the Delaunay neighbors of p are determined by computing a two dimensional Delaunay triangulation of p and
its projected neighbors The output of the algorithm consists of all triangles
with vertices in P whose vertices are mutual Delaunay neighbors.
" Complexity The complexity of the algorithm was not theoretically
ana-lyzed But is seems reasonable to assume that the local operations at eachsample point can be done in constant time each, which would amount to alinear time complexity in total But there are also the global operations ofdetermining the neighborhoods of the sample points and of consistently ori-enting the normals Although the latter operation is not really needed for thealgorithm to work