2D Computational Morphology
Remco C. Veltkamp
CWI
Department of Interactive Systems
P.O. Box 4079
1009 AB Amsterdam
The Netherlands
remco@cwi.nl
Eindhoven UniversityofTechnology
Department of Computer Science
P.O. Box 513
5600 MB Eindhoven
The Netherlands
remco@win.tue.nl
Computational Morphology is the analysis of form by computational means.
The present pap er is more sp ecically about the construction and manipu-
lation of closed object boundaries through a set of scattered p oints in 2D.
Results are developed in four successive stages of computational morphology:
1. impose a geometrical graph structure on the set of scattered p oints
2. construct a p olygonal b oundary curve from this geometrical graph
structure
3. build a hierarchy of p olygonal approximations together with localization
information
4. construct a geometric continuous object boundary.
The economic advantage of this approach is that there is no dep endency on
any sp ecic data source. It can be used forvarious types of data sources or
when the source is unknown.
1. Introduction
The work describ ed here deals with the computational asp ects of geometry
with resp ect to form or shap e information, that is, morphology. Morphol-
ogy is closely related to geometry a computational geometric approachtothe
analysis of form is called
Computational Morphology
20]. In particular, this
paper is ab out the construction and subsequent manipulation of closed ob ject
121
boundaries from a set of p oints in 2D. These p oints are scattered p oints, i.e.
no structural relationship between them is known in advance and they have
arbitrary position. We consider four stages of this task:
1. Given a set of scattered p oints, construct a geometric structure on the
points.
2. Given a geometric structure on a set of scattered points, construct an
ob ject b oundary by nding a closed polygon through all points.
3. Given a closed ob ject b oundary, construct a hierarchyof approximations
and lo calization information.
4. Given a closed p olygonal b oundary, construct a geometrically smo oth
(
G
1
-continuous) boundary curve.
In many applications in geometric mo deling, computer graphics, and ob ject
recognition, input data is available in the form of a set of 2D co ordinates that
are points on the b oundary of an
ob ject. Such points can be synthetic or
measured from the boundary of an existing ob ject. A collection of points,
however,isanambiguous representation of an ob ject, and can therefore not be
used directly in many applications. It is often essential to have a representation
of the whole boundary available that is unambiguously dening a valid ob ject.
The b oundary constructed from a set of p oints can for example b e used for the
initial design of an artifact, for numerical analysis, or for graphical display.
The way in which the boundary points are acquired may give useful in-
formation in order to construct the whole boundary, but can also make the
construction method very dependent on the sp ecic data source. If it is not
known how the data is obtained or if a single construction method is to be used
fordatafromvarious t
ypes of sources, then no structural relation b etween the
input p oints may be assumed, except that they all lie on the b oundary of an
ob ject. The order of the points in the input then provides no information on
their top ological relation to each other. In particular, they do not lie on a
regular grid, but are scattered points. In this paper we assume no a priori
knowledge ab out any structural relation b etween the p oints.
The simplest boundary through a set of points is one that consists of straight
segments, making a 2D simple closed polygonal curve. A simple closed polygon
must consist of
N
edges, with
N
the number of data points. A brute force
algorithm that tests all combinations of
N
edges out of all
;
N
2
possible edges
takes
;
N
2
N
!
time, which is at least (
N
N
). ((
f
(
N
)) can b e read as `order exactly
f
(
N
)',
O
(
f
(
N
)) as `order at most
f
(
N
)', which gives an upper bound, and (
f
(
N
)) as
`order at least
f
(
N
)', whichgives a lower b ound, all three `for large
N
'.) This is
122
clearly infeasible when
N
is large. We therefore exploit some geometric relation
between the data p oints. Section presents the
-Neighborhoo d Graph, which
describes the geometric structure on the set of data p oints. Section describes
how the
-Neighborhood Graph is exploited for the construction of a closed
polygonal ob ject b oundary.
In many real applications, a b oundary constructed from a set of p oints con-
sists of thousands of faces. An approximation of the ob ject, however, is often
sucient. Lo calization provides b ounding volume information, e.g. a sequence
of b ounding rectangles containing pieces of the b oundary. Such information is
useful for ecient op erations such as collision detection for rob ot motion plan-
ning. Approximation and localization can b e combined in a single scheme, and
several levels of approximation and lo calization can b e combined in a hierar-
chical way. Section presents a new hierarchical approximation and localization
scheme.
Polygonal b oundaries are only
C
0
-continuous at the vertices there the tan-
gentvectors instantly change direction. A smoother curve, consisting of curved
segments that in
terpolate the vertices and are smo othly connected, is often
desired. A curve that has a continuously changing tangent vector is called
G
1
-continuous. Section presentsascheme to make the curve
G
1
-continuous.
Finally, Section presents some concluding remarks.
2. Point set analysis
In order to perform any geometric analysis on a set of scattered points, some
geometric structure must be imp osed on it. Such a structure typically relates
points to each other if they satisfy some geometric prop erty, and is represented
by a graph. Examples of such geometric graphs are the Nearest Neighbor
Graph, the Euclidean Minimal Spanning Tree, the Innite Strip Graph 6],
the Sphere of Inuence Graph 21], the Relative Neighborhoo d Graph 14], the
Gabriel Graph 11], the Convex Hull, the Delaunay Triangulation 5] and its
dual Voronoi Diagram 27], the
-Shape 9], and the
-Skeleton 13]. Section
presents the
-Neighborho od Graph, a parameterized geometric graph which
unies a number of the before-mentioned ones. The
-Neighborhoo d Graph is
used in Section for the construction of an ob ject boundary through all data
points.
2.1. The
-NeighborhoodGraph
The
-Neighborhoo d Graph has b een introduced in 24]. Here we will consider
one particular form. To start with, let us consider the 2D Delaunay Trian-
gulation on a set of points. The Delaunay Triangulation is a lling of the
plane inside the Convex Hull of the point set by triangles with the
following
properties:
1. the vertices of each triangle are data points,
2. the disc touching the vertices of each triangle contains no other data point
in its interior.
123
Figure 1
. Example DelaunayTriangulation and a few `largest empty discs'.
(A disc is a lled circle.) A region that contains no data p oint in its interior
is called empty. If no more than three data points lie on any empty disc, the
Delaunay Triangulation is uniquely dened. Otherwise the triangulation on
these sp ecic p oints must b e non-overlapping.
The DelaunayTriangulation can equivalently b e dened as the collection of
all edges that have an empty disc touching its vertices. For all these edges
there are two largest possible empty discs touching their vertices, which either
touch a third p oint or haveaninnite radius. In this last situation the disc
degenerates to an empty half-plane, and the edge lies on the Convex Hull. Both
situations are illustrated in Figure 1. The radii of the two discs are a scaling
of the radius
r
of the smallest p ossible disc touching the twovertices. These
scaling factors are written as an expression 1
=
(1
;
c
), with 0
c
1. The two
radii are thus
r=
(1
;
c
0
)and
r=
(1
;
c
1
), with 0
c
0
c
1
1.
The DelaunayTriangulation is a particular instance of the
-Neighborhoo d
Graph, which discriminates between the case that the centersofthe discslie
at opp osite sides of the edge and the case that they lie at the same side. In the
latter case the parameter
c
0
is taken negative. So,
c
0
lies in the range
;
1
1],
dening a radius of
r=
(1
;j
c
0
j
). Parameter
c
1
still lies in the range 0
1].
The
-Neighborhoo d Graph that coincides with the DelaunayTriangulation is
denoted
(
;
1
1]
0
1]).
For each edge in the DelaunayTriangulation the
union
of two discs touching
its vertices is empty, i.e contains no data points. The
-Neighborhoo d Graph
also considers the case that only the
intersection
of two discs is empty. In that
case the parameter
c
1
is taken negative. So,
c
1
may lie in the range
;
1
1],
124
Figure 2
. Left: 77 vertices. Right: the corresponding
(
;
1
1]
0
1]) (Delau-
nayTriangulation).
dening a radius of
r=
(1
;j
c
1
j
). The graph
(
;
1
1]
d
1]) for some
d
2
;
1
0]
is the Delaunay Triangulation plus all edges for which there are
c
0
2
;
1
1]
and
c
1
2
d
0] such that the intersection of two discs touching their vertices
with radii
r=
(1
;j
c
0
j
) and
r=
(1
;j
c
1
j
) is empty. The smaller the value of
d
,
the smaller the area of the intersection, and the more edges are included in the
-Graph.
Each
d
2
;
1
0] yields one sp ecic
-Graph from the whole spectrum of
graphs
(
:::
]
:::
]). This sp ectrum unies a number of geometric graphs such
as the Convex Hull, the DelaunayTriangulation, the Gabriel Graph, and the
-Skeleton, into a continuum that ranges from the void to the complete graph.
The
-Graph provides a geometric structure for point pattern analysis and
can for example be used for geographics and network analysis. The graph
(
;
1
1]
d
1]),
d
2
;
1
0] is the
-Graph that we use for b oundary construc-
tion.
A formal denition and analysis and many examples of the
-Neighborhoo d
Graph are given in 24]. The 2D
(
;
1
1]
d
1]) can be constructed in time
(
N
log
N
) for
d
=0,which is optimal, and
O
(
N
2
)for
d<
0. Figure 2 gives
an example of a p oint set and the
(
;
1
1]
0
1]) on that set. The p ointsetis
taken from the silhouette of Ucello's chalice, which serves as the cover picture
of the journal `Computer Aided Geometric Design' 19].
3. Boundary construction
125
